Numerical Solution of a Parabolic Equation with a Weakly Singular Positive-type Memory Term
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We find a numerical solution of an initial and boundary value problem. This problem is 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 equation is discretized in space by linear finite elements, and in time by the backward-Euler method. We prove existence and uniqueness of the solution to the continuous problem, and demonstrate that some regularity is present. In addition, convergence of the discrete sequence of iterations is shown.