The variational iteration method for solving linear and nonlinear Volterra integral and integro-differential equations

The variational iteration method is used for solving the linear and nonlinear Volterra integral and integro-differential equations. The method is reliable in handling Volterra equations of the first kind and second kind in a direct manner without any need for restrictive assumptions. The method significantly reduces the size of calculations.     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

A collocation approach for solving systems of linear Volterra integral equations with variable coefficients

In this paper, a numerical method is introduced to solve a system of linear Volterra integral equations (VIEs). By using the Bessel polynomials and the collocation points, this method transforms the system of linear Volterra integral equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives an analytic solution when the exact solutions are polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with existing results. All of the numerical computations have been performed on computer using a program written in MATLAB v7.6.0 (R2008a).     

STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind

In this paper, single-term Walsh series (STWS) method is applied to obtain the numerical solutions of Hammerstein systems of nonlinear Volterra integral equations of second kind (HSNVIES). Using the properties of the STWS method, HSNVIES can be easily converted into solvable recursive system of algebraic equations. Solutions obtained from the recursive system of algebraic equations are the solutions of the HSNVIES. Illustrative examples are provided with numerical solutions and the efficiency of this STWS method is also compared with the existing methods.     

Approximate solution of a system of linear integral equations by the Taylor expansion method

A simple and efficient approximate technique is developed to obtain the solution to a system of linear integral equations. This technique is based on the Taylor expansion. The method has been successfully applied to determine approximate solutions of a system of Fredholm integral equations and Volterra integral equations of not only the second kind but also the first kind. The mth order approximation of the solution is exact up to a polynomial of degree equal to or less than m. Several illustrative examples are presented to show the effectiveness and accuracy of this method.     

Extrapolation for solving a system of weakly singular nonlinear Volterra integral equations of the second kind

This article discusses an extrapolation method for solving a system of weakly singular nonlinear Volterra integral equations of the second kind. Based on a generalization of the discrete Gronwall inequality and Navot's quadrature rule, the modified trapeziform quadrature algorithm is presented. The iterative algorithm for solving the discrete system possesses a high accuracy order O(h ^2+α). After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using extrapolation. An a posteriori error estimation is provided. Some numerical results are presented to illustrate the efficiency of our methods.     

Moving least squares collocation method for Volterra integral equations with proportional delay

In this work, we apply the moving least squares (MLS) method for numerical solution of Volterra integral equations with proportional delay. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. An error bound is obtained to ensure the convergence and reliability of the method. Numerical results approve the efficiency and applicability of the proposed method.     

Approximate solution of systems of Volterra integral equations with error analysis

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(?c N ^1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

Fixed point method for solving nonlinear quadratic Volterra integral equations

We consider a paper of Bana? and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of a quadratic Volterra integral equation. In this paper, we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic Volterra integral equation. Then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results.     

A reliable computational approach for approximate solution of Hammerstein integral equations of mixed type

The aim of this article is to present an efficient analytical and numerical procedure for solving the nonlinear Hammerstein integral equations of mixed type. Our method mainly depends on a Taylor expansion approach. Also, we obtain the approximate solution of the nonlinear Volterra–Hammerstein integral equations of mixed type in terms of the Taylor polynomials. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.     

A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions

In this paper, we use hat basis functions to solve the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs) of the second kind. This method converts the system of integral equations into a linear or nonlinear system of algebraic equations. Also, we consider the order of convergence of the method and show that it is O(h²). Application of the method on some examples show its accuracy and efficiency.     

Approximate solution of fractional integro-differential equations by Taylor expansion method

In this paper, Taylor expansion approach is presented for solving (approximately) a class of linear fractional integro-differential equations including those of Fredholm and of Volterra types. By means of the mth-order Taylor expansion of the unknown function at an arbitrary point, the linear fractional integro-differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions. This method gives a simple and closed form solution for a linear fractional integro-differential equation. In addition, illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

On the error analysis of the numerical solution of linear Volterra integral equations of the second kind

In the numerical solution of linear Volterra integral equations, two kinds of errors occur. If we use the collocation method, these errors are the collocation and numerical quadrature errors. Each error has its own effect in the accuracy of the obtained numerical solution. In this study we obtain an error bound that is sum of these two errors and using this error bound the relation between the smoothness of the kernel in the equation and also the length of the integration interval and each of these two errors are considered. Concluded results also are observed during the solution of some numerical examples.     

一种改进的基于深度神经网络的偏微分方程求解方法

偏微分方程求解是计算流体力学等科学与工程领域中数值分析的计算核心。由于物理的多尺度特性和对离散网格质量的敏感性,传统的数值求解方法通常包含复杂的人机交互和昂贵的网格剖分开销,限制了其在许多实时模拟和优化设计问题上的应用效率。提出了一种改进的基于深度神经网络的偏微分方程求解方法TaylorPINN。该方法利用深度神经网络的万能逼近定理和泰勒公式的函数拟合能力,实现了无网格的数值求解过程。在Helmholtz、Klein-Gordon和Navier-Stokes方程上的数值实验结果表明,TaylorPINN能够很好地拟合计算域内时空点坐标与待求函数值之间的映射关系,并提供了准确的数值预测结果。与常用的基于物理信息神经网络方法相比,对于不同的数值问题,TaylorPINN将预测精度提升了3~20倍。     

Modified multilag methods for singularly perturbed volterra integral equations

Modified multilag methods are used for numerical solution of singularly perturbed Volterra integral equations (SVIEs). In particular, the stability of the methods is investigated for a class of convolution kernels. The stability behavior is independent of the choice of the quadrature rules. Numerical results are considere.