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Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays

ABSTRACT In this paper, we study the spatial variable coefficient fractional convection–diffusion wave equation with the singe delay and multi-delay numerically when the exact solution satisfies a certain regularity. First, via the well-known exponential transformation, the delay problem can be greatly simplified, which allows us to use the variable coefficient four-order compact operator. Next, the numerical scheme is derived based on the compact operator and the reduction order method, followed by a linearized technique. Convergence of the full discrete numerical scheme is obtained with convergence order under the maximum norm by the energy argument. We prove the almost unconditional stability of the scheme under very mild conditions. Extending the numerical method to the multi-delay case is available. Extensive computational results are presented including single delay and double delay problems, which demonstrate the effectiveness and correctness of the developed schemes.