MODIFIED QUASI SIMPS0N 'S 3/8 RULE FOR SOLVING SYSTEM OF INTEGRAL EQUATION OF THE SECOND KIND LINEAR

Actually, it is possible to solve systems of integral equation by using many approaches. However, in this study, the modified quasi Simpson's 3/8 rule used to find the numerical solution of a system of linear Volterra integral equations of the second kind. This method solves systems of linear Volterra integral equations of the second kind in more accurate way than the modified trapezoidal rule. Some indicative examples are given to elaborate the accuracy of this method

Numerical Methods for Integral Equations

We first propose a multiscale Galerkin method for solving the Volterra integral equations of the second kind with a weakly singular kernel. Due to the special structure of Volterra integral equations and the ``shrinking support property of multiscale basis functions, a large number of entries of the coefficient matrix appearing in the resulting discrete linear system are zeros. This result, combined with a truncation scheme of the coefficient matrix, leads to a fast numerical solution of the integral equation. A quadrature method is designed especially for the weakly singular kernel involved inside the integral operator to compute the nonzero entries of the compressed matrix so that the quadrature errors will not ruin the overall convergence order of the approximate solution of the integral equation. We estimate the computational cost of this numerical method and its approximate accuracy. Numerical experiments are presented to demonstrate the performance of the proposed method. We also exploit two methods based on neural network models and the collocation method in solving the linear Fredholm integral equations of the second kind. For the first neural network (NN) model, we cast the problem of solving an integral equation as a data fitting problem on a finite set, which gives rise to an optimization problem. In the second method, which is referred to as the NN-Collocation model, we first choose the polynomial space as the projection space of the Collocation method, then approximate the solution of the integral equation by a linear combination of polynomials in that space. The coefficients of the linear combination are served as the weights between the hidden layer and the output layer of the neural network. We train both neural network models using gradient descent with Adam optimizer. Finally, we compare the performances of the two methods and find that the NN-Collocation model offers a more stable, accurate, and efficient solution

Approximate solution for the system of Non-linear Volterra integral equations of the second kind by using block-by-block method,”

Abstract: The aim of this paper is for finding the numerical solution (sometimes exact) for non-linear system of Volterra integral equations of the second kind (NSVIEK2) by using block-by-block method. W hich avoid the need for special starting procedures, but uses numerical quadrature rule. Also some illustrative examples are presented, to elucidate the accuracy of this method

Solving Volterra-Fredholm integral equations by natural cubic spline function

Using the natural cubic spline function, this paper finds the numerical solution of Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the natural cubic spline function of the unknown function at an arbitrary point and using the integration method to turn the VolterraFredholm integral equation into a system of linear equations concerning to the unknown function. An approximate solution can be easily established by solving the given system. This is accomplished with the help of a computer program that runs on Python 3.9

Cooperating Newton’s Method with Series Solution Method for Solving System of Linear Mixed Volterra-Fredholm Integral Equation of the Second Kind

In this paper, for the 1st time, we use Newton’s method with series solution method (SSM) for solving system of linear mixed Volterra-Fredholm integral equations of the second kind (SLMVFIE-2). In this work, we use a new technique for studying the numerical solutions for SLMVFIE-2 which is Newton’s method with SSM, first solving this system using SSM and after that cooperation Newton’s method with SSM, suggesting an algorithm for the technique. The new results are achieved and some new theorems have proved for convergence of the method, several numerical examples are tested for illustrating the usefulness of the technique; the numerical results are obtained and compared with the exact solution, computing the least square error, and running times which are criterion of discussion. For programming the technique, we use general Matlab program

Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method

A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation Au=f, is bounded and its image space is the reproducing kernel space W21[a,b]. It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. The n-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient

Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations

Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wavelets. The construction of most wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them. The properties of the Floor function are used to and a function which is one on the interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a; b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions. The main advantage of the wavelet technique for solving a problem is its ability to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential and Fredholm integral equations of the second kind. We also use collocation points and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. Illustrative examples are included to reveal the sufficiency of the technique. In linear integro-differential equations and Fredholm integral equations of the second kind cases, comparisons are done with CAS wavelets and differential transformation methods and it shows that the accuracy of these results are higher than them. Homotopy Analysis Method (HAM) is an analytic technique to solve the linear and nonlinear equations which can be used to obtain the numerical solution too. We extend the application of homotopy analysis method for solving Linear integro- differential equations and Fredholm and Volterra integral equations. We provide some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW, LLMW and CAS wavelets methods. For future studies, some problems are proposed at the end of this thesis