Numerical techniques for a parabolic equation subject to an overspecified boundary condition

• Published in: Applied mathematics and computation Vol. 132; no. 2; pp. 299 - 313
• Main Author: Dehghan, Mehdi
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 10.11.2002; Elsevier
• Subjects: Control parameter; Difference and functional equations, recurrence relations; Energy overspecification; Exact sciences and technology; Inverse problem; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical differentiation; Numerical integration–The cubic spline approximation; Ordinary differential equations; Pade approximant; Parabolic partial differential equations; Parallel schemes; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; The degree of accuracy; Numerical integration–The cubic spline approximation; Control parameter; Parallel schemes; Energy overspecification; Numerical differentiation; Pade approximant; The degree of accuracy; Inverse problem; Parabolic partial differential equations; Transcendental equation; Parallel algorithm; Semi linear equation; Differential equation; Recurrence relation; Initial value problem; Boundary condition; Difference equation; Exponential function; Matrix function; Partial differential equation; Cubic spline; Parabolic equation; Boundary value problem; Linear equation; Sequential method; Diffusion equation; Padé approximant; Recurrence equation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(01)00194-1

An inverse problem concerning diffusion equation with a control parameter is considered. A new numerical scheme is developed for obtaining approximate solutions to an initial-boundary-value problem for the semi-linear parabolic partial differential equation subject to an overspecified boundary condition. The space derivative in the PDE is approximated by finite difference replacements. The solution of the resulting system of first-order ordinary differential equations (ODEs) satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time can be controlled by choosing several Pade approximants to replace this matrix exponential term. A numerical technique is developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The results of a numerical experiment are presented, and the accuracy and Central Processor (CPU) time needed for this inverse problem are discussed.