Şuayip Yüzbaşı

Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials

In this paper, a collocation method based on the Bernstein polynomials is presented for the fractional Riccati type differential equations. By writing t->t^@a (0<@a<1) in the truncated Bernstein series, the truncated fractional Bernstein series is obtained and then it is transformed into the matrix form. By using Caputo fractional derivative, the matrix forms of the fractional derivatives are constructed for the truncated fractional Bernstein series. We convert each term of the problem to the matrix form by means of the truncated fractional Bernstein series. By using the collocation points, we have the basic matrix equation which corresponds to a system of nonlinear algebraic equations. Lastly, a new system of nonlinear algebraic equations is obtained by using the matrix forms of the conditions and the basic matrix equation. The solution of this system gives the approximate solution for the truncated limited N. Error analysis included the residual error estimation and the upper bound of the absolute errors is introduced for this method. The technique and the error analysis are applied to some problems to demonstrate the validity and applicability of the proposed method.

Laguerre approach for solving pantograph-type Volterra integro-differential equations

Abstract In this paper, a collocation method based on Laguerre polynomials is presented to solve the pantograph-type Volterra integro-differential equations under the initial conditions. By using the Laguerre polynomials, the equally spaced collocation points and the matrix operations, the problem is reduced to a system of algebraic equations. By solving this system, we determine the coefficients of the approximate solution of the main problem. Also, an error estimation for the method is introduced by using the residual function. The approximate solution is corrected in terms of the estimated error function. Finally, we give seven examples for the applications of the method on the problem and compare our results by with existing methods.

Numerical solutions of singularly perturbed one-dimensional parabolic convection-diffusion problems by the Bessel collocation method

In this paper, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. This method is based on the Bessel collocation method used for some problems of ordinary differential equations. In fact, the approximate solution of the problem in the truncated Bessel series form is obtained by this method. By substituting truncated Bessel series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Bessel coefficients can be computed. An error estimation technique is given for the considered problem and the method. To show the accuracy and the efficiency of the method, numerical examples are implemented and the comparisons are given by the other methods.

A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations

In this study, a collocation method that based on Bernstein polynomials is presented for nonlinear Fredholm-Volterra integro-differential equations (NFVIDEs). By means of the collocation method and the matrix operations, the problem is reduced into a system of the nonlinear algebraic equations. The approximate solutions are obtained by solving this nonlinear system. Error analysis is presented for the Bernstein series solutions of the nonlinear Fredholm-Volterra integro-differential equations. Several examples are given to illustrate the efficiency and implementation of the proposed method for solving the NFVIDEs. Comparisons are made to confirm the reliability of the method. Also error analysis is applied for the numerical examples.

Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation

In this study, the Bessel collocation method is presented for the solutions of system of linear Fredholm–Volterra integro-differential equations which includes the derivatives of unknown functions in integral parts. The Bessel collocation method transforms the problem into a system of linear algebraic equations by means of the Bessel functions of first kind, the collocation points and the matrix relations. Also, an error estimation is given for the considered problem and the method. Illustrative examples are presented to show efficiency of method and the comparisons are made with the results of other methods. All of numerical calculations have been made on a computer using a program written in Matlab.

Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction

In this study, a collocation method is presented to solve one-dimensional hyperbolic telegraph equation. The problem is given by hyperbolic telegraph equation under initial and boundary conditions. The method is based on the Bessel functions of the first kind. Using the collocation points and the operational matrices of derivatives, we reduce the problem to a set of linear algebraic equations. The determined coefficients from this system give the coefficients of the approximate solution. Also, an error estimation method is presented for the considered problem and the method. By using the residual function and the original problem, an error problem is constructed and thus the error function is estimated. By aid of the estimated function, the approximated solution is improved. Numerical examples are given to demonstrate the validity and applicability of the proposed method and also, the comparisons are made with the known results.

A collocation method to find solutions of linear complex differential equations in circular domains

In this study, we introduce a collocation approach for solving high-order linear complex differential equations in circular domain. By using collocation points defined in a circular domain and Bessel functions of the first kind, 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. Proposed method gives the analytic solution when the exact solutions are polynomials. Numerical examples are given to demonstrate the validity and applicability of the technique and the comparisons are made with existing results. The results obtained from the examples demonstrate the efficiency and accuracy of the present work. All of the numerical computations have been computed on computer using a code written in Matlab.

Exponential Collocation Method for Solutions of Singularly Perturbed Delay Differential Equations

This paper deals with the singularly perturbed delay differential equations under boundary conditions. A numerical approximation based on the exponential functions is proposed to solve the singularly perturbed delay differential equations. By aid of the collocation points and the matrix operations, the suggested scheme converts singularly perturbed problem into a matrix equation, and this matrix equation corresponds to a system of linear algebraic equations. Also, an error analysis technique based on the residual function is introduced for the method. Four examples are considered to demonstrate the performance of the proposed scheme, and the results are discussed.

A numerical scheme for solutions of a class of nonlinear differential equations

Abstract In this paper, a collocation method based on Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. First, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Second, by using these matrix forms, collocation points and the matrix operations, a nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of the assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and comparisons are made with existing results. The results show the efficiency and accuracy of the present work.

A Numerical Method for Solving Second-Order Linear Partial Differential Equations Under Dirichlet, Neumann and Robin Boundary Conditions

The aim of this paper is to give a collocation method to solve second-order partial differential equations with variable coefficients under Dirichlet, Neumann and Robin boundary conditions. By using the Bessel functions of the first kind, the matrix operations and the collocation points, the method is constructed and it transforms the partial differential equation problem into a system of algebraic equations. The unknown coefficients of the assuming solution are determined by solving this system. The algorithm of the proposed method is presented. Also, error estimation technique is introduced and the approximate solutions are improved by means of it. To show the validity and applicability of the presented method, we solve numerical examples and give the comparison of solutions and comparisons of the errors (actual and estimation).