High-order Linear Fredholm Integro-differential Equations Research Articles

Numerical Solution of High-Order Linear Fredholm Integro-Differential Equations by Lucas Collocation Method

In this paper, a useful matrix approach for high-order linear Fredholm integro-differential equations with initial boundary conditions expressed as Lucas polynomials is proposed. Using a matrix equationwhich is equivalent to a set of linear algebraic equations the method transforms to integro-differential equation. When compared to other methods that have been proposed in the literature, the numerical results from the suggested technique reveal that it is effective and promising. And also, error estimation of the scheme was derived. These results were compared with the exact solutions and the other numerical methods to the tested problems.

AN IMPROVED APPROACH FOR SOLUTIONS OF SYSTEMS OF LINEAR FREDHOLM INTEGRO DIFFERENTIAL EQUATIONS

In this paper, a numerical matrix method is used to solve the systems of high-order linear Fredholm integro-differential equations with variable coefficients under mixed conditions. The technique consists of collocation points and the Morgan-Voyce polynomials. The residual error functions of numerical solutions of the method are also presented. Firstly, the approximate solutions are formed and secondly, an error problem is constituted in favor of the residual error function. The numerical solutions are computed for this error problem by using the present method. The approximate solutions of the original problem and the error problem are the corrected Morgan-Voyce polynomial solutions. When the exact solutions of the problem are not known, the absolute errors can be approximately constructed through the approximate solutions of the error problem. Numerical examples are included to demonstrate the validity and the applicability of the technique, and also the results are compared with the different methods. All numerical computations have been performed using MATLAB v7.11.0 (R2010b).

High-order nonlinear Volterra–Fredholm-Hammerstein integro-differential equations and their effective computation

In a recent study (Turkyilmazoglu, 2014) high-order linear Fredholm integro-differential equations were solved by means of an elegant and accurate effective technique. This approach is extended here to obtain exact and analytic approximate solutions of high-order nonlinear Volterra–Fredholm–Hammerstein integro-differential equations. Difference from the earlier work is that the method involves solution of nonlinear algebraic equations in place of linear ones. High accuracy of the introduced technique is also observed for the nonlinear equations via comparisons with some of the available methods.

An effective approach for numerical solutions of high-order Fredholm integro-differential equations

We propose an effective method to solve high-order linear Fredholm integro-differential equations having a weak or strong kernel. The target is to construct fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on the power series representation via ordinary polynomials. Employing such polynomials leads to algebraic equations to be solved regarding the treated integro-differential equations. A mathematical proof for the numerical analysis is also provided. In practice, the convergence of the power series solution can be easily pursued via the ratio test. Exact solutions are obtained when the solutions are themselves polynomials. Better accuracies are also achieved within the method by increasing the number of polynomials. The introduced approach is applied to already worked problems in the literature by means of different numerical methods. Comparisons clearly show that our scheme is better and even more superior as compared to the existing ones.

A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals

This article develops an efficient direct solver for solving numerically the high-order linear Fredholm integro-differential equations (FIDEs) with piecewise intervals under initial-boundary conditions. A Bernoulli matrix approach is implemented for solving linear and nonlinear FIDEs with piecewise intervals under initial boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. A small number of Bernoulli polynomials is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for solving FIDEs with piecewise intervals.

A Chebyshev Method for a Class of High-Order Linear Fredholm Integro-Differential Equations

A Chebyshev Method for a Class of High-Order Linear Fredholm Integro-Differential Equations

Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases

In this paper, a numerical matrix method based on collocation points is presented for the approximate solution of the systems of high-order linear Fredholm integro-differential equations with variable coefficients under the mixed conditions in terms of the Bessel polynomials. Numerical examples are included to demonstrate the validity and the applicability of the technique and also the results are compared with the different methods. The results show the efficiently and the accuracy of the present work. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).

Taylor polynomial solution of hyperbolic type partial differential equations with constant coefficients

The purpose of this study is to give a Taylor polynomial approximation for the solution of hyperbolic type partial differential equations with constant coefficients. The technique used is an improved Taylor matrix method, which has been given for solving ordinary differential, integral and integro-differential equations [M. Gülsu and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82(5) (2005), pp. 629–642; M. Gülsu and M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput. 188 (2007), pp. 446–449; A. Karamete and M. Sezer, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math. 79(9) (2002), pp. 987–1000; N. Kurt and M. Çevik, Polynomial solution of the single degree of freedom system by Taylor matrix method, Mech. Res. Commun. 35 (2008), pp. 530–536; N. Kurt and M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Inst. 345 (2008), pp. 839–850; Ş. Nas, S. Yalçinbaş, and M. Sezer, A method for approximate solution of the high-order linear Fredholm integro-differential equations, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Edu. Sci. Technol. 25(5) (1994), pp. 625–633; M. Sezer, A method for approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Edu. Sci. Technol. 27(6) (1996), pp. 821–834; M. Sezer, M. Gülsu, and B. Tanay, A matrix method for solving high-order linear difference equations with mixed argument using hybrid Legendre and Taylor polynomials, J. Franklin Inst. 343 (2006), pp. 647–659; S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equation, Appl. Math. Comput. 127 (2002), pp. 196–206; S. Yalçinbaş and M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput. 112 (2000), pp. 291–308]. Some numerical examples, which consist of initial and boundary conditions, are given to illustrate the reliability and efficiency of the method. Also, the results obtained are compared by the known results; the error analysis is performed and the accuracy of the solution is shown.

A Hermite collocation method for the approximate solutions of high-order linear Fredholm integro-differential equations

In this study, a Hermite matrix method is presented to solve high-order linear Fredholm integro-differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011

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

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.

Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients

In this study, a practical matrix method is presented to find an approximate solution of high-order linear Fredholm integro-differential equations with constant coefficients 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.

A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations in the most general form

A Taylor method is developed for finding the approximate solution of high-order linear Fredholm integro-differential equations in the most general form under the mixed conditions. The problem is defined on the interval [−1, 1] and the solution is obtained in terms of Taylor polynomials about the origin. Transforming the interval [a, b] to the interval [−1, 1], a problem defined on [a, b] can also be solved using this method. Numerical examples are presented to illustrate the accuracy of the method.

A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations

ln this study, a matrix method is presented to solve (approximately) high-order linear Fredholm integro-differential equations under mixed conditions in terms of Taylor polynomials about any point. Also, the method can be applied to high-order linear differential and the Fredholm integral equations. To illustrate the pertinent features of the method, examples are presented and results compared.

A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations

A Taylor polynomial approach for solving high-order linear Fredholm integro-differential equations