# On the Solution of  Generalized Space Time Fractional Telegraph Equation

Authors: Badr S. Alkahtani; Vartika Gulati; Pranay Goswami
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We present the solution of generalized space time fractional telegraph equation by using  Sumudu variational iteration method which is the combination of variational iteration method and Sumudu transform. We tried to overcome the difficulties in finding the value of Lagrange multiplier by this new technique.


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## Body

## 1. Introduction

Differential calculus has its great importance in the field of Science and Technology [1]. The partial differential equations are one of the most powerful mathematical tools to describe, define, and explain the various phenomena of physics and chemistry and the process to solve them is a new interesting field to study. Telegraph equations are hyperbolic partial differential equations that gained remarkable interest due to their numerous applications in high-frequency transmission lines, propagation of electrical signals, optimization of guided communication system, and the propagation of quantum particles and many other physical and chemical phenomena. They are also applicable to designing high-voltage transmission lines.Recently generalized space time fractional telegraph equations were used to describe number of chemical and physical phenomena with remarkably interesting results. The solutions of generalized space time fractional telegraph equations were obtained by the various numerical methods. Numerous investigations have been done to solve fractional telegraph equations. The methods, namely, generalized differential transform, Fourier-Laplace transform, separations of variables, perturbation theory, and basic variational iteration, have been used to investigate the solution of homogeneous and nonhomogenous time fractional telegraph equation and the solutions are given in the form of Mittag-Leffler functions [2–6]. Nowadays variational iteration method is being used to solve many fractional differential equations and the presence of Sumudu transform is making the process rich.In this paper we modified the variational iteration method (VIM) by making use of the Sumudu transform and apply it to generalized space time fractional telegraph equations. This Sumudu variational iteration method enables us to overcome the difficulties that arise in finding the general Lagrange multipliers.


## 2. Mathematical Preliminaries

Definition 1.
The Caputo fractional derivative of orderα>0 is defined as [7] (1)Dtαft=∂αft∂tα=1Γm-α∫0t1t-τα-m+1∂mfτ∂τmdτ,m-1<α≤m,m∈N,∂mf∂tm,ifα=m∈N,where D0tα is called the Caputo derivative operator. By using Definition 1, we have the following result: (2)Dtαtβ=Γβ+1Γβ-α+1tβ-α,m-1<α≤m,β>m-1,β∈R,0m-1<α≤m,β≤m-1.Definition 2.
Forn-1<α<n, 0≤β≤1, n∈N, the generalized Hilfer fractional derivative [8, 9] is defined by (3)Da±α,βft=±Itβn-αddtIt1-βn-αft.In the definition above, type β allows Da±α,β to interpolate continuously between the classical Riemann-Liouville fractional derivative and the Caputo fractional derivative. Clearly, the case, β=0, reverts back to yield the classical Riemann-Liouville fractional derivative: (4)Da+α,0ft=DnItn-αft=Dtαft,0<α<1.Furthermore, in the case β=1, the Caputo fractional derivative is obtained: (5)Da+α,1ft=Itn-αDnft=Dtαft,0<α<1.Definition 3.
The Sumudu transform [10–12] over the set of functions(6)A=ft∣∃M,τ1,τ2>0,ft<Met/τi,ift∈-1j×0,∞is defined as(7)Gu=Sft=∫0∞fute-tdt,u∈-τ1,τ2.And the Sumudu transform of generalized Hilfer fractional derivative Da±α,βyx is given as follows:(8)SDa+α,βyt=u-αSytu-∑k=0n-1uk-n+βn-αlimt→a+⁡dkdtkIa+n-α1-βyt.Definition 4.
The two-parameter Mittag-Leffler functionEαz is defined [13] as(9)Eα,βz=∑k=0∞zkΓαk+β,where α,β∈C; Rα>0; Rβ>0; z∈C.
By using Definition4, we have (1)
E2,1z2=cosh⁡z,(2)
E2,2z2=sinh⁡z/z.Definition 5.
A generalization of Mittag-Leffler functionEα,βz of (9) is as follows [14]:(10)Eα,βγz=∑k=0∞γkΓαk+βzkk!,where α,β,γ∈C; Rα>0; Rβ>0; z∈C and γk denotes the familiar Pochhammer symbol or the shifted factorial, since (11)1k=k!k∈N0,γk=Γγ+kΓγ=1k=0;γ∈C∖0γγ+1⋯γ+k-1k∈N;γ∈CN0=N∪0=0,1,2,….


## 3. Variational Iteration Method

The analytical technique used by many researchers to study linear and nonlinear partial differential equations is commonly known as the variational iteration method. This technique of variational iteration was developed by Chinese mathematician He [15]. To illustrate the basic idea of this method, we consider the general nonlinear system of the following form:(12)Lux,t+Nux,t=fx,t,where u is the unknown function, L and N are linear and nonlinear operators, respectively, and f is the source term.The basic character of the method is to construct the following correction functional for (12):(13)un+1x,t=unx,t+∫0xλLunξ,t+Nunξ,t-fξ,tdξ,where λ is called the general Lagrange multiplier [16] and un is the approximate solution of nth order. If we have seen the whole process of the Lagrange multipliers in the case of an algebraic equation then solution of the algebraic equation fx=0 can be obtained by an iteration formula: (14)xn+1=xn+λfxn.The optimality condition for the extreme δxn+1/δxn=0 leads to(15)λ=-1f′xn,where δ is the classical variational operator.


## 4. Sumudu Variational Iteration Method (SVIM)

Consider the following generalized space time fractional telegraph equation:(16)D0+α,α1vx,t=a1∂βv∂tβx,t+a2∂γv∂tγx,t+a3vx,t+fx,t,where n-1<α<n; 0<α1≤1; β≤2; 0<γ≤1; x,t≥0; ∂β/∂tβ and ∂γ/∂tγ are Caputo derivatives defined by (1) and a1, a2, a3 are constants.Applying Sumudu transform with respect tox in both sides of (16),(17)SD0+α,α1vx,t=Sa1∂βv∂tβx,t+a2∂γv∂tγx,t+a3vx,t+fx,t.Using (8) with a=0 we get(18)u-αv-u,t-∑k=0m-1uk-n+α1n-αI0+n-α1-α1vk0,t-Sa1∂βv∂tβx,t+a2∂γv∂tγx,t+a3vx,t+fx,t=0.By the formula of (14), we get the iteration formula for the above as follows:(19)v-n+1u,t=v-nu,t+λu-αv-nu,t-∑k=0m-1uk-n+α1n-αI0+n-α1-α1vk0,t-Sa1∂βvn∂tβx,t+a2∂γvn∂tγx,t+a3vnx,t+fx,t.Putting λ=-uα, the Lagrange multiplier in (19) becomes(20)v-n+1u,t=uα∑k=0m-1uk-n+α1n-αI0+n-α1-α1vk0,t+uαSa1∂βvn∂tβx,t+a2∂γvn∂tγx,t+a3vnx,t+fx,t.Now taking inverse Sumudu transform on both sides of (20),(21)vn+1u,t=S-1uα∑k=0m-1uk-n+α1n-αI0+n-α1-α1vk0,t+uαSa1∂βvn∂tβx,t+a2∂γvn∂tγx,t+a3vnx,t+fx,t.This is the iteration formula for solving generalized space time fractional telegraph equation with the following initial iteration:(22)v0x,t=S-1∑k=0m-1uk-n+α1n-α+αI0+n-α1-α1vk0,t.


## 5. Main Results

Theorem 6.
Consider the space time fractional homogenous telegraph equation(23)D0+α,α1vx,t=∂2v∂t2x,t+∂v∂tx,t+vx,t,x,t≥0,1<α≤2,0<α1≤1,with initial conditions (24)I0+1-α1-α1v0,t=e-t,I0+2-α1-α1v′0,t=e-thas the solution in closed form as(25)vx,t=limn→∞⁡vnx,t=e-txα11-α+α-1Eα,α11-α+αxα+xα1Eα,α12-α+αxα.Proof.
We have the following space time fractional homogenous equation:(26)D0+α,α1vx,t=∂2v∂t2x,t+∂v∂tx,t+vx,t.By using iteration formula (21) derived above, we get the following iteration for (26):(27)vn+1x,t=S-1∑k=0m-1uk-n+α1n-α+αI0+n-α1-α1vk0,t+uαS∂2vn∂t2x,t+∂vn∂tx,t+vnx,t.The initial iteration is(28)v0x,t=S-1∑k=0m-1uk-n+α1n-α+αI0+n-α1-α1vk0,t,v0x,t=S-1u-1+α11-α+αe-t+u-1+α12-α+αe-t,v0x,t=xα11-α+α-1Γα11-α+αe-t+xα12-α+α-1Γα12-α+αe-t.Now by putting the values of “n” we get the iterations. For v1x,t put n=0 in (27); we find (29)v1x,t=S-1u-1+α11-α+αe-t+u-1+α12-α+αe-t+uαS∂2v0∂t2x,t+∂v0∂tx,t+v0x,t,(30)v1x,t=S-1u-1+α11-α+αe-t+u-1+α12-α+αe-t+uαSxα11-α+α-1Γα11-α+αe-t+xα12-α+α-1Γα12-α+αe-t,(31)v1x,t=S-1u-1+α11-α+αe-t+u-1+α12-α+αe-t+u-1+α11-α+2αe-t+u-1+α12-α+2αe-t.Taking inverse Sumudu transform of (31) we get(32)v1x,t=xα+α11-α-1Γα+α11-αe-t+xα+α12-α-1Γα+α12-αe-t+x2α+α11-α-1Γ2α+α11-αe-t+x2α+α12-α-1Γ2α+α12-αe-t.For n=1, v2x,t is given by the following iteration: (33)v2x,t=S-1u-1+α11-α+αe-t+u-1+α12-α+αe-t+uαS∂2v1∂t2x,t+∂v1∂tx,t+v1x,t,v2x,t=xα+α11-α-1Γα+α11-αe-t+xα+α12-α-1Γα+α12-α·e-t+x2α+α11-α-1Γ2α+α11-αe-t+x2α+α12-α-1Γ2α+α12-αe-t+x3α+α11-α-1Γ3α+α11-αe-t+x3α+α12-α-1Γ3α+α12-αe-t.Then the general term in successive approximation is given by(34)vnx,t=e-t∑k=0nxk+1α+α11-α-11Γk+1α+α11-α+xα1Γk+1α+α12-α.The solution in a closed form is given as follows:(35)vx,t=limn→∞⁡vnx,t=e-txα11-α+α-1Eα,α11-α+αxα+xα1Eα,α12-α+αxα.Theorem 7.
Consider the following generalized space time fractional nonhomogenous telegraph equation(36)D0+α,α1vx,t=∂2v∂t2x,t+∂vγ∂tγx,t+vx,t-x2-t+1,1<α≤2,0<γ≤1,0<α1≤1x,t≥0,with initial conditions(37)I01-α1-α1v0,t=t,I02-α1-α1vx0,t=0has the solution in closed form as (38)vx,t=limn→∞⁡vnx,t=x21-2Eα,3xα+1-t+t1-γΓ2-γEα,1xα-1+txα11-α+α-1Eα,α11-α+αxα+xα11-α+α-1Eα,α11-α+α1xα-Eα,11xα·t1-γΓ2-γ.Proof.
We have the following space time fractional nonhomogenous telegraph equation:(39)D0+α,α1x,t=∂2v∂t2x,t+∂vγ∂tγx,t+vx,t-x2-t+1.By using iteration formula (21) derived above, we get the following iteration for (39):(40)vn+1x,t=S-1∑k=0m-1uk-n+α1n-α+αI0+n-α1-α1vk0,t+uαS∂2vn∂t2x,t+∂γvn∂tγx,t+vnx,t-x2-t+1.The initial iteration v0x,t is given by(41)v0x,t=S-1∑k=0m-1uk-n+α1n-α+αI0+n-α1-α1vk0,t,v0x,t=S-1u-1+α11-α+αI0+1-α1-α1v0,t.Using initial conditions (37), we get(42)v0x,t=S-1u-1+α11-α+αt.Taking inverse Sumudu transform we find the initial iteration as(43)v0x,t=xα11-α+α-1Γα11-α+αt.Now by putting the values of “n” we get the iterations. For v1x,t put n=0 in (40); we have (44)v1x,t=S-1u-1+α11-α+αt+uαS∂2v0∂t2x,t+∂γv0∂tγx,t+v0x,t-x2-t+1.Taking Sumudu transform after solving the derivative terms we get(45)v1x,t=S-1uα11-α+α-1t+uα11-α+2α-1t1-γΓ2-γ+uα11-α+2α-1t-2u2+α-tuα+uα.Applying inverse Sumudu transform, (46)v1x,t=xα11-α+α-1Γα11-α+αt+xα11-α+2α-1Γα11-α+2αt1-γΓ2-γ+xα11-α+2α-1Γα11-α+2αt-2xα+2Γα+3-xαΓα+1t+xαΓα+1,v2x,t=xα11-α+α-1Γα11-α+αt+xα11-α+2α-1Γα11-α+2αt1-γΓ2-γ-x2αΓ2α+1t-xαΓα+1t+xα11-α+2α-1Γα11-α+2αt+xα11-α+3α-1Γα11-α+3αt+2xα11-α+3α-1Γα11-α+3αt1-γΓ2-γ-2x2α+2Γ2α+3+x2αΓ2α+1-2xα+2Γα+3+xαΓα+1-x2αΓ2α+1t1-γΓ2-γ.Then the general term in successive approximation is given by(47)vnx,t=txα11-α+α-1Γα11-α+α+t∑k=1nxα11-α+k+1α-1Γα11-α+k+1α-2x2∑k=1nxkαΓkα+3+t1-γΓ2-γ∑k=1nkxα11-α+k+1α-1Γα11-α+k+1α+1-t∑k=1nxkαΓkα+1-t1-γΓ2-γ∑k=1nk-1xkαΓkα+1.The solution in a closed form is given by(48)vx,t=limn→∞⁡vnx,t=x21-2Eα,3xα+1-t+t1-γΓ2-γEα,1xα-1+txα11-α+α-1Eα,α11-α+αxα+xα11-α+α-1Eα,α11-α+α1xα-Eα,11xα·t1-γΓ2-γ.


## 6. Special Cases

If we takeα1=1, in above Theorems 6 and 7, we have the results obtained by Alawad et al. [3].Corollary 8.
The space time fractional homogenous telegraph equation(49)∂αv∂xαx,t=∂2v∂t2x,t+∂v∂tx,t+vx,t,x,t≥0,1<α≤2,with the initial conditions(50)v0,t=e-t,vk0,t=e-thas the solution in closed form as (51)vx,t=limn→∞⁡vx,t=e-tEα,1xα+xEα,2xα.Remark 9.
If we putα=2 in (51) then the following solution is same as that obtained by Wazwaz [17]:(52)vx,t=e-tE2,1x2+xE2,2x2=e-tcosh⁡x+xsinh⁡xx=ex-t.Corollary 10.
The space time fractional nonhomogenous telegraph equation(53)∂αv∂xαx,t=∂2v∂t2x,t+∂vγ∂tγx,t+vx,t-x2-t+1,1<α≤2,0<γ≤1,x,t≥0,with initial conditions (54)v0,t=t,vx0,t=0has the solution as follows:(55)vx,t=limn→∞⁡vnx,t=t+x21-2Eα,3xα+1+t1-γΓ2-γEα,1xα-1.



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