Numerical solution of a parabolic equation with non-local boundary specifications

• Published in: Applied mathematics and computation Vol. 145; no. 1; pp. 185 - 194
• Main Author: Dehghan, Mehdi
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 20.12.2003; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Central processor time; Computer science; control theory; systems; Exact sciences and technology; Mathematics; Non-local boundary conditions; Numerical analysis; Numerical analysis. Scientific computation; Numerical integration procedure; One-dimensional parabolic equation; Ordinary differential equations; Pade approximant; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Sequential and parallel algorithms; Theoretical computing; Central processor time; Non-local boundary conditions; Sequential and parallel algorithms; One-dimensional parabolic equation; Numerical integration procedure; Pade approximant; Parallel algorithm; Numerical integration; Differential equation; Recurrence relation; Initial value problem; Numerical method; Exponential function; Matrix function; Partial differential equation; Parabolic equation; Boundary value problem; Numerical solution; Method of lines; Padé approximant; Sequential algorithm; Central processor; Non local theory; Finite difference method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(02)00479-4

The parabolic partial differential equations 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 will be 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.