The Chebyshev Collection Method for Solving Fractional OrderKlein-Gordon Equation

M. M. KHADERFaculty of Science, Benha University

Department of MathematicsBenha

EGYPTmohamedmbd@yahoo.com

N. H. SWETLAMFaculty of Science, Cairo University

Department of MathematicsGiza

EGYPTn−sweilam@yahoo.com

A. M. S. MAHDYFaculty of Science, Zagazig University

Department of MathematicsZagazigEGYPT

amr−mahdy85@yahoo.com

Abstract: In this paper, we are implemented the Chebyshev spectral method for solving the non-linear fractionalKlein-Gordon equation (FKGE). The fractional derivative is considered in the Caputo sense. We presented anapproximate formula of the fractional derivative. The properties of the Chebyshev polynomials are used to re-duce FKGE to the solution of system of ordinary differential equations which solved by using the finite differencemethod. Special attention is given to study the convergence analysis and estimate an upper bound of the error ofthe derived formula. The numerical results of applying this method to FKGE show the simplicity and the efficiencyof the proposed method.

Key–Words: Fractional Klein-Gordon equation; Caputo derivative; Chebyshev spectral method; Convergence anal-ysis.

1 Introduction

Fractional differential equations (FDEs) have recentlybeen applied in various areas of engineering, sci-ence, finance, applied mathematics, bio-engineeringand others. However, many researchers remain un-aware of this field. FDEs have been the focus of manystudies due to their frequent appearance in various ap-plications in fluid mechanics, viscoelasticity, biology,physics and engineering [1]. Consequently, consider-able attention has been given to the solutions of frac-tional differential equations of physical interest. MostFDEs do not have exact solutions, so approximateand numerical techniques ([2], [5], [25]-[28], [32]),must be used. Recently, several numerical methods tosolve FDEs have been given such as, variational itera-tion method [7], homotopy perturbation method ([4],[5]), Adomian decomposition method ([8], [24]), ho-motopy analysis method [6], collocation method ([9]-[17], [29]-[31]).

The Klein-Gordon equation plays a significantrole in mathematical physics and many scientific ap-plications, such as solid-state physics, nonlinear op-tics, and quantum field theory [33]. The equation hasattracted much attention in studying solitons ([20],[21]) and condensed matter physics, in investigatingthe interaction of solitons in a collisionless plasma, the

recurrence of initial states, and in examining the non-linear wave equations [3]. Wazwaz has obtained thevarious exact traveling wave solutions, such as com-pactons, solitons and periodic solutions by using thetanh method [33]. The study of numerical solutions ofthe Klein-Gordon equation has been investigated con-siderably in the last few years. In the previous studies,the most papers have carried out different spatial dis-cretization of the equation ([3], [33], [34]).

Representation of a function in terms of a se-ries expansion using orthogonal polynomials is a fun-damental concept in approximation theory and formthe basis of the solution of differential equations([9], [23]). Chebyshev polynomials are widely usedin numerical computation. One of the advantages ofusing Chebyshev polynomials as a tool for expansionfunctions is the good representation of smooth func-tions by finite Chebyshev expansion provided that thefunction x(t) is infinitely differentiable. The coeffi-cients in Chebyshev expansion approach zero fasterthan any inverse power in n as n goes to infinity.

We describe some necessary definitions andmathematical preliminaries of the fractional calculusfor our subsequent development.

Definition 1 The Caputo fractional derivative opera-tor Dα of order α is defined in the following form

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E-ISSN: 2224-2880 31 Volume 13, 2014

Dαf(x) =1

Γ(m− α)

∫ x

0

f (m)(t)

(x− t)α−m+1dt,

where α > 0, and m− 1 < α ≤ m, m ∈ N, x > 0.

Similar to integer-order differentiation, Caputofractional derivative operator is a linear operation

Dα (λ f(x) + µ g(x)) = λDα f(x) + µDα g(x),

where λ and µ are constants. For the Caputo’s deriva-tive we have [19]

DαC = 0, C is a constant, (1)

Dα xn =

0, for n ∈ N0 and n < ⌈α⌉Γ(n+1)

Γ(n+1−α)xn−α, for n ∈ N0and n ≥ ⌈α⌉.

(2)We use the ceiling function ⌈α⌉ to denote the small-est integer greater than or equal to α and N0 ={0, 1, · · · }. Recall that for α ∈ N, the Caputo dif-ferential operator coincides with the usual differentialoperator of integer order.

For more details on fractional derivatives defini-tions and its properties see ([18], [19]).

The main aim of the presented paper is con-cerned with the application of the Chebyshev collo-cation method to obtain the numerical solution of thenonlinear fractional Klein-Gordon equation and studythe convergence analysis of the proposed method.Khader [9] introduced a new approximate formula ofthe fractional derivative and used it to solve numeri-cally the fractional diffusion equation. Also, Khaderand Hendy in [13] used this formula to solve nu-merically the fractional delay differential equations.Chebyshev polynomials are well known family of or-thogonal polynomials on the interval [−1, 1] that havemany applications ([13], [23]). They are widely usedbecause of their good properties in the approximationof functions.

In this paper, we apply Chebyshev collocationmethod to obtain the numerical solution of the non-linear fractional Klein-Gordon equation of the form

∂2u(x, t)

∂t2+ aDαu(x, t) + b u(x, t) + c uγ(x, t)

= f(x, t), x ∈ (0, L), t > 0, α ∈ (1, 2], (3)

where Dα denotes the Caputo fractional derivative oforder α with respect to x, u(x, t) is unknown function,

and a, b, c and γ are known constants with γ ∈ R,γ = ±1.

We also assume the following initial conditions

u(x, 0) = g1(x), ut(x, 0) = g2(x), x ∈ (0, L) (4)

and the following boundary conditions

u(0, t) = u(L, t) = 0.

The organization of this paper is as follows. Inthe next section, derivation an approximate formulafor fractional derivatives using Chebyshev series ex-pansion. In section 3, the error analysis of the intro-duced approximate formula is given. In section 4, theprocedure of the solution using Chebyshev collocationmethod is given. The conclusion is given in section 5.

2 Derivation an approximate for-mula for fractional derivatives us-ing Chebyshev series expansion

The well known Chebyshev polynomials [23] are de-fined on the interval [−1, 1] and can be determinedwith the aid of the following recurrence formula

Tn+1(z) = 2zTn(z)− Tn−1(z),

T0(z) = 1,

T1(z) = z, n = 1, 2, · · ·

The analytic form of the Chebyshev polynomialsTn(z) of degree n is given by

Tn(z) = n

[n2]∑

i=0

(−1)i 2n−2 i−1 (n− i− 1)!

(i)! (n− 2 i)!zn−2 i

(5)where [n/2] denotes the integer part of n/2. The or-thogonality condition is

∫ 1

−1

Ti(z)Tj(z)√1− z2

dz =

π, for i = j = 0;

π2 , for i = j = 0;

0, for i = j.

In order to use these polynomials on the interval [0, L]we define the so called shifted Chebyshev polynomi-als by introducing the change of variable z = 2

L t −1. The shifted Chebyshev polynomials are defined as

T ∗n(t) = Tn(

2

Lt− 1) = T2n(

√t/L).

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The analytic form of the shifted Chebyshev poly-nomials T ∗

n(t) of degree n is given by

T ∗n(t) = n

n∑k=0

(−1)n−k 22k (n+ k − 1)!

Lk(2k)! (n− k)!tk (6)

for n = 2, 3, . . ..The function x(t), which belongs to the space

of square integrable functions in [0, L], may be ex-pressed in terms of shifted Chebyshev polynomials as

x(t) =

∞∑i=0

ci T∗i (t), (7)

where the coefficients ci are given by

c0 =1

π

∫ L

0

x(t)T ∗0 (t)√

Lt− t2dt,

ci =2

π

∫ L

0

x(t)T ∗i (t)√

Lt− t2dt, i = 1, 2, · · · (8)

In practice, only the first (m+1)-terms of shiftedChebyshev polynomials are considered. Then wehave

xm(t) =

m∑i=0

ci T∗i (t). (9)

The main approximate formula of the fractionalderivative of xm(t) is given in the following theorem.

Theorem 2 Let x(t) be approximated by Chebyshevpolynomials as (9) and also suppose α > 0, then

Dα(xm(t)) =m∑

i=⌈α⌉

i∑k=⌈α⌉

ciw(α)i, k tk−α, (10)

where w(α)i, k is given by

w(α)i, k = (−1)i−k 22k i (i+ k − 1)!Γ(k + 1)

Lk(i− k)! (2 k)! Γ(k+1−α).

(11)

Proof. Since the Caputo’s fractional differentiation isa linear operation we have

Dα(xm(t)) =

m∑i=0

ciDα(T ∗

i (t)). (12)

Employing Eqs.(1) and (2) we have

Dα T ∗i (t) = 0, i = 0, 1, · · · , ⌈α⌉ − 1, α > 0.

(13)

Also, for i = ⌈α⌉, ⌈α⌉+1, ...,m, and by using Eqs.(1)and (2), we get

Dα T ∗i (t) = i

i∑k=⌈α⌉

(−1)i−k 22k(i+ k − 1)!

Lk(i− k)! (2 k)!Dαtk

= i

i∑k=⌈α⌉

(−1)i−k 22k(i+ k − 1)!Γ(k + 1)

Lk(i− k)! (2 k)! Γ(k+1−α)tk−α.

(14)

A combination of Eqs.(13), (14) and (11) leads to thedesired result and then completes the proof of the the-orem. ⊓⊔

3 Error analysis

In this section, special attention is given to study theconvergence analysis and evaluate an upper bound ofthe error of the proposed formula.

Theorem 3 (Chebyshev truncation theorem) [23]The error in approximating x(t) by the sum of its firstm terms is bounded by the sum of the absolute valuesof all the neglected coefficients. If

xm(t) =

m∑k=0

ck Tk(t), (15)

then

ET (m) ≡ |x(t)− xm(t)| ≤∞∑

k=m+1

|ck|, (16)

for all x(t), all m, and all t ∈ [−1, 1].

Theorem 4 The Caputo fractional derivative of orderα for the shifted Chebyshev polynomials can be ex-pressed in terms of the shifted Chebyshev polynomialsthemselves in the following form

Dα(T ∗i (t)) =

i∑k=⌈α⌉

k−⌈α⌉∑j=0

Θi,j,k T∗j (t) (17)

where

Θi,j,k =

(−1)i−k 2i(i+k−1)! Γ(k−α+ 12)

hjΓ(k+12) (i−k)! Γ(k−α−j+1)Γ(k+j−α+1)Lk

j = 0, 1, · · · .

The proof please see [2].

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Theorem 5 The error

|ET (m)| = |Dαx(t)−Dαxm(t)|

in approximating Dαx(t) by Dαxm(t) is bounded by

|ET (m)| ≤∣∣∣ ∞∑i=m+1

ci

( i∑k=⌈α⌉

k−⌈α⌉∑j=0

Θi,j,k

)∣∣∣. (18)

Proof. A combination of Eqs. (7), (9) and (17) leadsto

|ET (m)| =∣∣∣Dαx(t)−Dαxm(t)

∣∣∣=

∣∣∣ ∞∑i=m+1

ci

( i∑k=⌈α⌉

k−⌈α⌉∑j=0

Θi,j,kT∗j (t)

)∣∣∣,but |T ∗

j (t)| ≤ 1, so, we can obtain

|ET (m)| ≤

∣∣∣∣∣∣∞∑

i=m+1

ci

( i∑k=⌈α⌉

k−⌈α⌉∑j=0

Θi,j,k

)∣∣∣∣∣∣ ,and subtracting the truncated series from the infiniteseries, bounding each term in the difference, summingthe bounds and hence completes the proof of the the-orem. ⊓⊔

4 Numerical implementation

In this section, we solve numerically the nonlinearfractional-order Klein-Gordon equation using the ap-proach Chebyshev spectral method. Some numeri-cal examples are presented to validate the solutionscheme. Symbolic computations are carried out usingMatlab 8. To achieve this propose we will considerthe following two examples ([4], [5]).

Example 6 Consider the fractional-order nonlinearKlein-Gordon problem

∂2u(x,t)∂t2

−Dαu(x, t) + u3(x, t) = f(x, t),

x ∈ (0, 1), t > 0, α ∈ (1, 2],(19)

with the following initial conditions

u(x, 0) = 0, ut(x, 0) = 0, (20)

and the following boundary conditions

u(0, t) = 0, u(1, t) = 0, (21)

where the source term f(x, t) is given by

f(x, t) = 2xα − Γ(α+ 1) t2 + x3α t6.

The exact solution of this problem is u(x, t) = xα t2.

In order to use the Chebyshev collocation method,we approximate u(x, t) with m = 3 as

u3(x, t) =

3∑i=0

ui(t)T∗i (x). (22)

From Eq.(19) and Theorem 2 we have

3∑i=0

ui(t)T∗i (x)−

3∑i=⌈α⌉

i∑k=⌈α⌉

ui(t)w(α)i,k x

k−α

+

(3∑

i=0

ui(t)T∗i (x)

)3

= f(x, t). (23)

We now collocate Eq.(23) at (m+1−⌈α⌉) points xp,p = 0, 1, · · · ,m− ⌈α⌉ as

3∑i=0

ui(t)T∗i (xp)−

3∑i=⌈α⌉

i∑k=⌈α⌉

ui(t)w(α)i,k x

k−αp

+

(3∑

i=0

ui(t)T∗i (xp)

)3

= f(xp, t). (24)

For suitable collocation points we use roots of shiftedChebyshev polynomial T ∗

m+1−⌈α⌉(x).In this case, the roots xp of shifted Chebyshev

polynomial T ∗2 (x), i.e.

x0 = 0.146447, x1 = 0.8872983.

Also, by substituting Eq.(22) in the boundary condi-tions (21) we can find

3∑i=0

(−1)i ui(t) = 0,

3∑i=0

ui(t) = 0. (25)

By using Eqs.(24) and (25) we obtain the followingnon-linear system of ODEs

u0(t) + k1 u1(t) + k2 u3(t)

=

3∑i=⌈α⌉

i∑k=⌈α⌉

ui(t)w(α)i,k x

k−α0

−

(3∑

i=0

ui(t)T∗i (x0)

)3

+ f0(t), (26)

u0(t) + k11 u1(t) + k22 u3(t)

=3∑

i=⌈α⌉

i∑k=⌈α⌉

ui(t)w(α)i,k x

k−α1

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E-ISSN: 2224-2880 34 Volume 13, 2014

−

(3∑

i=0

ui(t)T∗i (x1)

)3

+ f1(t), (27)

u0(t)− u1(t) + u2(t)− u3(t) = 0, (28)

u0(t) + u1(t) + u2(t) + u3(t) = 0, (29)

where

k1 = T ∗1 (x0), k2 = T ∗

3 (x0),

k11 = T ∗1 (x1), k22 = T ∗

3 (x1).

Now, to use finite difference method [22] for solvingthe system (26)-(29), we use the notations tn = n∆tto be the integration time 0 ≤ tn ≤ T,∆t = T/N, forn = 0, 1, ..., N. Define uni = ui(tn), fn

i = fi(tn),i = 1, 2. Then the system (26)-(29), is discretized andtakes the following form

un+10 − 2un0 + un−1

0

∆t2+ k1

un+11 − 2un1 + un−1

1

∆t2

+k2un+13 − 2un3 + un−1

3

∆t2

=3∑

i=⌈α⌉

i∑k=⌈α⌉

un+1i w

(α)i,k x

k−α0

−

(3∑

i=0

un+1i T ∗

i (x0)

)3

+ fn+10 , (30)

un+10 − 2un0 + un−1

0

∆t2+ k11

un+11 − 2un1 + un−1

1

∆t2

+k22un+13 − 2un3 + un−1

3

∆t2

=

3∑i=⌈α⌉

i∑k=⌈α⌉

un+1i w

(α)i,k x

k−α1

−

(3∑

i=0

un+1i T ∗

i (x1)

)3

+ fn+11 , (31)

un+10 − un+1

1 + un+12 − un+1

3 = 0, (32)

un+10 + un+1

1 + un+12 + un+1

3 = 0. (33)

This system presents the numerical scheme of the pro-posed problem (19) using the fractional finite differ-ence method. Solving this system using the Newtoniteration method yields the numerical solution of thenon-linear fractional Klein-Gordon equation (19).

At n = 1, we will evaluate the values of u0 =(u00, u

01, u

02, u

03) and u1 = (u10, u

11, u

12, u

13) using the

initial conditions (20). Therefore, we can obtain thesolutions

un = (un0 , un1 , u

n2 , u

n3 ), n = 2, 3, ..., N

using the numerical scheme (30)-(33).The obtained numerical results by means of the

proposed method are shown in table 1 and figures 1and 2. In the table 1, the absolute errors betweenthe exact solution uex and the approximate solutionuapprox, at m = 3, m = 5 and m = 7 with thefinal time T = 1 are given. But, in the figure 1we presented a comparison between the solution andthe approximate solution using the proposed methodat α = 1.8 for different values of the final timeTf = 0.25 and 0.5 at time step ∆t = 0.005 withm = 3. Also, in figure 2 we presented the behavior ofapproximate solution at t = 2 for different values ofα = 1.1, 1.4, 1.7 and 2 at time step ∆t = 0.005 withm = 3.

Figure 1. Comparison between the exact solution and theapproximate solution at α = 1.8 for different values of

time t.

Figure 2. The behavior of the approximate solution at thefinal time Tf = 2 for different values of α.

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E-ISSN: 2224-2880 35 Volume 13, 2014

The absolute error between the exact solution and the approximate solutionat m = 3, m = 5 and m = 7 with T = 1.

x |uex − uapprox| at m = 3 |uex − uapprox| at m = 5 |uex − uapprox| at m = 7

0.0 0.120849 e-03 0.274260 e-04 0.300445 e-050.1 0.021894 e-03 0.423794 e-04 0.417436 e-050.2 0.126809 e-03 0.373716 e-04 0.544455 e-050.3 0.321820 e-03 0.843125 e-04 0.617464 e-050.4 0.424838 e-03 0.323010 e-04 0.648473 e-050.5 0.429844 e-03 0.363133 e-04 0.639412 e-050.6 0.523805 e-03 0.193954 e-04 0.595429 e-050.7 0.623867 e-03 0.293780 e-04 0.531430 e-050.8 0.725877 e-03 0.493488 e-04 0.459438 e-050.9 0.726879 e-03 0.283224 e-04 0.379345 e-051.0 0.822821 e-03 0.773238 e-04 0.300045 e-05

Example 7 Consider the fractional non-linear Klein-Gordon problem

∂2u(x, t)

∂t2= Dαu(x, t)− 3

4 u(x, t)

+32 u

3(x, t) + f(x, t),

x ∈ (0, 5), t > 0, α ∈ (1, 2], (34)

with the following initial conditions

u(x, 0) = 0, ut(x, 0) = 0, (35)

and the following boundary conditions

u(0, t) = 0, u(5, t) = 0, (36)

where the source term f(x, t) is given by

f(x, t) = 6xα(5− x)t+ xα(5− x)t3

−1.5x3α(5− x)3t9

−(5α!− (α+ 1)!x).

The exact solution of this problem is

u(x, t) = xα(5− x) t3.

The obtained numerical results by means of theproposed method are shown in figures 3 and 4. Inthe figure 3, we presented a comparison between theexact solution and the approximate solution using theproposed method at α = 1.6 for different values ofthe final time Tf = 1.75 and 1.5 at time step ∆t =0.005 with m = 3. Also, in figure 4 we presented thebehavior of the approximate solution at Tf = 2 fordifferent values of α = 1.2, 1.5, 1.8 and 2 at time step∆t = 0.002 with m = 3.

Figure 3. Comparison between the exact solution and theapproximate solution at α = 1.6 for different values of

time t.

Figure 4. The behavior of the approximate solution at thefinal time Tf = 2 for different values of α.

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5 Conclusion and remarks

In this article, we are implemented Chebyshev spec-tral method for solving the non-linear fractionalKlein-Gordon equation. The fractional derivative isconsidered in the Caputo sense. The properties of theChebyshev polynomials are used to reduce the pro-posed problem to the solution of system of ordinarydifferential equations which solved by using finite dif-ference method. Special attention is given to study theconvergence analysis and estimate an upper bound ofthe error of the derived formula. The solution obtainedusing the suggested method is in excellent agreementwith the already existing ones and show that this ap-proach can be solved the problem effectively. It is ev-ident that the overall errors can be made smaller byadding new terms from the series (22). Comparisonsare made between approximate solutions and exact so-lutions to illustrate the validity and the great potentialof the technique. All computations in this paper aredone using Matlab 8.

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WSEAS TRANSACTIONS on MATHEMATICS M. M. Khader, N. H. Swetlam, A. M. S. Mahdy

E-ISSN: 2224-2880 38 Volume 13, 2014