Improved polynomial approximations for the solution of nonlinear integral equations

AbstractIn this paper, the solutions of nonlinear integral equations, including Volterra, Fredholm, Volterra–Fredholm of first and second kinds, are approximated as a linear combination of some basic functions. The unknown parameters of an approximate solution are obtained based on minimization of the residual function. In addition, the existence and convergence of these approximate solutions are investigated. In order to use Newton’s method for minimization of the residual function, a suitable initial point will be introduced. Moreover, to confirm the efficiency and accuracy of the proposed method, some numerical examples are presented. It is shown that there are considerable improvements in our results compared with the results of the existing methods. All numerical computations have been performed on a personal computer using Maple 12

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique

A numerical method for functional Hammerstein integro-differential equations

In this paper, a numerical method is presented to solve functional Hammerstein integro-differential equations. The presented method combines the successive approximations method with trapezoidal quadrature rule and natural cubic spline interpolation to solve the mentioned equations. The existence and uniqueness of the problem is also investigated. The convergence and numerical stability of the problem are proved, and finally, the accuracy of the method is verified by presenting some numerical computations

A Computational Method for System of Linear Fredholm Integral Equations

This paper focuses on developing a numerical method based on a cubic spline approach for the solution of system of linear Fredholm equations of the second kind. This method produces a system of algebraic equations. The efficiency and accuracy of the method are demonstrated by a numerical example and the mathematical software Matlab R2010a was used to carry out the necessary computations. Keywords: System of linear Fredholm integral equations, natural cubic splin

Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions

This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods

Chebyshev cardinal functions for solving volterra-fredholm integro- differential equations using operational matrices

Abstract In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique

The Generalized Laguerre Matrix Method or Solving Linear Differential-Difference Equations with Variable Coefficients

In this paper, a new and efficient approach based on the generalized Laguerre matrix method for numerical approximation of the linear differential-difference equations (DDEs) with variable coefficients is introduced. Explicit formulae which express the generalized Laguerre expansion coefficients for the moments of the derivatives of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. In the scheme, by using this approach we reduce solving the linear differential equations to solving a system of linear algebraic equations, thus greatly simplify the problem. In addition, several numerical experiments are given to demonstrate the validity and applicability of the method

COLLOCATION METHOD FOR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS NEHZAT EBRAHIMI a1 AND JALIL RASHIDINIA

ABSTRACT This paper introduces an approach for obtaining the numerical solution of the linear and nonlinear Volterra-Fredholm integro-differential equations based on quintic B-spline functions.The solution is collocated by quintic B-spline and then the integrand is approximated by 5-points Gauss-Tur´an quadrature formula with respect to the Legendre weight function.The main characteristic of this approach is that it reduces linear and nonlinear Volterra -Fredholm integro-differential equations to a system of algebraic equations, which greatly simplifying the problem. The error analysis of proposed numerical method is studied theoretically. Numerical examples illustrate the validity and applicability of the proposed method