Differential Equations with Applications to Industry · 2019. 8. 7. · Shaher Momani, Jordan Gaston M. N’Guerekata, USA Juan J. Nieto, Spain Sotiris K Ntouyas, Greece Donal O’Regan,

Differential Equations with Applications to IndustryGuest Editors: Ebrahim Momoniat, T. G. Myers, Mapundi Banda, and Jean Charpin

International Journal of Differential Equations

Differential Equations with Applicationsto Industry

International Journal of Differential Equations

Differential Equations with Applicationsto Industry

Guest Editors: Ebrahim Momoniat, T. G. Myers,Mapundi Banda, and Jean Charpin

Copyright q 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “International Journal of Differential Equations.” All articles are open access articles dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

Editorial BoardOm P. Agrawal, USABashir Ahmad, Saudi ArabiaChérif Amrouche, FranceSabri Arik, TurkeyDumitru Baleanu, TurkeyVieri Benci, ItalyElena Braverman, CanadaAlberto Cabada, SpainJinde Cao, ChinaDer-Chen Chang, USAYang Chen, UKGui Qiang Chen, USAFengde Chen, ChinaCharles E. Chidume, ItalyI. D. Chueshov, UkraineShangbin Cui, ChinaToka Diagana, USAJinqiao Duan, USAM. A. El-Gebeily, Saudi ArabiaA. M. El-Sayed, EgyptKhalil Ezzinbi, MoroccoZhaosheng Feng, USAGiovanni P. Galdi, USAD. D. Ganji, USAWeigao Ge, ChinaYoshikazu Giga, JapanJaume Giné, SpainJerome A. Goldstein, USAS. R. Grace, EgyptMaurizio Grasselli, ItalyTasawar Hayat, PakistanEmmanuel Hebey, FranceHelge Holden, NorwayMayer Humi, USAElena I. Kaikina, Mexico

Dexing Kong, ChinaQingkai Kong, USAM. O. Korpusov, RussiaA. M. Krasnosel’skii, RussiaMiroslav Krstic, USAP. A. Krutitskii, RussiaMin Ku, PortugalMustafa Kulenovic, USAKarl Kunisch, AustriaAlexander Kurganov, USAJose A Langa, SpainPhilippe G. Lefloch, FranceDaniel Franco Leis, SpainNikolai N. Leonenko, UKYuji Liu, ChinaFawang Liu, AustraliaWen Xiu Ma, USARuyun Ma, ChinaT. R. Marchant, AustraliaMarco Marletta, UKRoderick Melnik, CanadaS. A. Messaoudi, Saudi ArabiaStanisław Migórski, PolandA. Mikelic, FranceShaher Momani, JordanGaston M. N’Guerekata, USAJuan J. Nieto, SpainSotiris K Ntouyas, GreeceDonal O’Regan, IrelandJong Yeoul Park, KoreaKanishka Perera, USARodrigo Lopez Pouso, SpainRamón Quintanilla, SpainYoussef Raffoul, USAT. M. Rassias, Greece

Yuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaSamir H. Saker, EgyptMartin Schechter, USAWilliam E. Schiesser, USALeonid Shaikhet, UkraineZhi Qiang Shao, ChinaQin Sheng, USAJunping Shi, USAStevo Stevic, SerbiaIoannis G. Stratis, GreeceJian-Ping Sun, ChinaGuido Sweers, GermanyNasser-eddine Tatar, Saudi ArabiaRoger Temam, USAGunther Uhlmann, USAJ. van Neerven, The NetherlandsA. Vatsala, USAPeiguang Wang, ChinaLihe Wang, USAMingxin Wang, ChinaZhi-Qiang Wang, USAGershon Wolansky, IsraelPatricia J. Y. Wong, SingaporeJen-Chih Yao, TaiwanJingxue Yin, ChinaJianshe Yu, ChinaVjacheslav Yurko, RussiaQi S. Zhang, USASining Zheng, ChinaSongmu Zheng, ChinaFeng Zhou, ChinaYong Zhou, ChinaWenming Zou, ChinaXingfu Zou, Canada

Contents

Differential Equations with Applications to Industry, Ebrahim Momoniat, T. G. Myers,Mapundi Banda, and Jean CharpinVolume 2012, Article ID 491874, 2 pages

An Energy Conserving Numerical Scheme for the Dynamics of Hyperelastic Rods,Thorsten Fütterer, Axel Klar, and Raimund WegenerVolume 2012, Article ID 718308, 15 pages

Controlled Roof Collapse during Secondary Mining in Coal Mines, Ashleigh HutchinsonVolume 2012, Article ID 806078, 21 pages

On the Solutions Fractional Riccati Differential Equation with Modified Riemann-LiouvilleDerivative, Mehmet MerdanVolume 2012, Article ID 346089, 17 pages

Application of Heat Balance Integral Methods to One-Dimensional Phase Change Problems,S. L. Mitchell and T. G. MyersVolume 2012, Article ID 187902, 22 pages

Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations,Yusuke YamauchiVolume 2012, Article ID 417261, 16 pages

Numerical and Analytical Study of Bladder-Collapse Flow, M. Tziannaros and F. T. SmithVolume 2012, Article ID 453467, 14 pages

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2012, Article ID 491874, 2 pagesdoi:10.1155/2012/491874

EditorialDifferential Equations with Applicationsto Industry

Ebrahim Momoniat,1 T. G. Myers,2Mapundi Banda,3 and Jean Charpin4

1 Centre for Differential Equations, Continuum Mechanics and Applications,School of Computational and Applied Mathematics, University of the Witwatersrand,Johannesburg, Private Bag 3, Wits 2050, South Africa

2 Centre de Recerca de Matemàtica, Edifici C, Campus de Bellaterra, Bellaterra, 08193 Barcelona, Spain3 Applied Mathematics Division, Department of Mathematical Sciences, Stellenbosch University,Private Bag X1, Matieland 7206, South Africa

4 MACSI, Department of Mathematics & Statistics, College of Science & Engineering,University of Limerick, Limerick, Ireland

Correspondence should be addressed to Ebrahim Momoniat, [email protected]

Received 11 December 2012; Accepted 11 December 2012

Copyright q 2012 Ebrahim Momoniat et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This special issue is focused on the application of differential equations to industrial mathe-matics. Of particular interest is the role played by industrial mathematics in the developmentof new ideas and applications. We are particularly interested in industrial mathematicsproblems that come from industrial mathematics study group meetings, which take place reg-ularly at universities across the world. These study group meetings are motivated by solvingreal-world problems that are posed by industry representatives at the start of the meeting.Graduate students and academics then spend one week developing mathematical modelsthat simulate the problems presented. These mathematical models are then solved �usuallyafter some simplification�, and conclusions relevant to the real-world problem are made.

This special issue contains a paper that is based on a problem presented by the coalmining industry in South Africa at an industrial mathematics study group meeting. In thepaper, the author considers the possible collapse of the roof between the pillar to be minednext in secondary coal mining and the first line of pillar remnants called snooks. Here, theEuler-Bernoulli beam equation is used to model the roof rock between the pillars, which is theworking face between two pillars. The model predicts that the beam will break at the clampedend at the pillar. The failure of the beam for different values of the physical parameters isinvestigated computationally.

Many industrial mathematics problems contain an aspect of heat conduction. Thisspecial issue contains a paper in which a new error measure is proposed for the heat balance

2 International Journal of Differential Equations

integral method that combines a least-square error with a boundary immobilisation method.The authors show how an optimal heat balance formulation can be obtained by applying theirerror measure to three basic thermal problems. This new error measure combined with theheat integral method is then applied to two industrially important phase change problems.

Elastic rods are used in many industrial and engineering applications. This specialissue also contains another paper that develops a numerical method for special Cosserat rodsdeveloped for hyperelastic materials and potential forces. The numerical method preservesthe orthonormality of the directors and the conservation of energy of the system of partialdifferential equations modeling the elastic rods.

In an application to biological modeling, an article developing a mathematical modelfor collapsing bladder flow is presented in this special issue. The authors derive a coupledsystem of nonlinear equations derived from the Navier-Stokes equations modeling urinaryvelocities that depend on the shape of the bladder. Both computational work and special-configuration analysis are applied over a range of configurations including results for thecircle and sphere as basic cases. The authors also include models of more realistic bladdershapes as well as the end stage of the micturition process where the bladder is relativelysquashed down near the urethral sphincter.

The investigation of industrial mathematics problems sometimes leads to thedevelopment of new methods of solution of differential equations. This special issue containsa paper on the fractional variational iteration method to determine approximate analyticalsolutions of nonlinear fractional differential equations. The fractional variational iterationmethod is applied to the nonlinear fractional Riccati equation with a modified Riemann-Liouville derivative. The fractional variational iteration method is shown to be an efficientmethod for the solution of nonlinear fractional differential equations.

Modeling in industrial mathematics problems with parabolic equations is verycommon. This special issue also contains a survey paper in which the author investigatesthe blow-up phenomena for Fujita-type parabolic equations. The author then goes on todiscuss various results on the life span of positive solutions for several superlinear parabolicproblems.

This special issue has covered both the theoretical and applied aspects of industrialmathematics. Papers contain the development of new mathematical models or well-knownmodels applied to new physical situations as well as the development of new mathematicaltechniques. It is this multidisciplinary nature of industrial mathematics that makes it achallenging, fruitful, and exciting area of research.

Acknowledgment

We are grateful to all the authors who have made a contribution to this special issue.

Ebrahim MomoniatT. G. Myers

Mapundi BandaJean Charpin


Research ArticleAn Energy Conserving Numerical Scheme forthe Dynamics of Hyperelastic Rods

Thorsten Fütterer,1 Axel Klar,2 and Raimund Wegener1

1 Fraunhofer Institut für Techno- und Wirtschaftsmathematk, Fraunhofer-Platz 1,67663 Kaiserslautern, Germany

2 Fachbereich Mathematik, Technische Universität Kaiserslautern, 67653 Kaiserslautern, Germany

Correspondence should be addressed to Axel Klar, [email protected]

Received 12 January 2012; Revised 21 June 2012; Accepted 5 July 2012

Academic Editor: Mapundi Banda

Copyright q 2012 Thorsten Fütterer et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A numerical method for special Cosserat rods based on Antman’s description Antman, 2005 isdeveloped for hyperelastic materials and potential forces. This method preserves the relevantproperties of the underlying PDE system, namely, the orthonormality of the directors and theconservation of the energy.

1. Introduction

Elastic rods are considered in many fields of science and technology; see, for example, �1–3�.For the simulation of their dynamics, the correct description of the local and global physicalproperties and reasonable computation times are the essential requirements.

Over the past years many different approaches to the numerical simulation of elasticrods have been developed; see, for example, �4–9� for different approaches involving finiteelement methods, finite difference methods and discrete mechanics.

In this paper we use the description of the rod as a special one-dimensional Cosseratcontinuum following the formulation in Antman �10�. This is a geometrically correct one-dimensional description based on partial differential equations. This type of modeling fulfillsthe above requirements with respect to simplicity and correctness. Numerical schemes for thespecial case of Kirchhoff equations have been developed by Pai �1� �stationary� andWeber etal. �3� �instationary�. Energy conservation and the directors’ orthonormality are not strictlyfulfilled in these schemes. Schemes for structural mechanics problems conserving energy andfurther invariants of the equation are developed, investigated, and applied, for example, in�11–13�.


We formulate the kinematic and dynamic equations of the Cosserat rod theory inthe so-called director-basis. The rod’s curve r and the outer potential force f are the onlyfields described in an external Cartesian basis. For technical reasons we use a representationof the rotational group by unit-quaternions. We note that the first use of quaternions ingeometrically-exact rod models was in �14�, see also �15�. In Section 2 we present an overviewof the model resulting in the following system:

∂tr � D−1�q� · p,∂tq � Ω�w� · q,

∂tv � ∂sp � u × p � v × w,∂tu � ∂sw � u × w,

∂tp �1

(ρA)∂sn �

1(ρA)u × n � p × w � 1(

ρA)D�q� · f,

∂tw �(ρJ)−1 · ∂sm �

(ρJ)−1 · (u × m � v × n � ((ρJ) · w) × w).

�1.1�

Here, t is the time and s the parameter determining a material cross section of the rod. Theunit-quaternions q and the associated orthogonal matrix D�q� describe the transformationbetween the fixed external basis and the director-basis. The vector fields v, u, p, and w are thetangent of the curve, the generalized curvature, the velocity, and the angular velocity, respec-tively. Moreover, �ρA� is the rod’s line density and the positive definite matrix �ρJ� is definedby the moments of inertia of its cross sections. In this paper the contact force n and the contactcouple m are specified by a hyperelastic material law. In Section 3 we introduce the conceptof energy as a constant of motion. In Section 4 we develop a straightforward finite differencescheme for the above equations with appropriate boundary conditions. The scheme conservesthe energy and the orthonormality of the directors. In Section 5 the method is investi-gated for several examples using Timoshenko’s material law.

2. Model

Following Antman �10�, a special Cosserat rod in the three-dimensional Euclidian space E3

is geometrically characterized by three vector-valued functions r,d1,d2 : �sa, sb� × R → E3.The parameter s ∈ �sa, sb� ⊂ R identifies a material cross section �material point� of the rod,r�s, t� characterizes the position of this cross section at time t. The derivatives of the curve rwith respect to t and s,

p � ∂tr, v � ∂sr, �2.1�

are the velocity and the tangent field. The orthonormal directors d1,d2 characterize theorientation of the cross sections. Defining d3 � d1 × d2, we get a local orthonormal basisat all material points. Due to the orthonormality of the directors there exist vector-valuedfunctions u and w such that

∂tdk � w × dk, ∂sdk � u × dk �2.2�

International Journal of Differential Equations 3

for k ∈ {1, 2, 3}. We call w the angular velocity and u the generalized curvature. The defini-tions of the vector fields p, v, w, and u imply the compatibility conditions:

∂tv � ∂sp, ∂tu � ∂sw �w × u, �2.3�

that complete the kinematic equations of the the special Cosserat rod theory in an invariantnotation.

To rewrite these kinematic equations in an appropriate basis, we decompose anarbitrary vector field x of our rod theory in the director-basis dk as well as in a fixed externalCartesian basis ek, that is, x �

∑3k�1 xkdk �

∑3k�1 xkek ∈ E3. The corresponding component

triple,

x � �x1, x2, x3� ∈ R3, x � �x1, x2, x3� ∈ R3, �2.4�

are strictly to distinguish from the vector field x ∈ E3. The component triples of its derivativeswith respect to t and s in the director-basis are

�∂tx · dk�k�1,2,3 � ∂tx � w × x, �∂sx · dk�k�1,2,3 � ∂sx � u × x. �2.5�

The transformation between the director-basis and the external basis is given by an ortho-normal matrix D with components Dij � di · ej . We use a representation of D in unit-quat-ernions q � �q0, q1, q2, q3� ∈ R4, cf. �16�,

D � D�q� �

⎛

⎜⎜⎝

q20 � q21 − q22 − q23 2

(q1q2 − q0q3

)2(q1q3 � q0q2

)

2(q1q2 � q0q3

)q20 − q21 � q22 − q23 2

(q2q3 − q0q1

)

2(q1q3 − q0q2

)2(q2q3 � q0q1

)q20 − q21 − q22 � q23

⎞

⎟⎟⎠. �2.6�

Here, the orthonormality of the matrix D�q� is guaranteed by ‖q‖ � 1. For an arbitrary x ∈ E3with x �

∑3k�1 xkdk �

∑3k�1 xkek we obtain x � D�q� · x. Instead of the kinematic equation for

the directors, we formulate an equivalent equation for the quaternions, cf. �16�,

∂tq �12

⎛

⎜⎜⎝

0 w1 w2 w3−w1 0 −w3 w2w2 w3 0 −w1−w3 w2 w1 0

⎞

⎟⎟⎠ · q � Ω�w� · q. �2.7�

Initializing �2.7� with the unit-quaternions, the skew symmetry of Ω�w� guarantees the pre-servation of the norm of the quaternions in time, that is, the orthonormality of D�q� and hence


the orthonormality of the director-basis. Using the presented formalism we obtain our finalversion of the kinematic equations of the rod:

∂tr � D�q�−1 · p,∂tq � Ω�w� · q,

∂tv � ∂sp � u × p � v × w,∂tu � ∂sw � u × w.

�2.8�

The balance laws for momentum and angular momentum yield the dynamic equationsof the Cosserat rod theory, cf. �10�,

(ρA)∂tp � ∂sn � f,

∑

α,β�1,2

(ρJαβ

)∂t(dα × ∂tdβ

)� ∂sm � v × n � l. �2.9�

Here, �ρA� is the line-density and �ρJαβ� are the moments of inertia. These quantities are timeindependent, since they are defined in the reference configuration as Lagrangian quantities.We assume that they are constant with respect to s, too. The body force line density f has tobe specified in the applications. We assume l � 0 for the corresponding body couple density.The contact force n and contact couple m have to be defined by material laws. Usually, this isdone in the director-basis for reasons of objectivity. Thus, we decompose also the dynamicalequations in the director-basis:

∂tp �1

(ρA)∂sn �

1(ρA)u × n � p × w � 1(

ρA)D · f,

∂tw �(ρJ)−1 · ∂sm �

(ρJ)−1 · (u × m � v × n � ((ρJ) · w) × w).

�2.10�

The positive definite matrix �ρJ� is given by the moments of inertia:

(ρJ)�

⎛

⎜⎜⎝

(ρJ22

) −(ρJ12) 0−(ρJ21) (ρJ11) 0

0 0(ρJ11

)�(ρJ22

)

⎞

⎟⎟⎠. �2.11�

The body force, for example, gravity, is obviously defined in the external basis. Thus, �2.10�are coupled to the kinematic equation for the quaternions.

Remark 2.1. The special Cosserat rod theory describes the angular momentum as a linearfunction of the angular velocity. The choice of the representation of the vector fields in thedirector-basis leads to the time independent matrix �ρJ� characterizing this linear depend-ence. Besides the proper formulation of the material laws, this time independence is one ofthe major advantages of the choice of the director-basis.


In this paper we restrict ourselves to hyperelastic materials. That means there exists apotential R6 � �v, u� �→ ψ�v, u� ∈ R, such that

n � ∂vψ, m � ∂uψ. �2.12�

Moreover, we assume that only potential forces act on the rod. Thus there exists a functionR

3 � r �→ V �r� ∈ R, such that

f � −∂rV. �2.13�

Remark 2.2. The more general class of elastic materials are materials where n and m arefunctions of the so-called strain variables v and u. These functions may also depend explicitlyon s.

The kinematic equations �2.8� and the dynamic equations �2.10� together with therestriction to hyperelastic materials and potential forces constitute our rod theory, see also�1.1�. We consider system �1.1� with two types of boundary conditions defining a fixed or afree end. For simplicity we restrict our description to a fixed end at s � sa

p�sa, t� � 0, w�sa, t� � 0, �2.14�

and a free end at s � sb

n�sb, t� � 0, m�sb, t� � 0. �2.15�

The presentation of the energy conserving numerical algorithm in Section 4 deals withthe above general class of rods. For the numerical examples in Section 5 we specify the rod’sgeometry, a hyperelastic material law, and the potential forces. We consider a homogeneouscylinder with diameter d > 0, cross section area A � π/4d2, and moment of inertia I �π/64d4. In this case, the matrix of inertia is

(ρJ)�(ρI)

diag�1, 1, 2�. �2.16�

We use the material law of Timoshenko �17� for Poisson number μ � 1/2:

ψ �12�EA�

(13v21 �

13v22 � �v3 − 1�2

)�

12�EI�

(u21 � u

22 �

23u23

), �2.17�

where E is Young’s modulus. Additionally, we restrict to gravitational forces, that is,

V � −(ρA)geg · r, �2.18�

where g is the gravitational constant and eg is the direction of gravity in the external Cartesianbasis.


Remark 2.3. For hyperelastic materials �1.1� is an inhomogeneous hyperbolic system. In thespecial case of Timoshenko for a homogeneous cylinder the hyperbolic part is linear witheigenvalues 0 �sevenfold�, ± c �threefold�, and ± c/3 �threefold�, where c � √E/ρ is thespeed of sound. Computing the eigenvectors one can easily show that the fixed and free endboundary condition correctly handle the characteristic variables r and q corresponding to theeigenvalue 0. With respect to the remaining variables we do not prescribe characteristic vari-ables, but the correct number of variables on both sides of the rod.

3. Energy as a Constant of Motion

The system �1.1� for the state vector φ � �r, q, v, u, p,w� ∈ R19 can be written in the generalform of a conservation law as

∂tφ � ∂sf�φ� � h�φ� �3.1�

with flux-function f�φ� � �0, 0, p,w, �1/�ρA��n, �ρJ�−1 · m� and source term h�φ� that is easyto identify from �1.1�. We introduce the energy density:

ε�φ� �12(ρA)p2 �

12w · (ρJ) · w � ψ�v, u� � V �r�. �3.2�

and the symmetric function:

a(φ1,φ2

)�

12�n1 · p2 � m1 · w2 � p1 · n2 � w1 · m2� �3.3�

for arbitrary states φ1, φ2. The derivative of the energy density with respect to the state vectorφ is given by

∂φε �(−f, 0, n,m, (ρA)p, (ρJ) · w

), �3.4�

that leads to the properties:

∂φε · h � 0, 12∂φε(φ1) · f(φ2

)� a(φ1,φ2

), ∂φε · ∂sf � ∂sa�φ,φ�. �3.5�

We conclude the local energy balance:

∂tε � ∂sa�φ,φ�, �3.6�

that is, a�φ,φ� is the energy flux and there is no energy source term. For the presented fixedand free end boundary conditions �2.14�, �2.15� we have a vanishing energy flux at the bound-aries. Therefore, the total energy is a constant of motion:

ddt

E � ddt

∫sb

sa

εds′ � 0. �3.7�


4. Discretization

For the spatial discretization we use a simple finite difference scheme. We note that similarfinite difference schemes have been developed and their properties, in particular, the con-servation of invariants, have been investigated in �8, 12, 18�.

We discretize �sa, sb� with �N � 1� equidistant mesh points sj and denote the cor-responding length of the cells by Δs. The boundary points are s1 � sa and sN�1 � sb. Asusual, the numerical fluxes are denoted by Fj�1/2 for all j ∈ {1, . . . ,N}. The state vector andthe source terms are discretely given at the mesh points j ∈ {1, . . . ,N�1}, that is, φj φ�sj , t�and Hj h�φ�sj , t��. For the inner points j ∈ {2, . . . ,N} the spatial discretization results in

ddt

φj �1Δs[Fj�1/2 − Fj−1/2

]�Hj , Fj�1/2 12

[f(φj

)� f(φj�1

)]. �4.1�

Thereby, the numerical flux function is approximated by the arithmetic average of the fluxat neighboring mesh points. For the flux calculation at the boundaries we have to fulfill theDirichlet boundary conditions at the fixed end sa � s1:

p1 � 0, w1 � 0, �4.2�

and at the free end sb � sN�1:

nN�1 � 0, mN�1 � 0. �4.3�

Thus, for the remaining components our finite difference scheme reads at the fixed end sa �s1:

ddt

r1 � 0,ddt

q1 � 0,ddt

v1 �1Δs

p2,ddt

u1 �1Δs

w2, �4.4�

and at the free end sb � sN�1:

ddt

rN�1 � D−1�qN�1� · pN�1, ddtqN�1 � Ω�wN�1� · qN�1,

ddt

pN�1 � − 1Δs1ρA

nN � pN�1 × wN�1 � 1(ρA)D�qN�1� · fN�1,

ddt

wN�1 � − 1Δs(ρJ)−1 · mN �

(ρJ)−1 · (((ρJ) · wN�1

) × wN�1).

�4.5�


Now, we come up with our main point, the �semi�discrete energy conservation of thisscheme. The energy density is approximated locally at the mesh points j ∈ {1, . . . ,N�1}, thatis, εj ε�φj�. We obtain for the inner points j ∈ {2, . . . ,N}:

ddtεj � ∂φε

(φj

)· d

dtφj

�19�� ∂φε

(φj

)·[

1Δs(Fj�1/2 − Fj−1/2

)�Hj

]

�17��

1Δs

[a(φj ,φj�1

)− a(φj−1,φj

)].

�4.6�

For the boundaries we have to take into account the Dirichlet conditions:

ddtε1 � ∂φε

(φ1) · d

dtφ1

�16,20,22��

1Δs

�n1 · p2 � m1 · w2�

�15,20��

2Δs

a(φ1,φ2

),

ddtεN�1 � ∂φε

(φN�1

) · ddt

φN�1�16,20,23�

� − 1Δs

�nN · pN�1 � mN · wN�1�

�15,21�� − 2

Δsa(φN,φN�1

).

�4.7�

Applying the trapezoidal quadrature rule for the discrete energy,

E EDisc � Δs2ε1 � Δs

N∑

j�2

εj �Δs2εN�1, �4.8�

guarantees its conservation:

ddt

Edisc � Δs2

ddtε1 � Δs

N∑

j�2

ddtεj �

Δs2

ddtεN�1 � 0. �4.9�

This means, the chosen semidiscretization in space ensures that the discrete energy �4.8� is afirst integral of the ODE-system �4.1�–�4.5�.

For the time discretization any energy conserving method can be used. We choose aGauss method, that also guarantees the preservation of the norm of the quaternions. In thenumerical realization, we make use of the second order Gauss method, that is, the midpointrule, to obtain a temporal order that is consistent with the spatial one, at least at the innerpoints. For the discretization of space and time, and the use of the midpoint rule for the con-servation of certain properties we refer also to the above mentioned papers �12, 18�.

To solve the resulting nonlinear equations a Newton method is used. The strict con-servation of energy and orthogonality are the main advantages of the straightforward finitedifference scheme presented here.


Remark 4.1. The scheme described above concentrates on preserving the energy of the rod andthe orthonormality of the directors. In the sense of numerical methods for hyperbolic systemsthe scheme is not able to handle shocks properly. It does not have the usual properties likebeing a TVD scheme or satisfying the entropy condition.

Remark 4.2. Higher order discretizations are also possible. For example, we could considerthe following fourth order numerical flux function:

Fj�1/2 12[f(φj

)� f(φj�1

)]�

116

[−f(φj�2

)� f(φj�1

)� f(φj

)− f(φj−1

)]. �4.10�

Then, the time derivative of the discrete energy density at the inner mesh point j is given by

ddtεj �

1Δs

⎡

⎢⎢⎣a(φj ,φj�1

)

︸︷︷︸�O1

−a(φj−1,φj

)

︸︷︷︸�U1

−18a(φj ,φj�2

)

︸︷︷︸�O3

�14a(φj ,φj�1

)

︸︷︷︸�O2

−14a(φj−1,φj

)

︸︷︷︸�U2

�18a(φj−2,φj

)

︸︷︷︸�U3

⎤

⎥⎥⎥⎦.

�4.11�

The termsO1,O2,O3,U1,U2, andU3 are eliminated at the mesh points j�1, j�1, j�2, j−1, j−1,and j−2, respectively. Neglecting the boundary points this yields again the conservation of thediscrete energy. The discretization near the boundary points has to be considered separately.

5. Numerical Examples

In this section we present three numerical examples, restricting ourselves to Timoshenko’smaterial law for a homogeneous cylinder as discussed at the end of Section 2. Introducing atypical length, a typical time, and a typical mass:

styp � sb − sa, ttyp �styp√E/ρ

, mtyp � ρπ

4d2styp, �5.1�

the dimensionless parameters of the model are the slenderness ratio and the gravity number:

δ �d

styp, γ �

ρgstyp

E. �5.2�

In more details, we have �EA� � �ρA� � 1, �EI� � �ρI� � π/16δ2, and for the speed of soundc � 1. In the dimensionless form Timoshenko’s material law reads

n �(

13v1,

13v2, v3 − 1

), m �

π

16δ2(u1, u2,

23u3

). �5.3�


To simplify the formulation of the equations we define for x ∈ R3 and λ ∈ R:

�x�λ � �x1, x2, λx3�, x̂ � �−x2, x1, 0�. �5.4�

Then, the system �1.1� reads

∂tr � D−1�q� · p,∂tq � Ω�w� · q,

∂tv � ∂sp � u × p � v × w,∂tu � ∂sw � u × w,

∂tp �13∂s�v�3 �

13u × �v�3 � û � p × w − γD�q� · �0, 1, 0�,

∂tw � ∂s�u�1/3 �13u3û �

16π

1δ2

(1 − 2

3v3

)v̂ �w3ŵ.

�5.5�

As mentioned, the rod is fixed at one side,

p�0, t� � �0, 0, 0�, w�0, t� � �0, 0, 0�, �5.6�

and the other side is free, that is, for Timoshenkos’s material law:

v�1, t� � �0, 0, 1�, u�1, t� � �0, 0, 0�. �5.7�

The chosen initial configuration of a straight rod and direction of gravity eg � �0,−1, 0�can be seen in Figure 1. In the following examples different initial torsions will be considered.More precisely,

r�s, 0� � sd3,

d1�s, 0� � e2 cos(μ�s�

)� e3 sin

(μ�s�

),

d2�s, 0� � −e2 sin(μ�s�

)� e3 cos

(μ�s�

),

�5.8�

where μ : �0, 1� → R is a field of torsion angles that has to be defined. These conditions areequivalent to initial conditions for r�s, 0� and q�s, 0�. Moreover, due to the definitions of v andu we have

v�s, 0� � �0, 0, 1�, u�s, 0� �(0, 0, ∂sμ

). �5.9�

Finally we prescribe

p�s, 0� � �0, 0, 0�, w�s, 0� � �0, 0, 0�, �5.10�


e2

e1

e3

g

Figure 1: Initial situation.

that is, the rod is initially at rest. Our initial conditions are compatible to the boundary con-ditions if μ�0� � 0 and ∂sμ�1� � 0.

In all simulations we choose the CFL-number equal to 1. This implies Δt � Δs as forthe speed of sound c � 1 in the dimensionless form.

We remark that in all simulations energy is strictly conserved according to the aboveanalysis.

Example 5.1 �Torsional Oscillation without Gravity�. We choose γ � 0, δ � 10−3, and a field oftorsion angles fulfilling the compatibility condition:

μ�s� �2π

sin(π

2s). �5.11�

In this example, only the torsion u3 and the angular velocity w3 are involved, because of thevanishing gravity. The equations reduce to the homogenous wave-equation:

∂tu3 − ∂sw3 � 0,

∂tw3 − 13∂su3 � 0.�5.12�

Due to the chosen torsion angle we exactly initialize the fundamental mode of the wave equa-tion with a frequency

ωtheo �π√12. �5.13�

We use this example as a benchmark for the convergence properties of our scheme comparingthe computed frequencies with the analytical one for different grid sizes, see Figure 2. Com-paring the identified simulation frequency with the fundamental frequency of the analytical


0 200 400 600 800 10000.896

0.9

0.904

0.908

N

ω

ωsimωtheo

N

10

25

50

100

200

500

1000

ωsim

0.8976

0.9054

0.9078

0.9067

0.9067

0.9069

0.9069

Rel. error

3.4 · 10−1

1.6 · 10−3

1.1 · 10−3

2.2 · 10−4

2.2 · 10−4


0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

ω

ωsimωtheo

δ

ωsim

4.357 · 10−5

4.368 · 10−3

8.738 · 10−2

1.318 · 10−2

4.357 · 10−2

8.715 · 10−2

δ

10−3

5 · 10−3

10−2

1.5 · 10−2

5 · 10−2

10−1

Figure 3: Transversal oscillation-frequency ω versus slenderness δ.

Example 5.3 �Three-dimensional Problem Including all Strain Variables�. We choose γ � 10−4,δ � 10−2, and a field of torsion angles:

μ � 6πs�2 − s�. �5.15�

In this case, see Figure 4, the rod is initially twisted three times and the gravitational numberis comparatively high. That means all strain variables of the rod are excited during theevolution. The algorithm is able to deal with such a situation, in particular, from the pointof view of conservation of energy. The results are reasonable as long as the linear materiallaw is valid. One has to take into account that for large strain values Timoshenko’s materiallaw does not guarantee that the deformation of the rod preserves the orientation. See Antman�10� for a precise definition of the preservation of orientation.

6. Conclusion

In this paper we use the description of a hyperelastic rod in the formulation of Antman �10�.For the resulting hyperbolic system we developed a numerical method, which conserves theenergy of the rod as well as the orthonormality of the directors. However, the scheme is notable to handle shocks properly. It is neither a TVD scheme nor does it satisfy the entropycondition.

For the material law of Timoshenko �17� we illustrated the method using somenumerical examples. For these examples, the conservation properties are strictly fulfilledand very good agreement of the numerical and analytical results can be observed. Finally,we mention that for realistic three-dimensional problems a nonlinear material law has to beused, which can be easily incorporated in the scheme.


0 0.2 0.4 0.6 0.8 1

0

0.5−1

−0.9−0.8−0.7−0.6−0.5

−0.5

−0.4−0.3−0.2−0.1

0

e 2||g

e3e1

�a�

00.2

0.4 0.6 0.8 1

0

0.5−1

−0.9−0.8−0.7−0.6−0.5

−0.5

−0.4−0.3−0.2−0.1

0

e 2||g

e1

e3

�b�

00.2

0.4 0.6 0.8 1 −0.5

0

0.5−1

−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1

0

e 2||g

e1

e3

�c�

00.2

0.4 0.6 0.8 1

0

0.5−1

−0.9−0.8−0.7−0.6−0.5

−0.5

−0.4−0.3−0.2−0.1

0e 2||g

e1

e3

�d�

Figure 4: Rod’s curve at t � 0 �a�, t � 40 �b�, t � 80 �c�, and t � 120 �d�.

Acknowledgments

This work has been supported by Deutsche Forschungsgemeinschaft �DFG�, KL 1105/18-1 and WE 2003/3-1 and by Rheinland-Pfalz Excellence Center for Mathematical and Com-putational Modeling �CM�2.

References

�1� D. K. Pai, “Interactive simulation of thin solids using Cosserat models,” Computer Graphics Forum, vol.21, no. 3, pp. 347–352, 2002.

�2� J. Barbic and D. James, “Real-time subspace integration for St. Venant-Kirchhoff deformable models,”ACM Transactions on Graphics, vol. 24, no. 3, pp. 982–990, 2005.

�3� A. Weber, T. Lay, and G. Sobottka, “Stable integration of the dynamic Cosserat equations with appli-cation to hair modeling,” Journal of WSCG, vol. 16, pp. 73–80, 2008.

�4� J. C. Simo, “A finite beam formulation. The three dimensional dynamic problem. Part I,” ComputerMethods in Applied Mechanics and Engineering, vol. 49, pp. 55–70, 1985.

�5� J. C. Simo and L. Vu-Quoc, “A three dimensional finite-strain rod model. Part II: computationalaspects,” Computer Methods in Applied Mechanics and Engineering, vol. 58, pp. 79–116, 1986.


�6� J. C. Simo and L. Vu-Quoc, “On the dynamics in space of rods undergoing large motions—a geo-metrically exact approach,” Computer Methods in Applied Mechanics and Engineering, vol. 66, no. 2, pp.125–161, 1988.

�7� J. C. Simo, J. E. Marsden, and P. S. Krishnaprasad, “The Hamiltonian structure of nonlinear elasticity:the material and convective representations of solids, rods, and plates,” Archive for Rational Mechanicsand Analysis, vol. 104, no. 2, pp. 125–183, 1988.

�8� L. Vu-Quoc and X. Tan, “Optimal solid shells for nonlinear analyses of multilayer composites. part ii:dynamics,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 9-10, pp. 1017–1059,2003.

�9� E. Wittbrodt, I. Adamiec-Wojcik, and S. Wojciech, Dynamic of Flexible Multibody Systems, Springer,2006.

�10� S. S. Antman, Nonlinear Problems of Elasticity, Springer, New York, NY, USA, 2005.�11� J. C. Simo, N. Tarnow, and K. K. Wong, “Exact energymomentum conserving algorithms and sym-

plectic schemes for nonlinear dynamics,” Computer Methods in Applied Mechanics and Engineering, vol.100, no. 1, pp. 63–116, 1992.

�12� S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for thenonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839–1875,1995.

�13� L. Vu-Quoc and J. Simo, “On the dynamics of earth-orbiting flexible satellites with multibodycomponents,” AIAA Journal of Guidance, Control, and Dynamics, vol. 10, no. 6, pp. 549–558, 1987.

�14� L. Vu-Quoc, Dynamics of flexible structures performing large overall motions: a geometrically-nonlinearapproach, technical report no. UCB/ERL M86/36 [Ph.D. thesis], University of California at Berkeley, 1986.

�15� F. A. McRobie and J. Lasenby, “Simo-Vu Quoc rods using Clifford algebra,” International Journal forNumerical Methods in Engineering, vol. 45, no. 4, pp. 377–398, 1999.

�16� L. Mahadevan and J. B. Keller, “Coiling of flexible ropes,” Proceedings of the Royal Society A, vol. 452,no. 1950, pp. 1679–1694, 1996.

�17� S. P. Timoshenko, “On the correction for shear of differential equations for transverse vibrations ofprismatic bars,” Philosophical Magazine, vol. 6, no. 41, pp. 744–746, 1921.

�18� L. Vu-Quoc and S. Li, “Invariant-conserving finite difference algorithms for the nonlinear Klein-Gordon equation,” Computer Methods in Applied Mechanics and Engineering, vol. 107, no. 3, pp. 341–391,1993.

�19� C. Peterson, Dynamik der Baukonstruktionen, Vieweg, 1996.�20� S. P. Timoshenko, Schwingungsproblem der Technik, Springer, 1932.


Research ArticleControlled Roof Collapse during Secondary Miningin Coal Mines

Ashleigh Hutchinson

School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, PrivateBag 3, Wits 2050, South Africa

Correspondence should be addressed to Ashleigh Hutchinson, [email protected]

Received 26 December 2011; Accepted 10 April 2012

Academic Editor: Jean Charpin

Copyright q 2012 Ashleigh Hutchinson. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The problem considered is an investigation of the possible collapse of the roof between the pillarnext to be mined in secondary coal mining and the first line of pillar remnants called snooks. Theroof rock between the pillar, which is the working face, and the snook is modelled as an Euler-Bernoulli beam acted on at each end by a horizontal force and by its weight per unit length. Thebeam is clamped at the pillar and simply supported �hinged� at the snook. The dimensionlessdifferential equation for the beam and the boundary conditions depend on one dimensionlessnumber B. We consider the range of values of B before the displacement and curvature first becomesingular at B � B1. The model predicts that for all practical purposes, the beam will break at theclamped end at the pillar. The failure of the beam for values of B greater than B1 is investigatedcomputationally.

1. Introduction

We consider the challenge posed by coal mine pillar extraction �1, 2�. Secondary mininginvolves revisiting a mine and extracting coal from the pillars. The mining of these pillarscommences from the area furthest away from the point of entry of the mine. This exerciseinvolves cutting the existing pillars into smaller pillars called snooks. As each section ismined, the roof must collapse in a controlled manner in order to pose no safety risk to thoseminers operating underground. We analyse the behaviour of the roof of the mine betweenthe pillar next to be mined and the first line of snooks. This is the work area and must be safefor the miners.


Panel width130 –200 m

Tunnel width5 –7 m

Pillar10 –20 m wide

Pillars left tosupportoverburden

Figure 1: A mining panel showing the pillars before pillar extraction. Reproduced with permission fromN. van der Merwe.

2. Model

In Figure 1, a mining panel in shown prior to pillar extraction. The tunnels are excavated incoal which are approximately 5 m to 7 m wide. They are excavated in a fixed pattern crossingat right angles creating a checker board layout. The coal between the tunnels forms the pillarswhich support the overburden rock. The width of the pillars is approximately 10 m to 20 mwide and is a function of the depth of the mine. The height of the tunnels ranges from 3 m to4 m. Secondary mining is carried out in two stages. In the initial stage, approximately 5 to 10pillars are removed and the roof is left to collapse. This stage is modelled in �2�. Followingthis, adjacent pillars are mined and smaller sections are left to collapse. The purpose of thispaper is to model the second stage in the extraction process. Figure 2 shows the snooks afterpillar extraction. The pillars are cut to leave four snooks, approximately 2 m, one at eachcorner. The snooks have to be small enough to fail when the miners are a safe distance �aboutthe width of a pillar� from the working face but they have to be large enough to be stableright next to the unmined pillars.

The roof consists of horizontal layers of rock of approximate thickness 0.5 m to 20 m, asshown in Figure 3. The Euler-Bernoulli beam equation can be used to describe the horizontallayers of rock in the roof. The use of the Euler-Bernoulli beam equation assumes that the roofis thin compared with its horizontal extent and that only the horizontal direction is important.The horizontal extent of the beam is the distance from the next pillar to be mined to the firstline of snooks which is the width of the tunnel and is approximately 6 m. The ratio of thethickness of the beam to its length ranges from about 0.1 to about 3 and thus for the theory toapply the thickness of the beam should not exceed 2 m. The width of the mining panel rangesfrom about 130 m to 200 m. If we take the width of the mining panel as the width of the beamthen the ratio of the length of the beam to its width varies from about 0.05 to 0.03. Dependenceof the variables in the direction of the width of the beam can therefore be neglected. The useof the Euler-Bernoulli beam is therefore justified for a beam of thickness less than about 2 m.

In this paper we will investigate if roof collapse can occur between the next pillar tobe mined and the first line of snooks when these snooks are stable and do not fail. In order toachieve this, we consider the roof of the mine to be clamped at a pillar while at the adjacentsnook, the roof is simply supported or hinged. We consider the roof to be simply supported atthe snook since as secondary mining takes place, disturbances in the rock mass and also the


Snook

These snooks must fail

First line ofsnooks mustbe stable

Pillar

Figure 2: A mining panel showing the snooks after pillar extraction. The pillar is replaced by four snooks,one at each corner. Reproduced with permission from N. van der Merwe.

Goaf (brokenoverburden rock)

Collapsed snooksStable snook Unmined pillar

Layers of rock

Figure 3: A cross-section of the mine showing failed snooks and goaf, a stable snook, the next pillar to bemined, and the overburden which consists of horizontal layers of rock. Reproduced with permission fromN. van der Merwe.

roof collapse due to the failure of the neighbouring snooks could change the roof structurein the region where the snooks support the beam �3�. We model this by assuming that at thesnook the beam is no longer clamped and use instead that the beam is simply supported orhinged at the snook. We also consider the behaviour of this small section of the roof when adisturbance, such as a seismic event, causes a sudden increase in the horizontal force actingat each end of the section of the roof.

An analysis of the problem where both ends of the beam are clamped is presentedin �2�. The beam number B, which occurred in the dimensionless Euler-Bernoulli beamequation, was defined in �2� as follows:

B � L[P

EI

]1/2, �2.1�

where P is the horizontal axial force applied to the ends of the beam, L is the length, E is theYoung’s modulus of the beam, and I is a second moment of the cross-sectional area of thebeam. The beam number was the only dimensionless parameter in the problem. This number


x3

x3

x2

x

i

P

q

P k

j

1

= 0 x3 = L

SnookPillar

Figure 4: Beam with end at x � 0 clamped and end x � L simply supported �hinged�.

has arisen before in the literature, for example in �4�, but no name was assigned to it. Thedisplacement became singular when B � 2nπ where n � 1, 2, 3 . . .. The magnitude of thedisplacement was greatest at the centre of the beam for 0 < B < 2π due to the symmetry ofthe problem. For 0 < B < 2π the magnitude of the curvature was greatest at the endpoints ofthe beam and thus the beam collapsed at these points when the tensile strength of the beamwas exceeded. The problem of one end clamped and one simply supported is not symmetric.Our task is to solve and analyse this problem and to compare it with the problem with bothends clamped.

Previous work on roof failure due to the failure of snooks is reviewed in �5�. Usefultexts are �6–9�.

3. Derivation of the Differential Equation

The combined beam and strut is shown in Figure 4. We will use the notation and conventionsof Segal and Handelman �10�. The coordinate axes are defined in terms of the undeformedbeam. The x1- and x2-axes are along the axes of principle moments of inertia of the cross-section of the beam with the x1-axis vertically downwards. The x3-axis is horizontal andpasses through the centroid of each cross-section. The origin of the coordinate system is at thecentroid of the cross-section of the left end of the beam. Unit vectors, i, j, and k are directedalong each coordinate axis. For simplicity, we denote x3 by x.

An outline of the derivation of the differential equation for the Euler-Bernoulli beamwhen both ends are clamped is given in �2, 10�. The potential energy V of an elastic beam oflength L and Young’s modulus E is given by the following:

V �w� �∫L

0

⎡

⎣12EI

(d2w

dx2

)2− 1

2P

(dw

dx

)2− (q�x� � s�x�)w�x�

⎤

⎦dx, �3.1�

where w�x� is the displacement of the beam from the horizontal position x verticallydownwards in the direction of i, I is the second moment of area about the x2-axis, q�x� is


the magnitude of the body force per unit length, s�x� is the magnitude of the applied surfacetraction per unit length in the direction of i, and P is the horizontal force acting at each endof the beam. We assume that the simple support or hinge can oppose an axial force thusdisallowing any axial motion. The nonlinear strain tensor is used in part of the derivation ofV �w� in �10�.

The derivation of the beam equation depends on the boundary conditions. We showthat the boundary conditions for the present problem yield the same beam equation as in�2� and thus the only difference between the two problems is their boundary conditions.The boundary conditions with one end clamped and the other end simply supported are asfollows:

w�0� � 0,dw

dx�0� � 0, w�L� � 0,

d2w

dx2�L� � 0. �3.2�

At equilibrium, the potential energy is at an extremum. In order to impose thiscondition we let

w�x� � w0�x� � �w1�x�, �3.3�

where � is a constant parameter. Since w�x� satisfies boundary conditions �3.2� for all � itimplies that w0�x� and w1�x� must separately satisfy �3.2�. Thus we have,

d

d�V �w0 � �w1�|��0 �

∫L

0

[

EId2w0dx2

d2w1dx2

− P dw0dx

dw1dx

− (q�x� � s�x�)w1�x�]

dx, �3.4�

giving that, at equilibrium,

0 �∫L

0

[

EId2w0dx2

d2w1dx2

− P dw0dx

dw1dx

− (q�x� � s�x�)w1�x�]

dx. �3.5�

We use integration by parts and the boundary conditions to deduce the followingresults:

∫L

0

d2w0dx2

d2w1dx2

dx �∫L

0

d4w0dx4

w1�x�dx,

∫L

0

dw0dx

dw1dx

dx � −∫L

0

d2w0dx2

w1�x� dx.

�3.6�

Using �3.6� we can rewrite �3.5� as

0 �∫L

0

[

EId4w0dx4

� Pd2w0dx2

− q�x� − s�x�]

w1�x� dx. �3.7�


Since �3.7� holds for arbitrary w1�x�, we can deduce that

EId4w0dx4

� Pd2w0dx2

� q�x� � s�x�. �3.8�

In our model the roof is made of horizontal layers of rock each acting as a beam. Thehorizontal force P acting on each end of the beam arises as a result of the compressive stressesdue to the rock mass above. The quantity q is the weight per unit length of the beam which weassume is constant. The quantity s is the magnitude of the applied normal surface tractionper unit length due to the transfer of stresses from the adjoining layers. We assume that sis constant. The displacement depends on the weight and on the applied normal surfacetraction in the same way. We will therefore denote the combined forces, s and q, simply by q.The above derivation and in �2� differ from that of Segal and Handelman �10� by the inclusionof q�x� and s�x� in the analysis. Also, in �10�, the boundary conditions were as follows:

w�0� � 0, w′′�0� � 0, w�1� � 0, w′′�1� � 0, �3.9�

while in �2�, the boundary conditions were as follows:

w�0� � 0, w′�0� � 0, w�1� � 0, w′�1� � 0. �3.10�

We see that the Euler-Bernoulli beam equation remains valid for the boundaryconditions used in this paper. Equation �3.8� is now written in dimensionless form. Define�2�:

x �x

L, w �

w

S, S �

qL4

EI, �3.11�

where S is the characteristic displacement. Equation �3.8� becomes

d4w

dx4� B2

d2w

dx2� 1, �3.12�

where the beam number B is defined by �2.1�. The boundary conditions, �3.2�, whenexpressed in dimensionless variables become

w�0� � 0,dw

dx�0� � 0, w�1� � 0,

d2w

dx2�1� � 0. �3.13�

The bending moment M is �10� as follows:

M � EId2w

dx2, �3.14�


where, since it is assumed that the displacement is sufficiently small that linear theory applies�10�,

d2w

dx2� curvature of the neutral axis of the beam. �3.15�

Define �2�:

M �M

qL2. �3.16�

Then

M �d2w

dx2, �3.17�

and we will refer toM as both the bending moment and curvature of the beam. The overheadbar will be suppressed in the rest of the paper to keep the notation simple.

4. Mathematical Solution

Consider the model of the roof rock between a pillar at x � 0 which is the working face and asnook at x � 1 described by an Euler-Bernoulli beam with end x � 0 clamped and end x � 1simply supported or hinged. The displacement w�x� satisfies the differential equation

d4w

dx4� B2

d2w

dx2� 1, �4.1�

subject to the boundary conditions

w�0� � 0,dw

dx�0� � 0, w�1� � 0,

d2w

dx2�1� � 0. �4.2�

The solution for w�x� is

w�x� �x2

2B2− 1B4�sinB − B cosB�

[[(

1 �B2

2

)

sinB − B]

�1 − cos�Bx��

�

[

1 −(

1 �B2

2

)

cosB

]

�Bx − sin�Bx��]

,

�4.3�

provided that B does not satisfy

F�B� � sinB − B cosB � 0. �4.4�


2 4 6 8 10

5

B

−5

F(B

)

Figure 5: Graph of F�B� � sinB − B cosB against B for B in the range �0,10�.

2 4 6 8 10

5

10

By

−5

−10

Figure 6: Graph of y � tanB and y � B against B for B in the range �0,10�.

We can rewrite �4.4� as follows:

tanB � B. �4.5�

Equations �4.4� and �4.5� are plotted in Figures 5 and 6. The first five roots of �4.4� and�4.5� are

B0 � 0, B1 � 4.4934, B2 � 7.7253, B3 � 10.9041, B4 � 14.0662. �4.6�

At these values of B, the displacement becomes infinite. Since B � 4.4934 is the firstnonzero value of B for which the displacement becomes singular, our primary concern isin the interval 0 < B < 4.4934. In the next section, we discuss the solution �4.3� for thedisplacement and calculate the curvature of the beam which determines the location at whichthe beam will break.

For a beam with both ends clamped �2�, the displacement becomes singular for B �2nπ where n � 1, 2, 3, . . .. Comparing the first points at which the displacement becomessingular in the two models, we note that 4.4934 < 2π � 6.2832. Small displacements andsmall derivatives are used in the derivation of the Euler-Bernoulli beam equation. The theory


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

x

4

0.5

Figure 7: Graphs of the displacement w�x� for B � 0.5, 1.5, 2.5, 3.5, 4.

therefore breaks down in the neighbourhood of the points x � Bn where the curvature hassingular behaviour. A full nonlinear theory would need to be used in these regions. However,the beam will break when its tensile strength is exceeded which could be well before thesingularities in the curvature are reached.

5. Analysis of the Results

Graphs of the displacement w�x� for values of the beam number, B, in the range 0 < B <4.4934 are shown in Figure 7. The displacement has two stationary points which are locatedat x � 0 and at an interior point.

Consider first the beam for small values of B. The asymptotic expansion of w�x� asB → 0 is given by the following:

w�x� �x2�1 − x�

48

[

3 − 2x � B2

60

(4x3 − 11x2 � 4x � 6

)�O(B4)]

. �5.1�

The displacement is nonzero when B � 0 because of the weight per unit length, q,acting on the beam. Graphs of the displacement w�x� for small values of B are presented inFigure 8. From the graphs, we can see that �5.1� is a good approximation for the displacementfor B ≤ 0.9.

Denote by x0 the point of maximum displacement of the beam. In order to estimate x0for small B, consider, from �5.1�,

dw

dx�x

48

[

8x2 − 15x � 6 − B2

20

(8x4 − 25x3 � 20x2 � 2x − 4

)�O(B4)]

� 0, as B −→ 0.

�5.2�

The root of the quadratic equation

8x2 − 15x � 6 � 0 �5.3�


0

1

2

3

4

5

6

w

0.10.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

×10−3

Figure 8: Comparison of the asymptotic expansion �5.1� correct to order B2 �- - -� with the exact solution�4.3� �—� of the deflection w�x� for B � 0.1, 0.25, 0.5, 0.9. The curves for the exact solution and asymptoticsolution overlap.

in the range 0 < x < 1 is x � 0.578 and therefore,

x0 � 0.578 �O(B2), as B −→ 0. �5.4�

The root of

8x4 − 25x3 � 20x2 � 2x − 4 � 0 �5.5�

in the range 0 < x < 1 is x � 0.594 which is only 2.77% larger than 0.578. The maximumturning points of the curves in Figure 10 are all close to the zero order in B value. This showsthat �5.4� is a good approximation of x0 for small values of B. Substituting �5.4� into �5.1�gives

w�x0� � 0.0054 �O(B2), as B −→ 0, �5.6�

which from Figures 7 and 8 is a good approximation for B ≤ 0.5. In comparison, for a beamwith clamped ends, from symmetry, the magnitude of the deflection is a maximum at x0 � 0.5for 0 < B < 2π .

In order to gain insight into the possible failure of the beam, we need to determine thepoint at which the beam is under maximum stress. We assume that this is the point at whichthe magnitude of the curvature is greatest. The dimensionless curvature of the beam is givenby the following:

d2w

dx2�

1B2

− 1B2�sinB − B cosB�

[[(

1 �B2

2

)

sinB − B]

cos�Bx�

�

[

1 −(

1 �B2

2

)

cosB

]

sin�Bx�

]

.

�5.7�


0

0.05

0.15

0.25

0.35

0.45

|C

urva

ture

|

4

0 0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.6 0.7 0.8 0.9 1x

Figure 9: Graphs of the magnitude of the curvature for B � 0.5, 1.5, 2.5, 3.5, 4.

Graphs of the magnitude of the curvature for a range of values B < B1 are given inFigure 9.

The curvature vanishes at x � 1 because the end x � 1 is simply supported. Denoteby x1 the position of the local maximum of the magnitude of the curvature when 0 < x < 1.Since the sign of the curvature at x � 0 is opposite to that at x � x1, the curvature must vanishat a point, say x2, where 0 < x2 < x1. We can deduce from Figure 9 that x2 does not changesignificantly as B is increased. For the range of values used in Figure 9, the magnitude of thecurvature at x � 0 is greater than at x � x1. We will investigate later as to whether this isalways the case.

Consider first the curvature for small values of B. The asymptotic expansion of �5.7�as B → 0 is given by the following:

d2w

dx2�

18�1 − x�

[

1 − 4x � B2

30

(10x3 − 15x2 � 1

)�O(B4)]

. �5.8�

To zero order in B, the zeros of the curvature occur at x � 1 and x � 1/4 and therefore,

x2 � 0.25 �O(B2), as B −→ 0. �5.9�

The root of the cubic equation

10x3 − 15x2 � 1 � 0 �5.10�

in the range 0 < x < 1 is x � 0.287 which explains why x2 does not greatly depend on B. Theexpansions are in good agreement with Figure 9.

We now consider the turning point x � x1 of the curvature. From �5.7�,

d3w

dx3�

1B2�sinB − B cosB�

[[(

1 �B2

2

)

sinB − B]

sin�Bx�

−[

1 −(

1 �B2

2

)

cosB

]

cos�Bx�

]

.

�5.11�


0.2 0.4 0.6 0.8 1

B

100

200

300

400

|w”(x)|

Figure 10: Graph of the magnitude of the curvature for B � 4.4930.

which vanishes for x � x1, where

tan�Bx1� �P�B�Q�B�

,

P�B� � 1 −(

1 �B2

2

)

cosB,

Q�B� �

(

1 �B2

2

)

sinB − B.

�5.12�

We now examine R, the absolute value of the ratio of the curvature at x � 0 to thecurvature at x � x1 as follows:

R�B� �∣∣∣∣w′′�0�w′′�x1�

∣∣∣∣. �5.13�

For the range of values of B considered in Figure 9, R�B� > 1. However, as Bapproaches the first singular value B � B1 � 4.4934, the ratio R approaches and may exceedunity. This is illustrated in Figure 10 where B � 4.4930. We now investigate analytically theratio R�B� for 0 ≤ B ≤ B1.

Consider first the asymptotic behaviour as B → 0. Now

P�B� �5

24B4 �O

(B6),

Q�B� �13B3 �O

(B5),

�5.14�


and therefore, from �5.12�

x1 �58�O(B2), as B −→ 0. �5.15�

The value x1 � 0.625 for small B is consistent with the graphs in Figure 9. Also, from�5.8�, as B → 0,

w′′�0� �18

(1 �O

(B2)),

w′′�x1� � − 9128(

1 �O(B2)),

�5.16�

and therefore,

R�B� �169

�O(B2), as B −→ 0. �5.17�

A more detailed calculation yields the following result

R�B� �169

[1 − 47

2304B2 �O

(B4)], �5.18�

so that R�B� decreases initially as B increases.Consider next the limit as B → B1. The curvature w′′�x�, given by �5.7�, may be

expressed in terms of F�B�, P�B�, and Q�B� as follows:

d2w

dx2�

1B2F�B�

�F�B� −Q�B� cos�Bx� − P�B� sin�Bx��. �5.19�

Then,

w′′�0� �1

B2F�B��F�B� −Q�B��. �5.20�

Also, since x1 is defined by �5.12�,

cos�Bx1� �Q�B�

�P 2�B� �Q2�B��1/2, sin�Bx1� �

P�B�

�P 2�B� �Q2�B��1/2, �5.21�

and therefore,

w′′�x1� �1

B2F�B�

[F�B� −

(P 2�B� �Q2�B�

)1/2]. �5.22�


Hence,

R�B� �∣∣∣∣w′′�0�w′′�x1�

∣∣∣∣ �

∣∣∣∣∣

F�B� −Q�B�F�B� − �P 2�B� �Q2�B��1/2

∣∣∣∣∣. �5.23�

Now at B � B1, F�B1� � 0 and therefore, sinB1 � B1 cosB1. Thus,

Q�B1� � −B1P�B1�, �5.24�

and hence

R�B1� �B1

(1 � B21

)1/2 . �5.25�

Thus, R�B1� < 1. Also, using �5.12� and �5.24�, we find that in the limit B � B1, x1 isgiven by the following:

tan�B1x1� � − 1B1. �5.26�

Since B1 � 4.4934, it follows that

R�B1� � 0.9761, �5.27�

and that

x1�B1� � 0.66. �5.28�

The graph of R�B� against B for 0 ≤ B < B1 is presented in Figure 11. The analyticalresults for B → 0 and B → B1 agree with the graph. In comparison when the two ends ofthe beam are clamped, R�B� > 1 for 0 ≤ B < B1 and R�B1� � 1, where B1 � 2π .

6. Numerical Estimates

Consider first the beam number B defined by �2.1�. The total horizontal force P acting on eachend section is given by �2, 9, 10� the following:

P � kρgHbh, �6.1�

where H is the depth of the mine below the surface of the earth, ρ is the average density ofthe rock from the surface of the earth to the depth H, b is the breadth of the roof beam, h isthe thickness of the beam, and k is the lateral stress coefficient. The lateral stress coefficient isa function of the rock properties. The value k � 0 corresponds to a material that is completely


0 1 2 3 4B

0.5

1.0

1.5

Point of intersection at

B = 4.4324

Figure 11: Graph of the ratio R�B� for 0 ≤ B ≤ B1, where B1 � 4.4934. The point of intersection of the twocurves is B � 4.4324.

solid while k � 1 corresponds to a fluid in which the pressure is isotropic. Some models canpredict values of k > 1 and that k decreases with depth �7�. For the shallow coal mines whichwe will consider, we will take k � 2 �11�. The second moment of area about the x2-axis is �2�

I �bh3

12. �6.2�

The beam number �2.1� becomes

B � 3.46L

h

[kρgH

E

]1/2. �6.3�

For a beam with one end clamped and the other end simply supported, thedisplacement and curvature become infinite first at B � B1 � 4.4934. Thus we obtain theupper limit Lc for the length that a beam can have without collapsing:

L < Lc � 1.2971h[

E

kρgH

]1/2. �6.4�

As the beam becomes more fractured with time the Young’s modulus E will decreaseand Lc will decrease.

When both ends of the beam are clamped, B1 � 2π and the ratio of the maximumlength Lc�2� when one end is clamped and one is simply supported, to the maximum lengthLc�1� when both ends are clamped is

Lc�2�Lc�1�

�B1�2�B1�1�

� 0.7151. �6.5�

The critical length Lc�1� describes the initial stage of the process of pillar extraction whenseveral pillars are removed and the roof is left to collapse, while Lc�2� models the second


Table 1: Critical length Lc.

h�m� H � 120 mLc�m�

H � 500 mLc�m�

0.1 9.26 4.540.2 18.53 9.080.3 27.80 13.620.4 37.06 18.160.5 46.33 22.70.6 55.60 27.230.7 64.86 31.770.8 74.12 36.310.9 83.39 40.851 92.65 45.39

stage when adjacent pillars are mined and smaller sections of the roof are left to collapse. Wesee that Lc�2� < Lc�1� which is consistent with the two models.

We consider a beam made of sandstone. We use the following estimates:

E � 3 × 1010 Pa,h � 0.1 m, 0.2 m, . . . , 1 m,

k � 2,

ρ � 2.5 × 103 kg/m3,

g � 9.8 m/s2,

H � 120 m, 500 m, 1000 m.

�6.6�

Table 1 summarizes numerical estimates for Lc for a beam with one end clamped andthe other simply supported. In a coal mine the distance between the pillars ranges from 5 mand 7 m. When the bending moment or curvature exceeds the tensile strength of the beam theroof will collapse. This may occur for beam lengths L less than Lc since Lc provides only anupper limit on the length of the beam for collapse.

The axial force P may experience a sudden increase due to a seismic event which couldlast for a short time. Using �2.1�, the upper limit for the length, Lc, can be written as follows:

Lc � B1[EI

P

]1/2. �6.7�

If Lc is reduced below about 6 m due to an increase in P then a roof collapse may occur.


1 2 3 4 5 6

0.5

1

1.5

2

B

w”(0)

(2)(1)

Figure 12: Graphs of the curvature w′′�0� at x � 0 plotted against B for a beam with: �1� x � 0 clamped andx � 1 clamped, �2� x � 0 clamped and x � 1 simply supported.

We now compare the effect of hinged and clamped supports at the snook at x � 1 onthe curvature at the pillar at x � 0. When both ends of the beam are clamped �2�,

w′′1�0� �1B2

[1 − B

2 tan�B/2�

],

w′′1�0� �1

12�O(B2), as B −→ 0.

�6.8�

When the end x � 0 is clamped and the end x � 1 is simply supported �hinged�, from�5.7� and �5.8�,

w′′2�0� �1 − cosB − �1/2�B sinBB�sinB − B cosB� ,

w′′2�0� �18�O(B2), as B −→ 0.

�6.9�

Thus,

w′′2�0�w′′2�0�

|B�0 � 32 . �6.10�

In Figure 12 the graphs of w′′1�0� and w′′2�0� are plotted against B. We see that when

the end x � 1 is simply supported the curvature at x � 0 is greater for a given value of Bthan when it is clamped. The tensile strength of the beam will be exceeded at the end x � 0for lower values of B when the end x � 1 is simply supported than when it is clamped. Thesimply supported boundary condition has the effect of increasing the bending moment at theend x � 0 and causing the beam to break at x � 0 for lower values of B.


7. Values of the Beam Number B Greater Than B1

A graph of the magnitude of the curvature of the beam at the end x � 0, plotted against B, isgiven in Figure 13. It divides the values of B into the intervals,

I1 � �0, B1�, I2 � �B1, B2�, I3 � �B2, B3�, . . . , In � �Bn−1, Bn�, �7.1�

where B1, B2, B3,. . . are the values at which the displacement and curvature become infinite.We have only considered the first interval I1. Since, from �2.1�, we see that the beam numberB is proportional to L and P �1/2� the value of B would increase if either L or P were to increase.The length of the beam increases by a finite amount when a snook fails. In the second stageof the pillar extraction process, small sections of the mine collapse but if the snooks are toostrong they will support a longer section of the roof which will form a beam and collapsewhen the snooks fail. Another way in which B could increase suddenly is due to a seismicevent which may produce a discontinuous increase in P which could last for a short periodof time.

Consider first the displacement. The displacement for values of B in the first interval,I1, was considered in Figure 7. In Figure 14 graphs of the displacement for representativevalues of B in the intervals, I1 to I6, are presented. We see that as B increases through theintervals the number of turning points increases and that the displacement can take negativevalues beyond interval I1. The amplitude of the displacement will depend on how close B isto the singular end points of the interval.

Consider next the magnitude of the curvature. In Figure 9 graphs of the magnitudeof the curvature were plotted against x for values of B in the first interval I1. In Figure 15,graphs of the magnitude of the curvature are plotted for the same representative values usedto plot the displacement in Figure 14. For the values of B and n considered, the number oflocal maxima of the magnitude of the curvature in the nth interval In is n. The greatest localmaximum is not at the end x � 0 but at interior points. There may be several points for whichthe magnitude of the curvature has the maximum value. If the bending moment exceeds thetensile strength the beam will break at these interior points. The magnitude of the curvaturedepends on how close B is to the singular endpoints of the interval. Since B is proportionalto the length of the beam L we see from Figure 13 that if the value of B is in the range outsideof B1 then a longer beam could be less susceptible to failure than a shorter beam. This couldbe associated with the beam taking on a higher mode of bending. We see from Figure 14 thatthe displacement can be negative. For this to be possible in practice the beam would have tobe detached sufficiently from the layers above.

8. Conclusions

We investigated the possible roof collapse between the next pillar to be mined and the firstline of snooks by modelling the roof as an Euler-Bernoulli beam. The beam was simplysupported �hinged� at the snook and clamped at the pillar which was the working face. Themodel contained one dimensionless number—the beam number B.

Numerical estimates obtained for the critical length Lc are comparable to the expecteddistance between pillars which in a coal mine is 5–7 m thus making the model credible. Themodel also predicts that the roof may collapse at the clamped end at the pillar for 0 < B <4.4324. For the range 4.4324 < B < B1 � 4.4934, the model predicts that the roof may collapse


5 10 15 20

0.1

0.2

0.3

0.4

0.5

|C

urva

ture

|

B

Figure 13: Graph of the magnitude of the curvature at the end x � 0 plotted against B.

0

2

4

6

8

w

−8−6−4−2

0

w

−3−2−1

0

1

w

−1.5

−1

−0.5

0

w

0 0.2 0.4 0.6 0.8 1

−6−4−2

0

w

−6−4−2

0

w

26

12

19

9

16

x

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1x

0 0.2 0.4 0.6 0.8 1x

×10−3 ×10−3

×10−3 ×10−3

×10−4 ×10−4

Figure 14: Graphs of the displacement for representative values of the beam number B in the intervals I1to I6.

at an interior point closer to the snook than to the pillar where the magnitude of the curvatureattains its maximum value. However, this range contributes only 1.36 percent of the range�0, B1� and since it is likely that the threshold of the stress would have been exceeded forvalues of the beam number below B � 4.4324, for all practical purposes, the beam will breakat the clamped end if the threshold of its stress is exceeded. It will therefore break at the pillarwhich is the working face. However, the model showed that it is not necessary for the beamto break at the clamped end. The beam may break at interior points. In I1, the interior point


0

0.1

0.2

|C

urva

ture

|

0

0.1

0.2

|C

urva

ture

|

0

0.05

0.1

|C

urva

ture

|

0

0.02

0.04

0.06

|C

urva

ture

|0

0.02

0.04

|C

urva

ture

|

0 2 4 6 8 10

0.01

0.02

0.03

|C

urva

ture

|

2 6

9 12

1916

x0 2 4 6 8 1

x

0 2 4 6 8 1x

0 2 4 6 8 1x

0 2 4 6 8 1x

0 2 4 6 8 1x

Figure 15: Graphs of the magnitude of the curvature for representative values of the beam number B inthe intervals I1 to I6.

is unique for a given value of B. Other boundary conditions would need to be consideredand analyzed in order to determine whether the beam can collapse at an interior point forpractical values of B.

From Figure 12 we deduced that a beam which is simply supported �hinged� at thesnook produces a larger bending moment at the pillar �where it is clamped� than a beamwhich is clamped at the snook. The beam will break at a lower value of the beam numberwhen it is hinged at the snook.

The displacement and curvature become infinite at the zeros of F�B� defined by �4.4�which divides the value of B into intervals. We considered mainly the first interval 0 ≤ B ≤ B1.However, as the snooks fail B can increase discontinuously by finite amounts and may takevalues in the higher intervals. A preliminary computational investigation was undertakenof the displacement and curvature for values of B in these intervals. It was found that thedisplacement can take negative values. In the nth interval the magnitude of the curvature hadn local maxima. The maximum value of the magnitude of the curvature did not in generaloccur at the pillar but occurred instead at interior points. If the tensile strength of the beam isexceeded the beam could break at several interior points.

In practice, roof bolts are installed �12�.

Acknowledgments

The problem of pillar extraction in coal mines was submitted to the Mathematics in IndustryStudy Group 2011 �MISG� at the University of the Witwatersrand, Johannesburg. Theauthor thanks the referees for their valuable comments which improved this paper greatly.The author thanks Matthew Woolway, University of the Witwatersrand, Johannesburg,


for assisting in creating the diagrams. The author would like to express gratitude andappreciation to Professor Nielen van der Merwe and Dr. Halil Yilmaz of the School of MiningEngineering, University of the Witwatersrand, for submitting the problem. The author wouldparticularly like to acknowledge that she has profited greatly from the discussions held withProfessor Nielen van der Merwe. The author would like to acknowledge Professor ColinPlease of the University of Southhampton, England, who formulated the model for pillarextraction and roof collapse at the MISG. The author thanks Professor David Mason for hisvaluable comments, advice, and encouragement while she was writing this paper.

References

�1� J. N. van der Merwe, “Fundamental analysis of the interaction between overburden behaviour andsnook stability in coal mines,” Journal of the South African Institute of Mining and Metallurgy, vol. 105,pp. 63–73, 2005.

�2� C. Please, D. P. Mason, C. M. Khalique, J. M. T. Ngnotchouye, J. N. van der Merwe, and H. Yilmaz,“Coal mine pillar extraction,” Submitted to International Journal of Rock Mechanics and Mining Sciences.

�3� N. van der Merwe, “Private communication,” 2012.�4� C. Kimball and T.-W. Tsai, “Modeling of exural beams subjected to arbitrary end loads,” Journal of

Mechanical Design, vol. 124, pp. 223–235, 2002.�5� B. H. G. Brady and E. T. Brown, Rock Mechanics for Underground Mining, Chapman & Hall, London,

UK, 2nd edition, 1993.�6� L. Obert and W. L. Duvall, Rock Mechanics and the Design of Structures in Rock, Wiley & Sons, New

York, NY, USA, 1967.�7� E. P. Popov, Mechanics of Materials, Prentice Hall International, Englewood Cliffs, NJ, USA, 1978.�8� J. C. Jaeger and N. G. W. Cook, Fundamentals of Rock Mechanics, Methuen, London, UK, 1969.�9� J. C. Jaegar, N. G. W. Cook, and R. W. Zimmerman, Fundamentals of Rock Mechanics, Blackwell, Oxford,

UK, 4th edition, 2007.�10� L. A. Segal and G. H. Handelman, Mathematics Applied to ContinuumMechanics, Macmillan, New York,

NY, USA, 1977.�11� J. N. van der Merwe and B. J. Madden, in Rock Engineering for Underground Coal Mining, Special

Publications, Series 8, p. 135, Southern African Institute of Mining and Metallurgy, Johannesburg,South Africa, 2nd edition, 2011.

�12� J. N. van der Merwe and B. J. Madden, in Rock Engineering for Underground Coal Mining,Special Publications, Series 8, pp. 208–218, Southern African Institute of Mining and Metallurgy,Johannesburg, South Africa, 2nd edition, 2011.


Research ArticleOn the Solutions Fractional RiccatiDifferential Equation with ModifiedRiemann-Liouville Derivative

Mehmet Merdan

Department of Mathematics Engineering, Gümüşhane University, 29100 Gümüşhane, Turkey

Correspondence should be addressed to Mehmet Merdan, [email protected]

Received 10 December 2011; Accepted 6 March 2012

Academic Editor: Ebrahim Momoniat

Copyright q 2012 Mehmet Merdan. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Fractional variational iteration method �FVIM� is performed to give an approximate analyticalsolution of nonlinear fractional Riccati differential equation. Fractional derivatives are describedin the Riemann-Liouville derivative. A new application of fractional variational iteration method�FVIM� was extended to derive analytical solutions in the form of a series for these equations.The behavior of the solutions and the effects of different values of fractional order α areindicated graphically. The results obtained by the FVIM reveal that the method is very reliable,convenient, and effective method for nonlinear differential equations with modified Riemann-Liouville derivative

1. Introduction

In recent years, fractional calculus used in many areas such as electrical networks, controltheory of dynamical systems, probability and statistics, electrochemistry of corrosion,chemical physics, optics, engineering, acoustics, viscoelasticity, material science and signalprocessing can be successfully modelled by linear or nonlinear fractional order differentialequations �1–10�. As it is well known, Riccati differential equations concerned withapplications in pattern formation in dynamic games, linear systems with Markovian jumps,river flows, econometric models, stochastic control, theory, diffusion problems, and invariantembedding �11–17�. Many studies have been conducted on solutions of the Riccati differentialequations. Some of them, the approximate solution of ordinary Riccati differential equationobtained from homotopy perturbation method �HPM� �18–20�, homotopy analysis method�HAM� �21�, and variational iteration method proposed by He �22�. The He’s homotopyperturbation method proposed by He �23–25� the variational iteration method �26� and


Adomian decomposition method �ADM� �27� to solve quadratic Riccati differential equationof fractional order.

The variational iteration method �VIM�, which proposed by He �28, 29�, wassuccessfully applied to autonomous ordinary and partial differential equations and otherfields. He �30� was the first to apply the variational iteration method to fractional differentialequations. In recent years, a new modified Riemann-Liouville left derivative is suggested byJumarie �31–35�. Recently, the fractional Riccati differential equation is solved with help ofnew homotopy perturbation method �HPM� �23�.

In this paper, we extend the application of the VIM in order to derive analyticalapproximate solutions to nonlinear fractional Riccati differential equation:

Dα∗ y�x� � A�x� � B�x�y�x� � C�x�y2�x�, x ∈ R, 0 < α ≤ 1, t > 0, �1.1�

subject to the initial conditions

y�k��0� � dk, k � 0, 1, 2, . . . , n − 1, �1.2�

where α is fractional derivative order, n is an integer, A�x�, B�x�, and C�x� are known realfunctions, and dk is a constant.

The goal of this paper is to extend the application of the variational iteration methodto solve fractional nonlinear Riccati differential equations with modified Riemann-Liouvillederivative.

The paper is organized as follows: In Section 2, we give definitions related to thefractional calculus theory briefly. In Section 3, we define the solution procedure of thefractional variational iteration method to show inefficiency of this method, we presentthe application of the FVIM for the fractional nonlinear Riccati differential equations withmodified Riemann-Liouville derivative and numerical results in Section 4. The conclusionsare then given in the final Section 5.

2. Basic Definitions

Here, some basic definitions and properties of the fractional calculus theory which can befound in �31–35�.

Definition 2.1. Assume f : R → R, x → f�x� denote a continuous �but not necessarilydifferentiable� function, and let the partition h > 0 in the interval �0, 1�. Jumarie’s derivativeis defined through the fractional difference �34�:

Δ�α� � �FW − 1�αf�x� �∞∑

k�0

�−1�k(α

k

)

f�x � �α − k�h�, �2.1�

where FWf�x� � f�x � h�. Fractional derivative is defined as the following limit form �1, 7�:

f �α� � limh→ 0

Δα[f�x� − f�0�]

hα. �2.2�


This definition is close to the standard definition of derivatives �calculus for beginners�, andas a direct result, the αth derivative of a constant, 0 < α < 1, is zero.

Definition 2.2. The left-sided Riemann-Liouville fractional integral oper

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