Analytical Solution for the Time‐Fractional Telegraph Equation

Abstract

We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variabletandx,respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.

A New Technique of Laplace Variational Iteration Method for Solving Space-Time Fractional Telegraph Equations

“…Ansari [12] derived a formal solution of the time-fractional telegraph equation by applying a fractional exponential operator. Huang [13] considered the time-fractional telegraph equation for the Cauchy problem and signaling problem. He solved the problem by the combined Fourier-Laplace transforms.…”

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

“…The most frequent application area for fractional models includes anomalous diffusion and wave equations (see [5][6][7][8][9]), fractional advection-dispersion equations (see [10][11][12][13][14][15][16][17][18]), and fractional telegraph equations (see [19][20][21][22][23]). …”

“…Many authors have studied the properties of the fractional telegraph equation (see [19][20][21][22][23][27][28][29][30][31][32][33][34]). Ortigueira [19,20] considered time-and space-fractional telegraph equations and related fractional telegraph processes.…”

“…Chen et al [21] obtained an analytical solution of the time-fractional telegraph equation with different boundary conditions by using the method of separation of variables. Huang et al [22] provided the fractional Green's function for the time-fractional telegraph equation by employing Laplace and Fourier transforms. An efficient perturbative Laplace method [27] was used to obtain the approximate solutions of time-and space-fractional telegraph equations.…”

An approximate analytical solution of time-fractional telegraph equation

“…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.…”