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Numerical solution of a parabolic equation with non-local boundary specifications. (English) Zbl 1032.65104

Summary: The parabolic partial differential equations (PDEs) with non-local boundary specifications model various physical problems. Numerical schemes are developed for obtaining approximate solutions to the initial boundary-value problem for one-dimensional second-order linear parabolic partial differential equation with non-local boundary specifications replacing boundary conditions.
The method of lines semi-discretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The spatial derivative in the PDE is approximated by a finite-difference approximation. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function.
Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. The new algorithms are tested on two problems from the literature. The central processor unit times needed are also considered.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65Y05 Parallel numerical computation
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