A note on the numerical solution of non-linear parabolic equations in chebyshev series

A new method for the numerical solution of non linear parabolic equations is presented. The method is an extension of an existing algorithm for linear equations. Solutions are obtained in the form of a Chebyshev series, which is produced by approximating the partial differential equation by a set of ordinary differential equations over a small time interval. The method appears to be both accurate and economical.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.     

Coefficients perturbation methods for higher-order differential equations

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.     

Optimality of an error estimate of a one-step numerical integration method for certain initial value problems

In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.     

Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.