An efficient temporal approximation for weakly singular time-fractional semilinear diffusion-wave equation with variable coefficients

This paper deals with an efficient discretization in handling the discontinuous initial data and nonsmooth exact solution of the semilinear time-fractional diffusion-wave (TFDW) equation with variable coefficients to achieve the optimal convergence rate. We use nonuniform L 1 approach with half point discretization process to obtain the desired temporal accuracy in discretization of Caputo fractional derivative of order alpha is an element of ( 1 , 2). ) . The error analysis in approximation of the Caputo derivative is proved by assuming the weak singularity at t = 0. Then the mentioned model is transformed into a system of equations by using the developed nonuniform L 1 method and second order approximation of the space derivatives. We construct two linearized finite difference schemes in solving semilinear single-term and multi-term TFDW equations with min ( 3 - alpha, gamma (alpha - 1)) )) and min ( 3 - alpha(r), gamma (alpha(r) - 1)), )) , r = 0, , 1, , 2 convergence order, respectively where parameter gamma >= 1 is used in formation of nonuniform temporal grids. The alternating direction implicit (ADI) process is used in solving the two-dimensional semilinear TFDW equation with variable coefficients. Further, we prove the Von Neumann stability analysis for the developed scheme. To illustrate the theoretical findings, we provide four numerical examples in one and two-dimensions with smooth, nonsmooth and also discontinuous initial data. The presented numerical results validate that the proposed scheme are in agreement with theoretical findings for both cases nonsmooth solutions and discontinuous initial data.