Mehmet Sezer

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.

An improved Bessel collocation method with a residual error function to solve a class of Lane-Emden differential equations

In this study, the modified Bessel collocation method is presented to obtain the approximate solutions of the linear Lane-Emden differential equations. The method is based on the improvement of the Bessel polynomial solutions with the aid of the residual error function. First, the Bessel collocation method is applied to the linear Lane-Emden differential equations and thus the Bessel polynomial solutions are obtained. Second, an error problem is constructed by means of the residual error function and this error problem is solved by using the Bessel collocation method. By summing the Bessel polynomial solutions of the original problem and the error problem, we have the improved Bessel polynomial solutions. When the exact solution of the problem is not known, the absolute errors can be approximately computed by the Bessel polynomial solution of the error problem. In addition, examples that illustrate the pertinent features of the method are presented, and the results of this investigation are discussed.

Laguerre polynomial approach for solving Lane-Emden type functional differential equations

In this paper, a numerical method, which is called the Laguerre collocation method, for the approximate solution of Lane–Emden type functional differential equations in terms of Laguerre polynomials are derived. The method is based on the matrix relations of Laguerre polynomials and their derivatives, and reduces the solution of the Lane–Emden type functional differential equation to the solution of a matrix equation corresponding to system of algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples are included to demonstrate the validity and applicability of the proposed method.

Chelyshkov collocation method for a class of mixed functional integro-differential equations

In this study, a numerical matrix method based on Chelyshkov polynomials is presented to solve the linear functional integro-differential equations with variable coefficients under the initial-boundary conditions. This method transforms the functional equation to a matrix equation by means of collocation points. Also, using the residual function and Mean Value Theorem, an error analysis technique is developed. Some numerical examples are performed to illustrate the accuracy and applicability of the method.

A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations

In this study, an approximate method based on Bernoulli polynomials and collocation points has been presented to obtain the solution of higher order linear Fredholm integro-differential-difference equations with the mixed conditions. The method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations. The obtained matrix equation is based on the matrix forms of Bernoulli polynomials and their derivatives by means of collocations. The solutions are obtained as the truncated Bernoulli series which are defined in the interval [a,b]. To illustrate the method, it is applied to the initial and boundary values. Also error analysis and numerical examples are included to demonstrate the validity and applicability of the technique.

Hybrid Euler-Taylor matrix method for solving of generalized linear Fredholm integro-differential difference equations

The main purpose of this paper is to present a numerical method to solve the linear Fredholm integro-differential difference equations with constant argument under initial-boundary conditions. The proposed method is based on the Euler polynomials and collocation points and reduces the integro-differential difference equation to a system of algebraic equations. For the given method, we develop the error analysis related with residual function. Also, we present illustrative examples to demonstrate the validity and applicability of the technique.

A numerical approach with error estimation to solve general integro-differential-difference equations using Dickson polynomials

In this paper, a matrix method based on the Dickson polynomials and collocation points is introduced for the numerical solution of linear integro-differential-difference equations with variable coefficients under the mixed conditions. In addition, in order to improve the numerical solution, an error analysis technique relating to residual functions is performed. Some linear and nonlinear numerical examples are given to illustrate the accuracy and applicability of the method. Eventually, the obtained results are discussed according to the parameter-α of Dickson polynomials and the residual error estimation.

Laguerre Collocation Method for Solving Fredholm Integro-Differential Equations with Functional Arguments

Laguerre collocation method is applied for solving a class of the Fredholm integro-differential equations with functional arguments. This method transforms the considered problem to a matrix equation which corresponds to a system of linear algebraic equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, the approximate solutions are corrected by using the residual correction method.

Orthoexponential polynomial solutions of delay pantograph differential equations with residual error estimation

In this paper, a new matrix method based on orthogonal exponential (orthoexponential) polynomials and collocation points is proposed to solve the high-order linear delay differential equations with linear functional arguments under the mixed conditions. The convenience is that orthoexponential polynomials have shown to be effective in approximating a given function, fast and efficiently. An error analysis technique based on residual function is developed and applied to four problems to demonstrate the validity and applicability of the proposed method. It is confirmed that the present method yields quite acceptable results and the accuracy of the solution can significantly be increased by error correction and residual function.

A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays

In this paper, the Taylor collocation method has been used the integro functional equation with variable bounds. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. We have introduced the method to solve the functional integral equations with variable bounds. We have also improved error analysis for this method by using the residual function to estimate the absolute errors. To illustrate the pertinent features of the method numeric examples are presented and results are compared with the other methods. All numerical computations have been performed on the computer algebraic system Maple 15.