An approximate analytical solution of time-fractional telegraph equation

doi:10.1016/j.amc.2011.02.030

Applied Mathematics and Computation

Volume 217, Issue 18, 15 May 2011, Pages 7405-7411

https://doi.org/10.1016/j.amc.2011.02.030 Get rights and content

Abstract

In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1 < α ⩽ 2). By using initial values, the explicit solutions of telegraph equation for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.

Introduction

Many phenomena in engineering and applied sciences can be described successfully by developing models using fractional calculus, i.e. the theory of derivatives and integrals of fractional non-integer order [1], [2], [3], [4]. Fractional differential equations have gained much attention recently since fractional order system response ultimately converges to the integer order system response. No analytical method was available before 1992 for such equations even for linear fractional differential equations. The homotopy analysis method (HAM) was proposed by Liao [5] to solve fractional differential equations with great success. Recently, Hashim et al. [6] applied the homotopy analysis method (HAM) to fractional initial value problems and showed that HAM is the easy-to-use analytical method and it gives close to the exact solution for both the linear and nonlinear partial differential equations.

The general equation of the second order linear hyperbolic telegraph equation in one-dimension is represented by $\frac{\partial^{2} u}{\partial t^{2}} + α \frac{\partial u}{\partial t} + β u (x, t) = \frac{\partial^{2} u}{\partial x^{2}} + f (x, t), x \in [a, b], t \in [0, T],$ where α and β are known constants and u(x, t) is the unknown function. For α > β > 0, it represents telegraph equation. The equation is commonly used in the study of wave propagation of electric signals in a cable transmission line and also in wave phenomena. The equation is used in modeling reaction–diffusion in various branches of engineering sciences and biological sciences by many researchers like Mohebbi and Dehaghan [7], El-Azab and El-Glamel [8], Yousefi [9], Gao and Chi [10], Dehghan and Ghesmati [11] etc. Eq. (1) represents a damped wave motion for α > 0, β = 0. Recently, Das and Gupta [12] have solved the fractional hyperbolic PDE by using HAM.

In 2007, Atanackovic et al. [13] have analyzed diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains. In this article both the signaling and Cauchy problems are deduced for different particular cases. Orsingher and Beghin [14] have solved the fundamental solution to time fractional telegraph equations of different kind by using Fourier transform. Huang [15] applied Fourier–Laplace transforms during derivation of the fundamental solution for the Cauchy problem in a whole space domain and signaling problem in a half space domain. Momani [16] successfully applied ADM to find out the solution of space-time-fractional telegraph equation by considering two different kinds of examples. In 2008, Chen et al. [17] have derived the analytical solution of time fractional telegraph equation for Dirichlet, Neumann and Robin boundary conditions using method of separation of variables. But to the best of the authors’ knowledge the general telegraph equation with fractional time derivatives has not yet been solved by using the approximate analytical method HAM.

In 1992, Liao [5] proposed a mathematical tool based on homotopy, a fundamental concept in topology and differential geometry known as Homotopy Analysis Method. It is an analytical approach to get the series solution of linear and nonlinear partial differential equations (PDEs). The difference with the other perturbation methods is that this method is independent of small/large physical parameters. It also provides a simple way to ensure the convergence of series solution. Moreover the method provides great freedom to choose base function to approximate the linear and nonlinear problems [18], [19]. Another advantage of the method is that one can construct a continuous mapping of an initial guess approximation to the exact solution of the given problem through an auxiliary linear operator and to ensure the convergence of the series solution an auxiliary parameter is used. Liao and Tan [20] have shown that with the help of there, even complicated nonlinear problems are reduced to the simple linear problems. Recently Liao [21] has claimed that the difference from the other analytical methods is that one can ensure the convergence of series solution by means of choosing a proper value of convergence-control parameter.

In this paper, the homotopy analysis method is used to obtain the approximate analytical solution of telegraph equation with the fractional time derivative for both fractional Brownian motion and standard motion and the results are presented graphically for different particular cases.

Section snippets

Solution of the problem

The factional time derivative telegraph equation of order α (1 < α ⩽ 2) is given as $\frac{\partial^{α} u (x, t)}{\partial t^{α}} + \frac{\partial^{α - 1} u (x, t)}{\partial t^{α - 1}} + u (x, t) = \frac{\partial^{2} u (x, t)}{\partial x^{2}} + f (x, t)$ with the initial conditions $u (x, t) |_{t = 0} = 0 and {\frac{\partial u (x, t)}{\partial t}|}_{t = 0} = 0 .$ To solve Eq. (2) by HAM, we choose the initial approximation $u_{0} (x, t) = 0$ and the linear operator $L [ϕ (x, t; q)] = \frac{\partial^{α} ϕ (x, t; q)}{\partial t^{α}}, 1 < α ⩽ 2$ with the property $L [C_{1} + C_{2} t] = 0,$ where C₁ and C₂ are integral constants. Furthermore, in view of Eq. (2) we define a differential equation $N [ϕ (x, t; q)] = \frac{\partial^{α} ϕ (x, t; q)}{\partial t^{α}} + \frac{\partial^{α - 1} ϕ (x, t; q)}{\partial t^{α - 1}} + ϕ (x, t; q) - \partial$

Particular cases

Case I:
when $f (x, t) = \frac{t^{n}}{Γ (n + 1)} \sinh x,$
We obtain from Eq. (14) the values of u_m(x, t) for m = 1, 2, 3, … as $u_{1} (x, t) = \frac{- ℏ t^{n + α}}{Γ (n + α + 1)} \sinh x$ $u_{2} (x, t) = - ℏ (\frac{(1 + ℏ) t^{n + α}}{Γ (n + α + 1)} + \frac{ℏ t^{n + α + 1}}{Γ (n + α + 1)}) \sinh x,$ $u_{3} (x, t) = - ℏ (\frac{(1 + ℏ)^{2} t^{n + α}}{Γ (n + α + 1)} + \frac{2 ℏ (1 + ℏ) t^{n + α + 1}}{Γ (n + α + 2)} + \frac{ℏ^{2} t^{n + α + 2}}{Γ (n + α + 3)}) \sinh x,$ $u_{4} (x, t) = - ℏ (\frac{(1 + ℏ)^{3} t^{n + α}}{Γ (n + α + 1)} + \frac{3 ℏ (1 + ℏ)^{2} t^{n + α + 1}}{Γ (n + α + 2)} + \frac{3 ℏ^{2} (1 + ℏ) t^{n + α + 2}}{Γ (n + α + 3)} + \frac{ℏ^{3} t^{n + α + 3}}{Γ (n + α + 4)}) \sinh x$ and so on.
Preceding in this manner the components u_n, n ⩾ 1 of the Homotopy analysis method can be completely obtained, and the series solutions are thus entirely

Numerical results and discussion

In this section, numerical results of the probability density function u(x, t) for different fractional Brownian motions $α = \frac{4}{3}, \frac{3}{2}, \frac{5}{3}$ and also for standard motion α = 2 are calculated for various values of integer powers n of t and x for Cases I and II at x = 2, which are depicted through Fig. 1, Fig. 2 respectively. The three dimensional variations of u(x, t) vs. x and t at $α = \frac{3}{2}$ and n = 1 for Cases I and II are also shown through Fig. 3, Fig. 4, respectively.

It is seen from Fig. 1, Fig. 2 that for both

Conclusion

The most important part of the present investigation is that if f(x, t) is expressed as a polynomial of t, then u(x, t) decreases with the increase in the integral power of t but when it is expressed as a polynomial of x, then the reverse situation occurs. The same situation occurs for fractional hyperbolic equation (Das and Gupta [12]). So it can be concluded that the fractional telegraph equation is more stable in comparison with the other fractional hyperbolic equation. The authors believe

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An approximate analytical solution of time-fractional telegraph equation

Abstract

Introduction

Section snippets

Solution of the problem

Particular cases

Numerical results and discussion

Conclusion

Communications in Nonlinear Science and Numerical Simulation

Applied Mathematics and Computation

Applied Mathematics and Computation

Engineering Analysis with Boundary Elements

Applied Mathematics and Computation

Journal on Mathematical Analysis and Applications

Applied Mathematics and Computation

Communications in Nonlinear Science and Numerical Simulation

Chaos, Solitons and Fractals

Fractional calculus: some basic problems in continuum and statistical mechanics

Fractional Differential Equations