A Numerical Method for Fredholm Integro-Differential Equation using Positive Definite Radial Kernels and A Study of the Effect of The Shape Parameter (original) (raw)

Solving Fredholm integro–differential equations using reproducing kernel Hilbert space method

2013

In this study, the numerical solution of Fredholm integro-differential equation is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the initial condition of the problem is satisfied. The exact solution u x ð Þ is represented in the form of series in the space W 2 2 ½a; b. In the mean time, the n-term approxima te solution u n ðxÞ is obtained and is proved to converge to the exact solution uðxÞ. Furthermore, we present an iterative method for obtaining the solution in the space W 2 2 ½a; b. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear Fredholm integro-differential equations.

A Sinc–Collocation method for the linear Fredholm integro-differential equations

Zeitschrift für angewandte Mathematik und Physik, 2007

A Sinc-Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is O e −k √ N. Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.

Matrix Approach to the Direct Computation Method for the Solution of Fredholm Integro-Differential Equations of the Second Kind with Degenerate Kernels

CAUCHY Journal, 2020

In this paper, a matrix approach to the direct computation method for solving Fredholm Integro-Differential Equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. The result of this research is the solution of the second type Fredholm integro-differential equation (FIDE) with a numerically accurate kernel degenerate. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

Approximate solution of Fredholm Integro Differential equation using Quadrature Formulas methods

International Journal of Scientific Research in Science, Engineering and Technology, 2022

There are two reasons for this research, the first which is the main was to clarify the use of a closed quadrature formulas included ( Trapizoidal , Simpson’s 1/3 rule and Simpson’s 3/8 rule )which are the most familiar formula of numerical integration ,to evaluating the integral part to find the approximate solution of the 2nd kind of FIDE’s of the 1st order and reducing it to linear system of (n) equation with n unknowns of the solution sample value y(ti) ,i=0,1,2,3,…,n .The other reason was to explain the differences between three Quadrature formulas in solving equation according to the specified period , has been clarified through examples. Finally, Acomparison was made between the three methods ,programs for methods were written in MATLAB language and examples with satisfactory results are given .

Solution of System of Fredholm Integro-Differential Equations by RKHS Method

Int. J. Contemp. Math. Sciences, 2013

In this paper, an application of reproducing kernel Hilbert space (RKHS) method is applied to solve system of Fredholm integro-differential equations. The exact solutions are represented in the form of series in the reproducing kernel space. Moreover, the approximate solutions u n (x), v n (x) are proved to converge to the exact solutions u(x), v(x), respectively. The results reveal that the RKHS is simple and effective.

SOLVING FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS BY USING NUMERICAL TECHNIQUES

Kyungnam University Press, 2019

This paper mainly focuses on numerical techniques based on the Adomian Decomposition Method (ADM) and Direct Homotopy Analysis Method (DHAM) for solving Fredholm integro-differential equations of the second kind. The reliability of the methods and reduction in the size of the computational work give this methods wider applicability. Convergence analysis of the exact solution of the proposed methods will be established. Moreover, we proved the uniqueness of the solution. To illustrate the methods, an example is presented.

Legendre Collocation Method for Linear Second Order Fredholm Volterra Integro-Differential Equations

Journal of the Nigerian Mathematical Society, 2020

This paper discusses the development of a new numerical solution of second order linear Fredholm Volterra integro-dffierential equations by Legendre collocation method. The Fredholm Volterra integro-differential equation is rst converted into integral equation and then transformed into linear algebraic equations which are then solved using matrix inversion method. Numerical solution shows that the method gives better accuracy than the existing methods.