An Improved Method for the Solution of the Heat Equation in Chebyshev Series

A method is given for the numerical solution of the heat equationusing Chebyshev series. A feature of the method is that thematrices which have to be inverted (apart possibly from a few22 matrices) are all triangular. The method shows a considerableimprovement in accuracy over previous Chebyshev methods, particularlyas far as the approximation to the fundamental eigenvalue isconcerned. The extension of the method to equations of the form is also described.     

Chebyshev Methods for the Numerical Solution of Parabolic Partial Differential Equations in Two and Three Space Variables

Chebyshev methods for the numerical solution of parabolic partialdifferential equations in a region which can be transformedto either a square or a circular cylinder are developed. Theseprocedures are an extension of the method of Knibb & Scraton(1971). To illustrate the technique the solution of the heatconduction equation within an elliptical region is consideredin detail. The Chebyshev method given for this problem requiresconsiderably less computer time than the method of Dew &Scraton (1973). In the case when the space operators commutea highly efficient alternating direction Chebyshev method isgiven.     

A Generalized Chebyshev Method for Non-linear Parabolic Equations in One Space Variable

The derivation and implementation of a generalized Chebyshevmethod is described for the numerical solution of non-linearparabolic equations in one space dimension. The solution isobtained by using the method of lines and is approximated inthe space variable by piecewise Chebyshev polynomial expansions.These expansions are normally few in number and of high order.It is shown that the method can be derived from a perturbedform of the original equation. A numerical example is givento illustrate its performance compared with the finite elementand finite difference method. A comparison of various Chebyshev methods is made by applyingthem to two-point eigenproblems. It is shown by analysis andnumerical examples that the approach used to derive the generalizedChebyshev method is comparable, in terms of the accuracy obtained,with existing Chebyshev methods.     

A Note on the Numerical Solution of Quasi-linear Parabolic Partial Differential Equations with Error Estimates

A method is considered for the numerical solution of quasi-linearpartial differential equations. The partial differential equationis reduced to a set of ordinary differential equations usinga Chebyshev series expansion. The exact solution of this setof ordinary differential equations is shown to be the solutionof a perturbed form of the original equation. This enables errorestimates to be found for linear and mildly non-linear problems.     

The Solution of the Heat Equation in Two Space Variables using Chebyshev Series

A numerical solution of the heat conduction equation within a closed curve, is obtained in the form of a combinedFourier/Chebyshev series. The method is an extension of themethods for the one-dimensional heat equation given by Knibb& Scraton (1971) and Dew & Scraton (1972).