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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.

On the approximation of time-fractional telegraph equations using localized kernel-based method