2357 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
Volume 09 Issue 08 August 2021, Page no. β 2357-2370
Index Copernicus ICV: 57.55, Impact Factor: 7.184
DOI: 10.47191/ijmcr/v9i8.01
Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff
Ordinary Differential Equations
Pius Tumba1, Sunday Babuba2, A.I. Bakari3 1Department of Mathematics, Federal University, Gashua, Nigeria 2,3Department of Mathematics, Federal University, Dutse, Nigeria
ARTICLE INFO ABSTRACT
Published Online:
07 August 2021
Corresponding Author:
Pius Tumba
In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of
stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of π₯ = π₯π+π , π =
0,1,2,3,4,5 πππ6 and its first derivative was also evaluated at off-grid point π₯ = π₯π+π , π =15
2 and grid point π₯ = π₯π+π , π = 8. The method is suitable
for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and
are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.
KEYWORDS: Backward Differentiation Formula, Second Derivative, off-step points, Stiff ODEs
1. INTRODUCTION
In sciences, researchers are not desired in all solutions to a differential equation, but only in
those extra conditions satisfied solutions. For instance, in the case of Newtonβs second law
of motion for a point particle, one could be actually interested only in solutions such that the
particle that specifically position at some initial time. Such condition is called an initial
condition.
A first order ordinary differential equations (ODEs) on the unknown π¦ is
π¦β²(π‘) = π(π‘, π¦(π‘)) and π¦(π‘0) = π¦0 π β€ π‘ β€ π (1)
where f is continuous within the interval of integration. is sufficiently differentiable and
satisfies a Lipschitz condition (1).
Moreover, Awoyemi and Idowu (2005) derived a class of hybrid collocation methods for
thirdβorder ordinary differential equations. Hybrid block scheme were constructed to defeat
the Dahlquist (1956) barrier theorem in accordance with the conventional linear multistep
method was changed to include off-step points of derivation process (see Lambert (1973)
and Gear (1965)).
Similarly, Researchers like Butcher (2003), Zarina et al. (2005), Awoyemi et al. (2007),
Areo etal. (2011), Ibijola et al. (2011) have all constructed linear multistep methods (LMMs)
to generate numerical solution to (1).
Nasir et al. (2011), Zawawi et al. (2012), Akinfenwa et al. (2013), construct Block
Backward Differentiation Formulas for Solving Ordinary Differential Equations. This
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2358 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
method possesses the requirement for stiffly stable and suitable to solve stiff problems. The
numerical results were presented to verify the efficiency of this method as compared to the
Classical Backward Differentiation Formula (BDF) method and ode15s in Matlab. The
method outperformed the BDF method and ode15s in terms of maximum error and execution
time.
2. CONSTRUCTION OF UNIFORM ORDER ELEVEN OF EIGHT-STEP
HYBRID
In this research, we constructed eight-step hybrid block backward differentiation formula.
The method will be applied in block form by Onumanyi et al (1994,1999) where π β π π‘ππ
multistep collocation method with π‘ interpolation points and π collocation points were
obtained as follows:
π¦(π₯) = β πΌπ(π₯)π¦π+π + β β π½π(π₯)ππ+π
π β1
π=0
πβ1
π=0
(2)
and its extension it to the second derivative gives the general form of the continuous k βstep 2nd derivative linear multistep method as:
π¦(π₯) = β πΌπ(π₯)π¦π+π + β βπ½π(π₯)ππ+π + β2 β πΎπ(π₯)ππ+π
π‘β1
π=0
π β1
π=0
πβ1
π=0
(3)
Here the coefficients πΌπ(π₯), π½π(π₯) and πΎπ(π₯) are polynomials of degree (π β 1) .
where
πΌπ(π₯) = β πΌπ.π+1π₯π
π+π +π‘β1
π=0
(4)
βπ½π(π₯) = β βπ½π.π+1π₯π
π+π +π‘β1
π=0
(5)
β2π½π(π₯) = β β2πΎπ.π+1π₯π
π+π +π‘β1
π=0
(6)
2.1. Specification of the Hybrid Block Methods
From equation (2) - (6) we propose continuous k-step of extended block hybrid backward differentiation formula of the form
π¦(π₯) = βπΌπ(π₯)π¦π+π + βπ½πβ1(π₯)ππ+πβ1 + βπ½2πβ12
ππ+
2πβ12
+ βπ½π(π₯)ππ+π + β2πΎπβ1(π₯)ππ+πβ1 + β2πΎπ(π₯)ππ+π
π
π
(7)
2.1 .2 Eight Step Method with One Off-Step Point
In this case = 8 , π = 8, π‘ = 8 and (7) becomes
π¦(π₯) = πΌ0(π₯)π¦π + πΌ1(π₯)π¦π+1 + πΌ2(π₯)π¦π+2 + πΌ3(π₯)π¦π+3 + πΌ4(π₯)π¦π+4 + πΌ5(π₯)π¦π+5 + πΌ6(π₯)ππ+6 + πΌ7(π₯)π¦π+7 = β[π½0(π₯)ππ + π½1(π₯)ππ+1 + π½2(π₯)ππ+2 + π½3(π₯)ππ+3 + π½4(π₯)ππ+4 +
π½5(π₯)ππ+5 + π½6(π₯)ππ+6 + π½7(π₯)ππ+7 + π½15
2
(π₯)ππ+
15
2
+ π½8(π₯)ππ+8] + β2[πΎ0(π₯)ππ + πΎ1(π₯)ππ+1 + πΎ2(π₯)ππ+2 + πΎ3(π₯)ππ+3 + πΎ4(π₯)ππ+4 + πΎ5(π₯)ππ+5 + πΎ6(π₯)ππ+6 + πΎ7(π₯)ππ+7 +
πΎ8(π₯)ππ+8] (8)
and interpolating (4)at π₯ = π₯π , π₯π+1, π₯π+2 , π₯π+3, π₯π+4, π₯π+5, π₯π+6, π₯π+7 and collocating (5) at π₯ = π₯π+7, π₯π+15
2,π₯π+8 to give the systems of equation.
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2359 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
π¦(π₯) = βπππ₯π = π¦π (9)
6
π=0
π¦β²(π₯) = βππππ₯πβ1 = ππ+π (10)
6
π=1
π¦β²β²(π₯) = βπ(π β 1)πππ₯πβ2 = ππ+π (11)
6
π=2
Expressing the system of equations (9) -(11) in the form π΄π = π as:
A=
[ 1 π₯π π₯π
2 π₯π3 π₯π
4 π₯π5 π₯π
6 π₯π7 π₯π
8 π₯π9 π₯π
10 π₯π11 π₯π
12
1 π₯π+1 π₯π+12 π₯π+1
3 π₯π+14 π₯π+1
5 π₯π+16 π₯π+1
7 π₯π+18 π₯π+1
9 π₯π+110 π₯π+1
11 π₯π+112
1 π₯π+2 π₯π+22 π₯π+2
3 π₯π+24 π₯π+2
5 π₯π+26 π₯π+2
7 π₯π+28 π₯π+2
9 π₯π+210 π₯π+2
11 π₯π+212
1 π₯π+3 π₯π+32 π₯π+3
3 π₯π+34 π₯π+3
5 π₯π+36 π₯π+3
7 π₯π+38 π₯π+3
9 π₯π+310 π₯π+3
11 π₯π+312
1 π₯π+4 π₯π+42 π₯π+4
3 π₯π+44 π₯π+4
5 π₯π+46 π₯π+4
7 π₯π+48 π₯π+4
9 π₯π+410 π₯π+4
11 π₯π+412
1 π₯π+5 π₯π+52 π₯π+5
3 π₯π+54 π₯π+5
5 π₯π+56 π₯π+5
7 π₯π+58 π₯π+5
9 π₯π+510 π₯π+5
11 π₯π+512
1 π₯π+6 π₯π+62 π₯π+6
3 π₯π+64 π₯π+6
5 π₯π+66 π₯π+6
7 π₯π+68 π₯π+6
9 π₯π+610 π₯π+6
11 π₯π+612
1 π₯π+7 π₯π+72 π₯π+7
3 π₯π+74 π₯π+7
5 π₯π+76 π₯π+7
7 π₯π+78 π₯π+7
9 π₯π+710 π₯π+7
11 π₯π+712
0 1 2π₯π+7 3π₯π+72 4π₯π+7
3 5π₯π+74 6π₯π+7
5 7π₯π+76 8π₯π+7
7 9π₯π+78 10π₯π+7
9 11π₯π+710 12π₯π+7
11
0 1 2π₯π+
15
2
3π₯π+
15
2
2 4π₯π+
15
2
3 5π₯π+
15
2
3 6π₯π+
15
2
5 7π₯π+
15
2
6 8π₯π+
15
2
7 9π₯π+
15
2
8 10π₯π+
15
2
9 11π₯π+
15
2
10 12π₯π+
15
2
11
0 1 2π₯π+8 3π₯π+82 4π₯π+8
3 5π₯π+83 6π₯π+8
5 7π₯π+86 8π₯π+8
7 9π₯π+88 10π₯π+8
9 11π₯π+810 12π₯π+8
11
0 0 2 6π₯π+7 12π₯π+72 15π₯π+7
3 30π₯π+73 42π₯π+7
5 56π₯π+76 72π₯π+7
7 90π₯π+78 110π₯π+7
9 132π₯π+710
0 0 2 6π₯π+8 12π₯π+82 15π₯π+8
3 30π₯π+83 42π₯π+8
5 56π₯π+87 72π₯π+8
7 90π₯π+86 110π₯π+8
9 132π₯π+810
]
[ π0
π1
π2
π3
π4
π5
π6
π7
π8
π9
π10
π11
π12]
=
[
π¦π
π¦π+1
π¦π+2
π¦π+3
π¦π+4
π¦π+5
π¦π+6
π¦π+7
ππ+7
ππ+
15
2
ππ+8
ππ+7
ππ+8 ]
(12)
Using Maple software and inverting the matrix in (12) is yields the elements of the matrix π΄β1`.
The columns of π΄β1` give the continuous coefficients of (4)-(6) as:
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2360 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
πΌ0(π₯) = 1 β95958313819129π₯
29479449569430β+
111379502927091629π₯
24762737638321200β2
2
β14752082604856619623π₯
4160139923237961600β3
3
+5978817275676899711π₯
3328111938590369280β4
4
β 607983743988885997π₯
978856452526579200β5
5
+ 10064346687661109357π₯
66562238771807385600πβ6
6
β 8316257821028201π₯
316963041770511360β7
7
+ 10247925246994591π₯
3169630417705113600β8
8
β 1151437056841919π₯
4160139923237961600β9
9
+ 1042639590961571π₯
66562238771807385600β10
10
β17550288818501π₯11
33281119385903692800β11+
31325547497π₯12
3915425810106316800β12
πΌ1(π₯) =5999869653050π₯
421134993849ββ
162362039569331π₯
5053619926188β2
2
+19376892463308499π₯
606434391142560β3
3
β135118095989390719π₯
7277212693710720β4
4
+ 1002553554614321π₯
142690444974720β5
5
β 53054436242457737π₯
29108850774842880β6
6
+ 11911219801561π₯
35936852808448β7
7
β 82149948379561π₯
1940590051656192β8
8
+ 4520601070249π₯
1212868782285120β9
9
β 6280178983223 π₯
29108850774842880β10
10
+35892089683π₯11
4851475129140480β11
β195148837π₯12
1712285339696640β12
πΌ2(π₯) = β6875640670346π₯
140378331283β+
5682835861917011π₯
42113499384900β2
2
β77232977678555219π₯
5053619926188000β3
3
+5898372469497008023π₯
60643439114256000β4
4
β 140381714892468973π₯
3567261124368000β5
5
+ 2606002022753332429π₯
242573756457024000β6
6
β 3288459697500179π₯
1617158376380160β7
7
+ 289136258920171π₯
1078105584253440β8
8
β 122500746419197π₯
5053619926188000β9
9
+ 347843988763331π₯
242573756457024000β10
10
β6073810272629π₯11
121286878228512000β11
+11180057209π₯12
14269044497472000β12
πΌ3(π₯) =124156819047625π₯
842269987698ββ
8705963111552455π₯
20214479704752β2
2
+84000083720233513π₯
161715837638016β3
3
β75366803439211717π₯
215621116850688β4
4
+ 5641881259818595π₯
38050785326592β5
5
β 108983688887548537π₯
2587453402208256β6
6
+ 3555263960924639π₯
431242233701376β7
7
β 2894393499959887π₯
2587453402208256β8
8
+ 33503499322231π₯
324331675276032β9
9
β 5396554991149 π₯
862484467402752β10
10
+287842984985π₯11
1293726701104128β11
β538239565π₯12
152203141306368β12
πΌ4(π₯) = β324847870920125π₯
842269987698β+
23427907792376395π₯
20214479704752β2
2
β70097644067670879π₯
485147512914048β3
3
+5858213737831411585π₯
5821770154968576β4
4
β 50360748668263645π₯
114152355979776β5
5
+ 3008740914626408279π₯
23287080619874304β6
6
β 33633230900175019π₯
1293726701104128β7
7
+ 28069177569117659π₯
7762360206624768β8
8
β 166103439749567π₯
485147512914048β9
9
+ 491256957909977π₯
23287080619874304β10
10
β2963717454581π₯11
3881180103312384β11
+16893432587π₯12
1369828271757312β12
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2361 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
πΌ5(π₯) =698233548697762π₯
701891656415ββ
127982190512855483π₯
42113499348900β2
2
+3907504681261828703π₯
1010723985237600β3
3
β6679512172825498303π₯
2425737564570240β4
4
+ 881694260693648351π₯
713452224873600β5
5
β 17972507404872403181π₯
48514751291404800β6
6
+ 123294391180745231π₯
1617158376380160β7
7
β 58404991154460107π₯
5390527921267200β8
8
+ 2115964283425693π₯
2021447970475200β9
9
β 3188461778077843 π₯
48514751291404800β10
10
+58725123939133π₯11
24257375645702400β11β
π₯
2853808899494400β12
πΌ6(π₯) = β1591448100803050π₯
421134993849β+
58973616807690011π₯
5053619926188β2
2
β9125523287132968219π₯
606434391142560β3
3
+26407945574782487893π₯
2425737564570240β4
4
β 709173203422711231π₯
142690444974720β5
5
+ 14717766075275137199π₯
9702950258280960β6
6
β 102837002696707733π₯
323431675276032β7
7
+ 3307845701319711π₯
71873705616896β8
8
β 2745685566667217π₯
606434391142560β9
9
+ 280733838723136π₯
9702950258280960β10
10
β52594481758343π₯11
4851475129140480β11+
103239426419π₯12
570761779898880β12
πΌ7(π₯) =18043919361064657π₯
5895889913886ββ
234408889565975928593π₯
24762737638321200β2
2
+254426896842248963872279π₯
20800699616189808000β3
3
β2214071776927267887465043π₯
249608395394277696000β4
4
+ 2207889379530816131509π₯
543809140292544000β5
5
β 1240587114881513356098389π₯
998433581577110784000β6
6
+ 82789381930492078319π₯
316963041770511360β7
7
β 72105563826067360439π₯
1901778250623068160β8
8
+ 155606351177055440329π₯
41601399232379616000β9
9
β 239203987064516185771 π₯
998433581577110784000β10
10
+499457292157396621π₯11
55468532309839488000β11β
8845925643617369π₯12
58731387151594752000β12
π½7(π₯) = β30797607491730π₯
140378331283+
12252449651037159π₯
19652966379620β
2
β3924857667428473659π₯
5502830586293600β2
3
+28805166480229703303π₯
66033967035523200β3
4
β 1824665680566703409π₯
11653053006268800β4
5
+ 26167954478105592707π₯
792407604426278400β5
6
β 2623989574274443π₯
754673908977408β6
7
+ 64286272312943π₯
1509347817954816β7
8
β 2052021323393473π₯
33016983517761600β8
9
+ 697442591071403π₯
88045289380697600β9
10
β179476384395293π₯11
396203802213139200β10+
479316757153π₯12
46612212025075200β11
π½152
(π₯) = β428190539055104π₯
140378331283+
6685811626819584π₯
701891656415β
2
β78708148640651264π₯
6317024907735β2
3
+11583149024190976π₯
1263404981547β3
4
β 1585503861215744π₯
371589700455β4
5
+ 8402119605809536π₯
6317024907735β5
6
β 120118667889920π₯
421134993849β6
7
+ 89048602174336π₯
2105674969245β7
8
β 26946704038912π₯
6317024907735β8
9
+ 1766885496448π₯
63170249907735β9
10
β67963410176π₯11
6317024907735β10
+68496256π₯12
371589700455β11
π½8(π₯) =102709446821625π₯
140378331283β
1285663550076535π₯
561513325132β
2
+40470004012999603π₯
13476319803168β2
3
β358511358722606365π₯
1617158337638016β3
4
+ 9851655119649403π₯
9512696331648β4
5
β 209718963335245531π₯
646863350552064β5
6
+ 7530890237766749π₯
107810558425344β6
7
β 748235136449605π₯
71873705616896β7
8
+ 14230248199867π₯
13476319803168β8
9
β 45054856385429π₯
646863350552064β9
10
+872012211011π₯11
323431675276032β10
β1769443135π₯12
38050785326592β11
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2362 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
πΎ7(π₯) =252474550097700π₯β
140378331283β
1570774572740333 π₯
280756662566
2
+1717354357900665499π₯
235835596555440β
3
β15074283296663869543π₯
2830027158665280β2
4
+ 45540634791210947π₯
18496909533760β3
5
β 8622045232427895329π₯
11320108634661120β4
6
+ 2910438299459199π₯
17968426404224β5
7
β 2565992829113863π₯
107810558425344β6
8
+ 1121624842813669π₯
471671193110880β7
9
β 1747468970243231π₯
11320108634661120β8
10
+3695515319961π₯11
628894924147840β9β
66323138549π₯12
665888743215360β10
πΎ8(π₯) = β16888540189500π₯β
140378331283+
52898108169060 π₯
140378331283
2
β1111375015828707π₯
2246053300528β
3
β3286416400864111π₯
8984213202112β2
4
+ 904578847981169π₯
528483129536β3
5
β 1929273942531457π₯
35936852808448β4
6
+ 208278204400469π₯
17968426404224β5
7
β 62226870194839π₯
35936852808448β6
8
+ 395507042623π₯
2246053300528β7
9
β 418590251343π₯
35936852808448β8
10
+8126500505π₯11
17968426404224β9β
16544781π₯12
2113932518144β10
Evaluating (8) at the following points at
π₯π , π₯π+1, π₯π+2, π₯π+3, π₯π+4, π₯π+5, π₯π+6, π₯π+15/3 πππ π₯π+8 yields the following discrete methods which constitute the new eight step block method.
π¦π β795573993889721900
111379502927091629π¦π+1 +
3341507486807202468
111379502927091629π¦π+2 β
10664804811651757375
111379502927091629π¦π+3 +
28699187045661083875
111379502927091629π¦π+4 β
75253528021559024004
111379502927091629π¦π+5
+288970722357681053900
111379502927091629π¦π+6 β
234408889565975928593
111379502927091629π¦π+7
=1260β
111379502927091629[12252449651037159ππ+7 + 187202725550948352π
π+152
β 44998224252678725ππ+8] +88200β2
111379502927091629[140378331283ππ
+ 1570774572740333ππ+7 β 105796216338120ππ+8]
π¦π +25231414254947640
4965391873495939π¦π+1 β
918591857070157676
24826959367479695π¦π+2 +
585503967350684025
4965391873495939π¦π+3 β
4482510795788159225
14896175620487817π¦π+4 +
3773303822956282636
4965391873495939π¦π+5
β14112172589850215860
4965391873495939π¦π+6 +
171015849451373356153
74480878102439085π¦π+7
=28β
4965391873495939[32832731120396717ππ+7 + 399847878563266560π
π+152
β 95591127585482025ππ+8] +1960β2
496539187349 5939[5615133251320ππ+1
β 3388606915725319ππ+7 + 224219376062145ππ+8]
π¦π β7222751481023540
206294902600341π¦π+1 +
8110158627648812
5157372565008525π¦π+2 +
19088416297961325
68764967533447π¦π+3 β
172698241297211525
206294902600341π¦π+4 +
423413680921142884
206294902600341π¦π+5
β512403409953341620
68764967533447π¦π+6 +
30898039789388169313
5157372565008525π¦π+7
=28β
343824837667235[7185554611961157ππ+7 + 70180046374502400π
π+152
β 16669945678353625ππ+8] β392β2
68764967533447[7861186551848ππ+2
+ 601842327156471ππ+7 β 38993163236825ππ+8]
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2363 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
π¦π β2070320941916780
88165482138851π¦π+1 +
180752455135885332
440827410694255π¦π+2 β
41538978652452925
88165482138851π¦π+3 β
85609840632881075
88165482138851π¦π+4 +
333063784390344924
88165482138851π¦π+5
β1210325056048167220
88165482138851π¦π+6 +
4851208776878785793
440827410694255π¦π+7
=252β
88165482138851[440062565156559ππ+7 + 3569446989660160π
π+152
β 841909412290625ππ+8] +17640β2
88165482138851[1965296637962ππ+3
β 30992386067543ππ+7 + 1963324978713ππ+8]
π¦π β535551344432860
26631670316133π¦π+1 +
10225338751074412
44386117193555π¦π+2 β
827110478267325
306111153059π¦π+3 +
1210127800927725
306111153059π¦π+4 +
21449184720713764
8877223438711π¦π+5 β
190583150657241580
8877223438711 π¦π+6
+2342265532264503639
133158351580665π¦π+7
=28β
8877223438711[667521725849171ππ+7 + 5151530052157440π
π+152
β 1207191039323325ππ+8] β1960β2
8877223438711[9826483189810ππ+4
+ 45078599072657ππ+7 β 2806256440575ππ+8]
π¦π β28732889416700
1558701718659π¦π+1 +
14647518007276
82036932561π¦π+2 β
231791360980375
173189079851π¦π+3 +
18773267530814125
1558701718659π¦π+4 β
3928035699532
219938157π¦π+5 β
3126320160999700
173189079851 π¦π+6
+39036596515149631
1558701718659π¦π+7
=140β
519567239553[7012323610059ππ+7 + 98186615783424π
π+152
β 22800086843975ππ+8] +9800β2
519567239553[462422738344ππ+5 β 855181579233ππ+7
+ 52722651695ππ+8]
π¦π β393848612015180
22694743922483π¦π+1 +
17002139804752068
113473719612415π¦π+2 β
21584615528729575
22694743922483π¦π+3 +
120848531925713725
22694743922483π¦π+4 β
931785054474996708
22694743922483π¦π+5
+669436923774516380
22694743922483π¦π+6 +
799974701053192307
113473719612415π¦π+7
β252β
22694743922483[415729917509259ππ+7 β 5120456609300480π
π+152
+ 1122075341897825ππ+8] β17640β2
22694743922483[39305932759240ππ+6
+ 47323702151687ππ+7 β 256515380385ππ+8]
π¦π+
152
+1385420343021
8243051549790568448π¦π β
816784820775
294394698206806016π¦π+1 +
6675149415753
294394698206806016π¦π+2 β
148545520998875
1177578792827224064π¦π+3 +
675021185821125
1177578792827224064π¦π+4
β779956749745173
294394698206806016π¦π+5 +
6268503501535275
294394698206806016π¦π+6 β
8400581608123449459
8243051549790568448π¦π+7
=6435β
294394698206806016[14072591027781ππ+7 + 8522418946048π
π+152
β 512303316525ππ+8] +289864575β2
147197349103403008[24728719ππ+7 + 768915ππ+8]
π¦π+8 β40335
140378331283π¦π +
654800
140378331283π¦π+1 β
5230064
140378331283π¦π+2 +
281410092
140378331283π¦π+3 β
121250500
140378331283π¦π+4 +
507886960
140378331283π¦π+5 β
3191134800
140378331283π¦π+6 β
137597358436
140378331283π¦π+7 =
2520β
140378331283[12281666ππ+7 + 32112640π
π+15
2
+ 12283227ππ+8] β352800β2
140378331283[491ππ+7 + 5692ππ+8] (13)
3. CONVERGENCE AND STABILITY ANALYSIS OF THE METHOD
In this section the analysis of the newly constructed methods is carried by analyzing the order, error constant consistency, convergence and plotting the regions of absolute stability.
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2364 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
The new method in equation (3) is expressed in the form:
β(πΌππ¦π+π β βπ½πππ+π β β2ππππ+π) = 0
π
π=0
( 14)
Which is written symbolically as:
π(πΈ)π¦π β β2π(πΈ) = 0 (15)
πΈ being the shift operator and is defined as πΈππ¦π = π¦π+π , π(πΈ)and π(πΈ) are respectively the first and second characteristics polynomial of the LMM such that
π(πΈ) = β πΌππΈππ
π=0 , π (πΈ) = β π½ππΈππ
π=0 (16)
Following (Fatunla, 1991) and (Lambert,1973) we defined the local truncation error associated with (3) to be linear difference operator
[π¦(π₯); β] = β πΌππ¦π+π β βπ½πππ+πππ=0 β β2πΎπππ+π . (17)
Assuming that π¦(π₯) is sufficiently differentiable, we can we expand the terms in (17) as a Taylor series and comparing the coefficients of β gives
πΏ[π¦(π₯); β] = π0π¦(π₯) + π1βπ¦β²(π₯) + π2β2π¦β²β²(π₯) + β― + ππβππ¦π(π₯) + β― (18)
Where the constants πΆπ, π = 0, 1,2β¦ . , π = 1,2, β¦ π.
Using the concept above, the hybrid block method are obtained with the help of MAPLE 18 SOFTWARE have the following uniform order and error constants.
3.1 Table 1 Order and Error Constants of the Eight -Step
π¦π 11 11097820685016920015
764508908091556941456
π¦π+1 11 β
1412674980352712279
102247349459028375888
π¦π+2 11 β
14601896648447213
472002737149580208
π¦π+3 11 β
29213909016714541
605167869401073264
π¦π+4 11 β
12085140283692917
182799785049936912
π¦π+5 11 β
922206057457405
10698928596875376
π¦π+6 11 β
1575458937383051
14161520207629392
π¦π+
152
11 β
463663785585
18841260685235585024
π¦π+8 11 11276615
240889216481628
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2365 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
3.2 Zero Stability of the Eight -Step Method
The zero stability of the eight-step new method is determined using the approach of Ehigie et al. (2014) in which Maple18 software was used to yields the stability polynomial of the method as
π(π) = det (ππ΄ β π΅) (19)
=det
[
π
[ 795573993889721900
111379502927091629
3341507486807202468
111379502927091629
10664804811651757375
111379502927091629
282699187045661083875
111379502927091629
β75253528021559024004
111379502927091629
288970722357681053900
111379502927091629
2344088895659759289593
1113795029270916290 0
25231414254947460
4965391873495939
β918591857070157676
24826959367479695
585503967350684025
4965391873495939
β4482510795788159225
14896175620487817
3773303822956282636
4965391873495939
β14112172589850215860
4965391873495939
171015849451373356153
744808781024390850 0
β7222751481023540
206294902600341
8110158627648812
5157372565008525
19088416297961325
68764967533447
β172698241297211525
206294902600341
423413680921142884
2062949π΄02600341
β512403409953341620
68764967533447
30898039789388169313
51573725650085250 0
β2070320941916780
88165482138851
180752455135885332
440827410694255
β41538978652452925
88165482138851
85609840632881075
88165482138851
333063784390344924
88165482138851
β1210325056048167220
88165482138851
4851208776878785793
4408274106942550 0
535551344432860
26631670316133
10225338751074412
44386117193555
827110478267325
306111153059
1210127809027725
306111153059
21449184720713764
8877223438711
β190583150657241580
306111153059
2342265532284503639
1331583515806650 0
β28732889416700
1558701718659
14647518007276
82036932561
β231791360980375
173189079851
18773267530814125
1558701718659
3928035699532
219938157
β3126320160999700
173189079851
39036596515149631
15587017186590 0
β393848612015180
22694743922483
17302139804752068
113473719612415
β21584615528729575
22694743922483
120848531925713725
22694743922483
β931785054474996708
22694743922483
669436923774516380
22694743922483
799974701053192307
1134737196124150 0
816784820775
294394698206806016
6675149415753
294394698206806016
β148545520998875
1177578792827224064
675021185821125
1177578792827224064
β779956749745173
294394698206806016
6268503501535275
294394698206806016
8400581608123449459
82430515497905684481 0
654800
140378331283
β5230064
140378331283
28141092
140378331283
β121250500
140378331283
507886960
140378331283
β3191134800
140378331283
β137597358436
1403783312830 1]
β
[ 0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0 β1
0 0 0 0 0 0 0 0β1385420343021
8243051549790568448
0 0 0 0 0 0 0 040335
140378331283 ]
]
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2366 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
= πππ‘
[ 795573993889721900
111379502927091629π
3341507486807202468
111379502927091629π
10664804811651757375
111379502927091629π
282699187045661083875
111379502927091629π
β75253528021559024004
111379502927091629π
288970722357681053900
111379502927091629π
2344088895659759289593
111379502927091629π 0 1
25231414254947460
4965391873495939π
β918591857070157676
24826959367479695π
585503967350684025
4965391873495939π
β4482510795788159225
14896175620487817π
3773303822956282636
4965391873495939π
β14112172589850215860
4965391873495939π
171015849451373356153
74480878102439085π 0 1
β7222751481023540
206294902600341π
8110158627648812
5157372565008525π
19088416297961325
68764967533447π
β172698241297211525
206294902600341π
423413680921142884
206294902600341π
β512403409953341620
68764967533447π
30898039789388169313
5157372565008525π 0 1
β2070320941916780
88165482138851π
180752455135885332
440827410694255π
β41538978652452925
88165482138851π
85609840632881075
88165482138851π
333063784390344924
88165482138851π
β1210325056048167220
88165482138851π
4851208776878785793
440827410694255π 0 1
535551344432860
26631670316133π
10225338751074412
44386117193555π
827110478267325
306111153059π
1210127809027725
306111153059π
21449184720713764
8877223438711π
β190583150657241580
306111153059π
2342265532284503639
133158351580665π 0 1
β28732889416700
1558701718659π
14647518007276
82036932561π
β231791360980375
173189079851π
18773267530814125
1558701718659π
3928035699532
219938157π
β3126320160999700
173189079851π
39036596515149631
1558701718659π 0 1
β393848612015180
22694743922483π
17302139804752068
113473719612415π
β21584615528729575
22694743922483π
120848531925713725
22694743922483π
β931785054474996708
22694743922483π
669436923774516380
22694743922483π
799974701053192307
113473719612415π 0 1
816784820775
294394698206806016π
6675149415753
294394698206806016π
β148545520998875
1177578792827224064π
675021185821125
1177578792827224064π
β779956749745173
294394698206806016π
6268503501535275
294394698206806016π
8400581608123449459
8243051549790568448π π
β1385420343021
8243051549790568448654800
140378331283π
β5230064
140378331283π
28141092
140378331283π
β121250500
140378331283π
507886960
140378331283π
β3191134800
140378331283π
β137597358436
140378331283π 0 π β
40335
140378331283 ]
= π9 β π8 = 0
π1 = 1, π2 = π3 = π4 = π5 = π6 = π7 = π8 = π9 = 0 < 1
Hence, the block method (13) is zero stable and is of order π = 11 and hence by Henrici (1962) it is convergent.
3.3 Absolute Stability Region of the Eight-Step Method
The coefficient of the block method (13) expressed in the form
π΄(1)ππ€+1 = π΅(0)ππ€ + βπΆ(1)πΉπ€ + β2π·(1)πΊπ€ (20)
Where
π¦π€+1 = (π¦π+1,π¦π+2,π¦π+3, β¦ . . π¦πβπβ1,, π¦π+π)π
π¦π€ = (π¦πβπ+1,π¦πβπ+2,π¦πβπ+3, β¦ . . π¦πβ1,, π¦π)π
πΉπ€ = (ππ+1,ππ+2,π¦π+3, β¦ . . ππβπβ1,, ππ+π)π
For π€ = 0,β¦. and π = 0, π,β¦π β π.
Applying the test problem
π¦β² = ππ¦, π < 0 to yield
ππ€+1 = π·(π§)ππ€, π§ = πβ
Where the matrix π·(π§) = (π(π΄ β πΆπ§ β π·1π§2) β π΅) (21)
The matrix π·(π§) has eigenvalues {π1, π2, π3, β¦β¦ β¦ . . , ππ} = {0,0,0, β¦β¦ β¦ . . ππ}
Where the dominant eigenvalue ππ is the stability function π (π§): β β β which is the rational function with real coefficients.
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2367 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
Figure 1. Region of absolute stability of newly constructed method for k=8
4. NUMERICAL EXPERIMENT
The Solution curves of stiff system plotted were compared with those of MATLAB ODE solver
Ode 23.
Problem 1
π¦1β² = 0.01π¦1 β π¦2 + π¦3
π¦2β² = π¦1 β 100.005π¦2 + 99.995π¦3
π¦3β² = 2π¦1 + 99.995π¦2 β 100.005π¦3
Exact π¦1(π₯) = πβ0.01π₯(πππ 2π₯ + π ππ2π₯) π¦1(0) = 1
π¦2(π₯) = πβ0.01π₯(πππ 2π₯ + π ππ2π₯) +πβ200π₯ π¦2(0) = 1
π¦2(π₯) = πβ0.01π₯(πππ 2π₯ + π ππ2π₯) -πβ200π₯ π¦3(0) = 1
π€ππ‘β β = 0.1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Re(z)
Im
(z)
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2368 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
Figure 2: Solution Curve of Problem 1 Solved with eight-step
Problem 2
Robertson's Equations
π¦1β² = β0.04π¦1 + 10000π¦2π¦3 π¦1(0) = 1
π¦2β² = 0.04π¦1 β 10000π¦2π¦3 β 3000 0000π¦2
2 π¦2(0) = 0
π¦3β² = 3000 0000π¦2
2 π¦3(0) = 0
0 β€ π₯ β€ 70 and β = 0.1
βUniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equationsβ
2369 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
Figure 3: Solution Curve of Problem 2 Solved with eight-step
5. CONCLUSION
In this research, the uniform order eleven of eight-step hybrid block backward differentiation formulae for the solution of stiff ordinary differential equations was proposed and the scheme with
one off-step point were analyzed and to be consistent and zero-stable and convergence. The newly constructed was tested on both linear and nonlinear stiff systems and the results obtained
performed favorably compared to exact and ODE23 Solver.
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2370 Pius Tumba1, IJMCR Volume 09 Issue 08 August 2021
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