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Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term. (English) Zbl 1160.65068

Summary: New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order backward differentiation formula methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the \(L^2\) norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the \(L^{\infty}\) and \(H^1\)norms and superconvergence phenomena.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
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