Optimal convergence of an Euler and finite difference method for nonlinear partial integro-differential equations

Authors:
Q. Sheng;T. Tang
Affiliations:
Department of Mathematics, National University of Singapore 10 Kent Ridge Crescent, Singapore 0511;Department of Mathematics and Statistics, Simon Fraser University Burnaby, BC, Canada V5A 1S6
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
1995

Citing 6
Cited 1

Bifurcation with memory

SIAM Journal on Applied Mathematics
A numerical method for a partial integro-differential equation

SIAM Journal on Numerical Analysis
Finite element methods for parabolic and hyperbolic partial integro-differential equations

Non-Linear Analysis
Convolution quadrature and discretized operational calculus I.

Numerische Mathematik
A difference scheme for a nonlinear partial integrodifferential equation

SIAM Journal on Numerical Analysis
A finite difference scheme for partial integro-differential equations with a weakly singular kernel

Applied Numerical Mathematics

Numerical approaches to fractional calculus and fractional ordinary differential equation

Journal of Computational Physics

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Abstract

Fully discretized Euler method in time and finite difference method in space are constructed and analyzed for a class of nonlinear partial integro-differential equations emerging from practical applications of a wide range, such as the modeling of physical phenomena associated with non-Newtonian fluids. Though first-order and second-order time discretizations (based on truncation errors) have been investigated recently, due to lack of the smoothness of the exact solutions, the overall numerical procedures do not achieve the optimal convergence rates in time. In this paper, however, by using the energy method, we prove that it is possible for the scheme to obtain the optimal convergence rate O(@t). Numerical demonstrations are given to illustrate our result.