Journal
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Volume -, Issue -, Pages -Publisher
WILEY
DOI: 10.1002/mma.9788
Keywords
accurate second order; product integration rule; regularity; stability and convergence; Volterra integrodifferential equation; weakly singular kernel
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This paper proposes a second-order accurate scheme for the temporal discretization of abstract Volterra integrodifferential equation with a weakly singular kernel. The scheme is constructed using the Crank-Nicolson method for the time derivative approximation and the product integration rule for the integral term approximation. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at t=0$$ t equal to 0 $$. Stability and convergence are established by the energy argument, and numerical experiments validate the theoretical estimate.
In this paper, a second-order accurate scheme is considered for the temporal discretization of the abstract Volterra integrodifferential equation with a weakly singular kernel. The time discrete scheme is constructed by the Crank-Nicolson method for approximating the time derivative and product integration (PI) rule for approximating the integral term. The proposed scheme employs a graded mesh for time to compensate for the singular behavior of the exact solution at t=0$$ t equal to 0 $$. Under the suitable assumptions, the stability and convergence are established by the energy argument, and the error is of order k2$$ {k} circumflex 2 $$, where k$$ k $$ is the parameter for the time grids. Numerical experiments validate the theoretical estimate.
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