Abstract
In the present work, a hybrid transform-based localized meshless method is constructed for the solution of time-fractional telegraph equations. In the first step the Laplace transform is applied to the time-fractional telegraph equation, which reduces the problem to a finite set of elliptic equations which are solved with the help of local radial basis functions method in parallel. Finally, the solution is represented as an integral along a smooth curve in the complex plane. The integral is then evaluated by quadrature rule. The advantage of this method is that it does not suffer from time instability that may occur in a time stepping procedure. A clear improvement is observed in terms of stability, accuracy and ill-conditioning.
Highlights
Fractional calculus is the generalization of differentiation and integration to non-integer orders
Liu et al [15] derived the analytical solution of the nonhomogeneous time-fractional telegraph equation by considering three types of nonhomogeneous boundary conditions using the method of separation of variables
3 Analysis of the method we propose a meshless method based on Laplace transform for timefractional telegraph equation
Summary
Fractional calculus is the generalization of differentiation and integration to non-integer orders. The solution of space–time-fractional telegraph equation in a bounded domain is obtained in terms of Mittage-Leffler functions by the method of generalized differential transform [11]. Das et al [12] used a homotopy analysis method in approximating an analytical solution for the time-fractional telegraph equation and different particular cases have been derived. In [13] Jiang and Lin obtained a series solution for the time-fractional telegraph equation with Robin boundary value conditions using the reproducing kernel theorem. The Laplace transform is coupled with localized kernel-based method, and the resulting hybrid method is investigated for solving telegraph equations of fractional order. 6.1 Problem 1 Here we apply our proposed numerical method to the one dimensional time-fractional telegraph equation [13], CD2t αu(x, t). The local nature of the method makes it more attractive for such a type of problems
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