Volterra Fredholm Research Articles

A semi-analytical approach to solve integro-differential equations

In this work, we present a semi-analytical method for solving integro-differential equations under multi-point or mixed boundary conditions. The proposed method solves linear and nonlinear Fredholm–Volterra integro-differential equations. A convergence analysis of the proposed method is directly examined. Numerical examples are worked out to demonstrate the main results. Moreover, proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme. We show that with a few number of obtained approximating terms, we achieve a high accuracy level of the obtained results. However, increasing the number of approximating terms, yields a significant decrease of the error of the approximation. The proposed method is very useful, reliable, and flexible for solving different kinds of integro-differential equations.

Numerical method for a system of integro-differential equations by Lagrange interpolation

In this paper, we present the Lagrange polynomial solutions to system of higher-order linear integro-differential Volterra–Fredholm equations (IDVFE). This method transforms the IDVFE into the matrix equations which is converted to a system of linear algebraic equations. Some numerical results are given to illustrate the efficiency of the method.

Solution of systems of integro-differential equations using numerical treatment of fixed point

We approximate the solution of a system of nonlinear mixed Fredholm–Volterra integro-differential equations of the second kind, using fixed point techniques and Schauder bases in certain Banach spaces. The convergence and error are studied. Several numerical examples are given, and the obtained numerical approximations are compared with the corresponding exact solutions.

Solving system of Volterra–Fredholm integral equations with Bernstein polynomials and hybrid Bernstein Block-Pulse functions

This work approximates the unknown functions based on the Bernstein polynomials and hybrid Bernstein Block-Pulse functions, in conjunction with the collocation method for the numerical solution of system of Fredholm–Volterra integral equations. In both methods, the system of integral equations is approximated by the Gauss quadrature formula with respect to the Legendre weight function. The proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods. Moreover, the convergence analysis of these algorithms will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods.

Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three‐dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical solution of nonlinear integral equations using alternative Legendre polynomials

In this paper, the operational matrices of integration and the product for the alternative Legendre polynomials (ALPs) are first derived. Then, using these operational matrices and the collocation method, the nonlinear Volterra–Fredholm–Hammerstein integral equations are reduced to a set of nonlinear algebraic equations with unknown ALP coefficients. Some error estimations are provided and the efficiency and accuracy is verified by applying the method to some examples chosen from other literature.

Sinc-Galerkin Technique for the Numerical Solution of Fractional Volterra–Fredholm Integro-Differential Equations with Weakly Singular Kernels

AbstractThe sinc-Galerkin method is developed to approximate the solution of fractional Volterra–Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the sinc function approximation. Usually, this type of integral equations is very difficult to solve analytically as well as numerically. The present method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by sinc-Galerkin method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method.

The numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis

In this paper, we present a numerical method for solving two-dimensional nonlinear Volterra–Fredholm integral equations of the second kind. The method approximates the solution by the discrete collocation method based on radial basis functions (RBFs) constructed on a set of disordered data. The proposed method is meshless, since it does not require any background mesh or domain elements. Error analysis of this method is also investigated. Numerical examples which compare the proposed method with 2D-TFs method [4] approve its supremacy in terms of accuracy and computational cost. Using various RBFs we have concluded that MQ-RBF is the best choice for the proposed method.

A numerical approach for solving nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

In this paper, a numerical approximation based on Legendre wavelets has been developed to solve nonlinear fractional Volterra–Fredholm integro-differential equations. Legendre wavelets are generated by dilation and translation of Legendre polynomials. The properties of the Legendre wavelets are presented in the paper. The proposed wavelet method transforms the integral equations to a system of nonlinear algebraic equations and this algebraic system has been solved numerically by Newton’s method. Convergence analysis of the proposed method has been discussed in this paper. Some examples have been illustrated to show the applicability and accuracy of the present method.

Evolutionary computational intelligence in solving a class of nonlinear Volterra–Fredholm integro-differential equations

In this paper, a stochastic computational intelligence technique for solving a class of nonlinear Volterra–Fredholm integro-differential equations with mixed conditions is presented. The strength of feed forward artificial neural networks is used to accurately model the integro-equation. Comparisons with the exact solution and other numerical techniques are presented to show the efficiency of the proposed method. Theoretical and numerical results are presented. Analysis for the presented method is given.

Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via Tau-collocation method with convergence analysis

In this paper, we consider the nonlinear Volterra–Fredholm–Hammerstein integral equations. The approximate solution for the nonlinear Volterra–Fredholm–Hammerstein integral equations is obtained by using the Tau-Collocation method. To do this, the nonlinear Volterra–Fredholm–Hammerstein integral equations is transformed into a system of nonlinear algebraic equations in matrix form. Thus by solving this system unknown coefficients are obtained. The spectral rate of convergence for the proposed method is established in the L2-norm. The numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the Tau-collocation method is of high accuracy, more convenient and efficient for solving nonlinear Volterra–Fredholm–Hammerstein integral equations.

Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra–Fredholm integral equations

In this paper, we used the three-dimensional triangular functions (3D-TFs) for the numerical solution of three-dimensional nonlinear mixed Volterra–Fredholm integral equations. First, 3D-TFs and their properties are described. Then the properties of 3D-TFs together with their operational matrix are used to reduce the problem to a nonlinear system of algebraic equations. Furthermore, existence and uniqueness of the solution of three-dimensional nonlinear mixed Volterra–Fredholm integral equations are proved. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique. Also, some interesting comparisons between proposed method, block-pulse functions (BPFs) method and modified block-pulse functions (MBPFs) method are presented.

Numerical study of stochastic Volterra–Fredholm integral equations by using second kind Chebyshev wavelets

Abstract In this paper, we present a computational method for solving stochastic Volterra–Fredholm integral equations which is based on the second kind Chebyshev wavelets and their stochastic operational matrix. Convergence and error analysis of the proposed method are investigated. Numerical results are compared with the block pulse functions method for some non-trivial examples. The obtained results reveal efficiency and reliability of the proposed wavelet method.

Numerical solution of Volterra–Fredholm integral equations via modification of hat functions

In this paper, a new numerical approach is developed for solving linear and nonlinear Volterra–Fredholm integral equations. The fundamental structure of the presented method is based on the modification of hat functions. The properties of modification of hat functions (MHFs) are first presented. After implementation of our scheme, the solution of the main problem would be transformed into the solution of a system of linear or nonlinear algebraic equations. Also, an error analysis is provided under several mild conditions. In addition, examples are presented to illustrate the pertinent features of the method and the results are discussed.

Solving nonlinear Fredholm Volterra integro-differential equation by the $(G'/G)$ -expansion method

In this paper, we apply $(G'/G)$ -expansion methods for solving the interested nonlinear Fredholm-Volterra integro-differential equation.

Numerical solution of Volterra–Fredholm integral equations based on [formula omitted]-SVR method

In this paper, we try to study the numerical methods for solving integral equations from a new perspective—machine learning method. By means of the idea of kernel ε-support vector regression machine (ε-SVR), we construct an optimization modeling for a class of Volterra–Fredholm integral equations and propose a novel numerical method for solving them. The proposed method has a certain versatility and can be used to solve some other kinds of integral equations. In order to verify the effectiveness of the proposed method, we perform a series of comparative experiments with six specific Volterra–Fredholm integral equations and a method proposed in Wang et al. (2014). Experimental results show that the proposed method has a good approximation property.

Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations

In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation form in the Hilbert space \( W_{2}^{2} \left( \varOmega \right) \oplus W_{2}^{2} \left( \varOmega \right) \). Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.

Application of Fibonacci collocation method for solving Volterra–Fredholm integral equations

In this paper, a new matrix method based on Fibonacci polynomials and collocation points is proposed for numerically solving the Volterra–Fredholm integral equations. In fact, the approximate solution of the problem in the truncated Fibonacci series form is obtained by this method. Also, convergence analysis of the proposed method is provided under several mild conditions. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

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.

A new numerical method for solving two-dimensional Volterra–Fredholm integral equations

In this paper, a numerical method for solving nonlinear two-dimensional Volterra–Fredholm integral equations is presented. The approximate solution is expressed as expansion of two-dimensional delta basis functions (2D-DFs). Afterward, using the properties of 2D-DFs and their operational matrix of integration together with collocation method the numerical solution of these equations is reduced to the solution of a nonlinear system of algebraic equations. Moreover, it is proved in a theorem that the method is convergence and error is \(O(h^2)\). Furthermore, error analysis of the proposed method is provided under several mild conditions. Finally, the effectiveness of the method is illustrated in some numerical experiments.