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Numerical solution of a parabolic equation with a weakly singular positive-type memory term. (English) Zbl 0898.65102

The author considers a numerical solution of an initial and boundary value problem for a parabolic integro-differential equation whose integral is the convolution product of a positive definite weakly singular kernel with the time derivative of the solution. The author describes a product integration method for the discretization of the Volterra term in the equation. The equation is discretized in space by linear finite elements, and in time by the backward-Euler method. The author proves existence and uniqueness of a solution to the continuous problem, demonstrates that some regularity is present, and establishes convergence of the approximation scheme to a solution that satisfies mile regularity.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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