Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

A collocation method to solve higher order linear complex differential equations in rectangular domains

In this article, a collocation method is developed to find an approximate solution of higher order linear complex differential equations with variable coefficients in rectangular domains. This method is essentially based on the matrix representations of the truncated Taylor series of the expressions in equation and their derivates, which consist of collocation points defined in the given domain. Some numerical examples with initial and boundary conditions are given to show the properties of the method. All results were computed using a program written in scientific WorkPlace v5.5 and Maple v12. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Implicit Extension of Taylor Series Method with Numerical Derivatives

The Taylor series method is one of the earliest analytic‐numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher order derivatives using well‐known technique for the partial differential equations. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which led to the implicit R‐K algorithms [1].     

Numerical solution of a class of complex differential equations by the Taylor collocation method in elliptic domains

An approximate method for solving higher‐order linear complex differential equations in elliptic domains is proposed. The approach is based on a Taylor collocation method, which consists of the matrix represantation of expressions in the differential equation and the collocation points defined in an elliptic domain. Illustrative examples are included to demonstrate the validity and applicability of the technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

High order compact solution of the one‐space‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving one‐space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high‐order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one‐space‐dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of (Mohanty, 3 .© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

High order implicit collocation method for the solution of two‐dimensional linear hyperbolic equation

In this article, we introduce a high‐order accurate method for solving the two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives of linear hyperbolic equation and collocation method for the time component. The resulted method is unconditionally stable and solves the two‐dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. Numerical results show that the compact finite difference approximation of fourth order and collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A collocation approach for solving linear complex differential equations in rectangular domains

In this paper, a collocation method is presented to find the approximate solution of high‐order linear complex differential equations in rectangular domain. By using collocation points defined in a rectangular domain and the Bessel polynomials, this method transforms the linear complex differential equations into a matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. The proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on a computer using a program written in MATLAB v7.6.0 (R2008a). Copyright © 2012 John Wiley & Sons, Ltd.     

A Taylor polynomial approach for solving the most general linear Fredholm integro‐differential‐difference equations

In this study, a matrix method is developed to solve approximately the most general higher order linear Fredholm integro‐differential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This technique reduces the problem into the linear algebraic system. The method is valid for any combination of differential, difference and integral equations. An initial value problem and a boundary value problem are also presented to illustrate the accuracy and efficiency of the method. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical approach for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13 and we have the coefficients of Taylor expansion. In addition, numerical results are presented to demonstrate the effectiveness of the proposed method.     

A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation

A Taylor matrix method is proposed for the numerical solution of the two-space-dimensional linear hyperbolic equation. This method transforms the equation into a matrix equation and the unknown of this equation is a Taylor coefficients matrix. Solutions are easily acquired by using this matrix equation, which corresponds to a system of linear algebraic equations. As a result, the finite Taylor series approach with three variables is obtained. The accuracy of the proposed method is demonstrated with one example.     

A new numerical method for delay and advanced integro-differential equations

A general formulation is constructed for Jacobi operational matrices of integration, product, and delay on an arbitrary interval. The main purpose of this study is to improve Jacobi operational matrices for solving delay or advanced integro–differential equations. Some theorems are established and utilized to reduce the computational costs. All algorithms can be used for both linear and nonlinear Fredholm and Volterra integro-differential equations with delay. An error estimator is introduced to approximate the absolute error when the exact solution of a given problem is not available. The error of the proposed method is less compared to other common methods such as the Taylor collocation, Chebyshev collocation, hybrid Euler–Taylor matrix, and Boubaker collocation methods. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of delay integro‐differential equations by using Taylor collocation method

This paper is concerned with the numerical solution of delay integro‐differential equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Taylor polynomials for the numerical solution of delay integro‐differential equations. It is shown that this method is convergent. Numerical illustrations confirm our theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

A new Euler matrix method for solving systems of linear Volterra integral equations with variable coefficients

This article develops an efficient solver based on collocation points for solving numerically a system of linear Volterra integral equations (VIEs) with variable coefficients. By using the Euler polynomials and the collocation points, this method transforms the system of linear VIEs into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Euler coefficients. A small number of Euler polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving VIEs with variable coefficients.     

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution of high‐order linear Fredholm integro‐differential equations with piecewise intervals

In this study, a practical matrix method is presented to find an approximate solution for high‐order linear Fredholm integro‐differential equations with piecewise intervals under the initial boundary conditions in terms of Taylor polynomials. The method converts the integro differential equation to a matrix equation, which corresponds to a system of linear algebraic equations. Error analysis and illustrative examples are included to demonstrate the validity and applicability of the technique. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 27: 1327–1339, 2011     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.     

On a least-squares collocation method for linear differential-algebraic equations

Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.     

A new collocation method for solution of mixed linear integro-differential-difference equations

Numerical solution of mixed linear integro-differential-difference equation is presented using Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving mixed linear integro-differential-difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms mixed linear integro-differential-difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple10.     

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

In this paper, we consider a piecewise linear collocation method for the solution of a pseudo‐differential equation of order r=0, ?1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three‐point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low‐order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low‐order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]³) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N^?(2?r)/2). Note that, in contrast to well‐known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón–Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd.