# Conformable Sumudu Transform of Space-Time Fractional Telegraph Equation

Authors: Amjad E. Hamza; Mohamed Z. Mohamed; Eltaib M. Abd Elmohmoud; M. Magzoub
Journal: Abstract and Applied Analysis
 (2021)

---

## Abstract

This paper intends to obtain accurate and convergent numerical solutions of linear space-time matching telegraph fractional equations by means of a double Sumudu matching transformation method. Moreover, the numerical model is equipped to explain the work, the accuracy of the work, and sobriety in its presentation method, and as a result, the proposed method shows an effective and convenient way, to employ proven problems in science and engineering.


---

## Body

## 1. Introduction

In recent years, a large extent of progress has been made in the differential equation with frequent appearance in physics, chemical and industrial mathematics, processing and control theory, and fluid mechanics. For example, it acquires space-time telegraph fraction equation of the equation classic telegraph by replacing the derivative terms of space-time from the partial derivatives.The conformable telegraph equations have a wide variety of applications in science and engineering, ideally in optimizing propagation-oriented and propagating electrical communication systems [1, 2]. The solutions obtained from the telegraph equations of fraction of space-time are the terms Mittag-Leffler functions by mixing the Laplace transform and variation iteration method [3]. Recently, many new FDOs have been developed with the exponential and Mittag-Leffler kernels to analyze the systems and models with better memory characteristics. In this sequence, the latest work includes the investigations of the exothermic model and vibration equation of fractional order with the inclusion of the power law and the Mittag-Leffler law by Kumar et al. [4, 5]. The researchers used the Laplace transform and the Elzaki transform to solve fractional differential equations, in addition to a system of partial differential equations [6, 7]. Caputo-Fabrizio and other fractional operators have been utilized to examine the solution behavior of fractional-order models. In the year 2018, Evirgen and Yavuz [8] investigated an alternative scheme for a nonlinear optimization problem containing the CF operator. With the utilization of the semianalytical schemes HPSTM and HASTM, the solutions obtained via HPSTM are in good correspondence with the semianalytic solutions obtained by using HASTM [9]. This indicates that one can try to solve partial differential equations using the same techniques for conformable fractional derivatives (CFD) suggested by Khalil [10]. Analytical solutions have been extracted from this conformable Sumudu transform, and conformable fractional derivatives were solved to construct the new exact traveling wave solutions of the three special form of time fractional WKB equations [11–13], such as the time fractional approximate long-wave equations, the time fractional variant Bossiness equations, and the time fractional Wuzhang system of equations using the generalized expϕξ expansion [14] method with a conformable derivative sense. They have presented a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions, and aerofoil flow theory [15]. A finite difference scheme which depends on a new approximation based on an extended cubic B-spline for the second-order derivative is used to calculate the numerical outcomes of time fractional Burgers Equation (22). We have discussed the new definition of double Sumudu transformation and new conformable fractional derivative for converting the nonlinear fractional differential equations into the ODEs [16]. The conformable fractional double Sumudu transform method was applied to solve the time-space of conformable fractional telegraph differential equations. Thus, the aim of this study is to suggest an analytical solution for the one singular conformable fractional telegraph equation by using the conformable double Sumudu decomposition method.The methodology of this paper is as follows. In Section2, we present a double definition of Sumudu transformation, its properties, a description of conformable fractional, and fractional space-time telegraph equations. In Section 3, we obtained the exact solutions of the conformable fractional space-time telegraph equations. The conclusion from this paper is in Section 4.


## 2. Conformable Double Sumudu Transform and Some Properties

Now, we take into account a few definitions and properties of the CDST (conformable double Sumudu transform) which may allow for the finding of even more transformed pairsfxβ/β,tα/α,Fu,v rather than having to consider the following.Definition 1.
Let us assume thatf:0,∞→Ɍ, then the CFD (conformable fractional derivative) of f having order β can be represented by
(1)dβdxβfxββ=limϵ→0fxβ/β+ϵx1−βϵ,xββ>0,0<β≤1.See [4, 7, 17].Definition 2 (see [5]).
If the given functionfxβ/β,tα/α:R×0,∞→R, then the CSFPD (Conformable Space Fractional Partial Derivative) with order β, a function fxβ/β,tα/α, is represented by
(2)∂β∂xβfxββ,tαα=limϵ→0fxβ/β+ϵx1−β,tα/α−fxβ/β,tα/αϵ,xββ,tαα>0,0<α,β≤1.
The CSFPD with orderα a function fxβ/β,tα/α is represented by
(3)∂α∂tαfxββ,tαα=limσ→0fxβ/β,tα/α+σt1−α−fxβ/β,tα/ασ,xββ,tαα>0,0<α,β≤1.Definition 3.
Letfx,t be a piecewise continuous function on the given interval 0,∞×0,∞ having the exponential order. Consider for some, ∈Ɍsupxβ/β,tα/α>0,efxβ/β,tα/α/axβ/β+btα/α. According to these conditions, the CDST is represented by
(4)StαSxβfxββ,tαα=Fu,v=1uv∫0∞∫0∞e−xβ/uβ−tα/vαfxββ,tααdtdx,where u,v∈₵,0<α,β≤1 and the integrals are known to be the CFI (conformable fractional integral) with respect to xβ/β and tα/α, respectively.Definition 4 (single conformable Sumudu transform of a function with two variables).
CST (conformable Sumudu transform) with respect toxβ/β of fxβ/β,tα/α can be represented as
(5)Sxβfxββ,tαα=Fu,t=1u∫0∞e−xβ/uβfxββ,tααdx,where the integral is in the conformable sense with respect to x.The expressionSxβfxβ/β,tα/α reveals the conformable integral of (5); in the given expression, the subscript xβ/β on S indicates for which variable the CST can be applied.Homogenously, the CST of the same functionfxβ/β,tα/α with respect to variable tα/α is defined as
(6)Stαfxββ,tαα=Fx,v=1v∫0∞e−tα/vαfxββ,tααdt;due to these definitions, the successive transformation is represented by StαSxβfxβ/β,tα/α presented in (4). If it is assumed that the function fxβ/β,tα/α gives adequate contraints (Love 1970), the order of transformation can be altered, as
(7)1uv∫0∞∫0∞e−xβ/uβ−tα/vαfxββ,tααdtdx=1vu∫0∞∫0∞e−tα/vα−xβ/uβfxββ,tααdxdt,and symbolically, it can be represented as
(8)SxβStαfxββ,tαα=StαSxβfxββ,tαα=Fu,v.Theorem 5.
Let us assume that the two functionsfxβ/β,tα/α,gxβ/β,tα/α have CDST, then
(1)
SxβStαk1fxβ/β,tα/α+k2gxβ/β,tα/α=k1SxβStαfxβ/β,tα/α+k2SxβStαgxβ/β,tα/α,where k1 and k2 are constants
(2)
SxβStαfγxβ/β,σtα/α=1/ruvF1/γu,1/σv, where r=γβσα(3)
SxβStαe−dxβ/β−ctα/αfxβ/β,tα/α=1/uvF1/u+d,1/v+c(4)
−1m+nSxβStαxmβ/βmtnα/αnfxβ/β,tα/α=1/uv∂m+n/∂um∂vnF1/u,1/vLemma 6.
The CDST ofβth and αth order FPDs (Fractional Partial Derivatives) are defined as follows:
(9)SxβStαDxβfxββ,tαα=Fu,vu−F0,vu,(10)SxβStαDtαfxββ,tαα=Fu,vv−Fu,0v,where Dxβfxβ/β,tα/α, Dtαfxβ/β,tα/α means the βth and αth order CFPDs (Conformable Fractional Partial Derivatives), respectively. Similarly, the CDST of mixed FPDs can be expressed.Theorem 7.
Let0<α,β≤1 and m,n∈Ν such that fxβ/β,tα/α∈ClɌ+,Ɍ+,l=maxm,n. Also, assume that the CLT (Conformable Laplace Transforms) of the given functions Dxβifxβ/β,tα/α and Dtαjfxβ/β,tα/α,i=1,⋯,m,j=1,⋯,n exist. Then,
(11)SxβStαDxβmfxββ,tαα=Fu,vum−F0,vum−∑i=1m−1StαDxβif0,tum−i,SxβStαDtαnfxββ,tαα=Fu,vvn−Fu,0vn−∑j=1n−1SxβDtαjfx,0vn−j.
Similarly, the CDLT (Conformable Double Laplace Transform) of mixed partial derivative(12)SxβStαDxβmDtαnfxββ,tαα=1umvnFu,v−Fu,0−F0,v−∑j=1fx,tn−1vjSxβDtαjfx,0−∑i=1m−1uiStαDxβif0,t+∑i=1m−1∑j=1n−1vjuiDtαjDxβif0,0+umvnF0,0,where Dxβm and Dtαn represent m,n times conformable fractional derivatives of function fxβ/β,tα/α having order β and α, respectively.


## 3. Conformable Double Sumudu Transform

Now, we will study the following general time-space conformable fractional telegraph equation:(13)∂β∂xβfxββ,tαα=a∂α∂tαfxββ,tαα+b∂γ∂tγfxββ,tαα+cfxββ,tαα+hxββ,tαα,(14)1<α,β<2,0<γ<1,xββ,tαα>0,having initial condition:
(15)fxββ,0=k1xββ,Dtαfxββ,0=k2xββ,and boundary:
(16)f0,tαα=r1tαα,Dxβf0,tαα=r2tαα,where hxβ/β,tα/α is the given function and a,b,c are constants.By taking conformable double Sumudu transform for bout aids of Equation (13),
(17)1u2Fu,v−F0,vu2−1uStαDxβf0,tαα=a1v2Fu,v−Fu,0v2−1vSxβDtαfxββ,0+bFu,vv−Fu,0v+cFu,v+h´u,v.Secondly, by using the conformable single Sumudu transform from Equations (15) and (16), one can obtain
(18)F0,v=K1u,DtαF0,v=K2uFu,0=R1v,DxβFu,0=R2v.By replacing (9) in (8) and by direct calculation, one can obtain
(19)Fu,v−B1´vu2−1uB2´v=a1v2Fu,v−f1´uv2−1vf2´u+bFu,vv−f1´uv+cFu,v+h´u,v,1u2−av2−bv−cFu,v=B1´vu2+B2´vu−af1´uv2−af2´uv−bf1´uv+h´u,v,Fu,v=v2B1´v+uv2B2´v−au2f1´u−au2vf2´u−bu2vf1´u+u2v2h´u,vv2−au2−bu2v−cu2v2.Applying the inverse, CDST.Example 8.
Consider the homogenous fractional telegraph equation given by [3, 18]:
(20)Dxβ2fxββ,tαα−Dtα2fxββ,tαα−Dtαfxββ,tαα−fxββ,tαα=0,with the conditions
(21)f0,tαα=e−tα/α,fxββ,0=exβ/βDxβf0,tαα=e−tα/α,Dtαfxββ,0=−exβ/β,where Dxβ2 and Dtα2 mean two times conformable fractional derivative of function fxβ/β,tα/α. Applying conformable double Sumudu transform for () yields
(22)1u2Fu,v−F0,vu2−1uDxβf0,v+1v2Fu,v+Fu,0v2−1vDtαfu,0−Fu,vv+Fu,0v−Fu,v=0.
Applying conformable single Sumudu transform to conditions expressed in Equation (22) yields
(23)F0,v=11+v,Fu,0=11−uDxβF0,v=11+v,DtαFu,0=−11−u.
Making some algebraic manipulations in Equation (22), we have
(24)Fu,v=11−u1+v.
Thus, we get the solution of Equation (22) as (see Figure 1)
(25)fxββ,tαα=exβ/β−tα/α.Example 9.
Regard nonhomogenous space-time fractional telegraph equation stated as [17]
(26)Dxβ2fxββ,tαα−Dtα2fxββ,tαα−Dtαfxββ,tαα−fxββ,tαα=−2exβ/β+tα/α,with the constraints
(27)f0,tαα=etα/α,fxββ,0=exβ/βDxβf0,tαα=etα/α,Dtαfxββ,0=exβ/β,where Dxβ2 and Dtα2 mean two times conformable fractional derivative of function fxβ/β,tα/α. Applying conformable double Sumudu transform for () yields
(28)1u2Fu,v−F0,vu2−1uDxβf0f,v+1v2Fu,v+Fu,0v2−1vDtαfu,0−Fu,vv+Fu,0v−Fu,v=−21−u1−v.
Again, employing conformable single Sumudu transform to constraints stated in Equation (28), one can obtain
(29)F0,v=11−v,Fu,0=11−uDxβF0,v=11−v,DtαFu,0=11−u;after some mathematical computation in Equation (28), we get
(30)Fu,v=11−u1−v.
So, we are going to get the solution of Equation (28) as (see Figure 2)
(31)fxββ,tαα=exβ/β+tα/α.Example 10.
Let us use the telegraph equation [(13)]:
(32)Dxβ2fxββ,tαα=Dtα2fxββ,tαα+4Dtαfxββ,tαα+4fxββ,tαα,with the constraints
(33)f0,tαα=1+e−2tα/α,fxββ,0=1+e2xβ/βDxβf0,tαα=2,Dtαfxββ,0=−2,where Dxβ2 and Dtα2 mean two times conformable fractional derivative of function fxβ/β,tα/α. Applying conformable double Sumudu transform for () produces
(34)1u2Fu,v−F0,vu2−1uDxβf0f,v=1v2Fu,v−Fu,0v2−1vDtαfu,0+4Fu,vv−4Fu,0v+4Fu,v.
Again, employing conformable single Sumudu transform to constraints stated in Equation (34) yields
(35)F0,v=1+11+2v,Fu,0=1+11−2uDxβF0,v=2,DtαFu,0=−2;after some mathematical computation in Equation (34), one can obtain
(36)Fu,v=11−2u+11+2v.
So, we are going to get the solution of Equation (34) (see Figure 3):
(37)fxββ,tαα=e2xβ/β+e−2tα/α.Figure 1
The figure shows the solution of (20), when t=0.01,0≤x≤1.Figure 2
The figure shows the solution of (26), when t=0.01,0≤x≤1.Figure 3
The figure shows the solution of (26), when t=0.01,0≤x≤1.


## 4. Conclusion

The primary goal is the successful utilization of double Sumudu transforms to the conformable fractional-order and interesting and linear space-time conformable fractional telegraph equations. We have discussed the definition of double Sumudu transformation and new conformable fractional derivative for converting the linear fractional differential equations into the ODEs over time; we presented the general analytical solution of a linear space-time conformable fractional telegraph equations conformable fractional double Sumudu transform. Also, our suggested method is applied successfully for obtaining the general solutions of several linear and nonhomogeneous conformable fractional telegraph equations. Finally, the effects show that our resolved method is efficient and can be applied for finding the general solutions of all cases related to the conformable fractional differential equations.



---

*Source: 6682994-2021-04-09.xml*