Huang considered the combine Fourier-Laplace transform to solve time fractional telegraph equation. implemented separation of variables method for deriving the analytical solutions of the nonhomogeneous time fractional telegraph equation under Dirichlet, Neumann, and Robin boundary conditions. Momani and Garg and Sharma used Adomian decomposition method developed by Adomian in for solving homogeneous and nonhomogeneous space-time fractional telegraph equation. Orsingher and Xuelei and Orsingher and Beghin considered the space and time fractional telegraph equations and obtained the Fourier transform of their fundamental solution. There are various methods developed to solve fractional telegraph equations. A space-time fractional telegraph equation is obtained from the classical telegraph equation by replacing the time and space derivative terms by fractional derivatives.
In recent years, great interest has been developed in fractional differential equation because of its frequent appearance in fluid mechanics, mathematical biology, electrochemistry, and physics. Its applications arise in signal analysis for transmission and propagation of electrical signals and also modelling reaction diffusion. The telegraph equation developed by Oliver Heaviside in 1880 is widely used in Science and Engineering. Furthermore, we give illustrative examples to demonstrate the efficiency of the method. Double Laplace transform method is applied to find exact solutions of linear/nonlinear space-time fractional telegraph equations in terms of Mittag-Leffler functions subject to initial and boundary conditions.
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