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In this paper, we study the two-dimensional Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain. The memory kernel of this equation makes it not easy to construct an efficient numerical scheme and perform theoretical analysis. The numerical method is considered by the finite difference approach for spatial discretization and Crank–Nicolson (CN) alternating direction implicit (ADI) scheme in the time direction. The integral terms are approximated by the fractional convolution quadrature. We prove that the proposed method is unconditional stable and derive error estimates innorm. The literature reported on the ADI finite difference method of this model is extremely sparse. Numerical results support the theoretical analysis.

An ADI finite difference method for the two-dimensional Volterra integro-differential equation with weakly singular kernel