Newton-Kantorovich approximations to nonlinear singular integral equation with shift

The paper is concerned with the applicability of some new conditions for the convergence of Newton-kantorovich approximations to solution of nonlinear singular integral equation with shift of Uryson type. The results are illustrated in generalized Holder space.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

Differential transform method for solving Volterra integral equation with separable kernels

In this paper, Volterra integral equations with separable kerenels are solved using the differential transform method. The approximate solution of this equation is calculated in the form of a series with easily computable terms. Exact solutions of linear and nonlinear integral equations have been investigated and the results illustrate the reliability and the performance of the differential transform method.     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

Spectral method for multidimensional Volterra integral equation with regular kernel

This paper is concerned with obtaining an approximate solution for a linear multidimensional Volterra integral equation with a regular kernel. We choose the Gauss points associated with the multidimensional Jacobi weight function ω(x)=∏di=1(1-xi)^α(1+xi)^β,-1<α,β<1/d-1/2 (d denotes the space dimensions) as the collocation points. We demonstrate that the errors of approxima te solution decay exponentially. Numerical results are presen ted to demonstrate the effectiveness of the Jacobi spectral collocation method.     

Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation

The operational Tau method, a well-known method for solving functional equations, is employed to solve a system of nonlinear Volterra integro-differential equations with nonlinear differential part. In addition, an error estimation of the method is presented. Some cases of the mentioned equations are solved as examples to illustrate the ability and reliability of the method. The results reveal that the method is very effective and convenient.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

Asymptotic properties of solutions to some nonlinear integral equations of convolution type

For a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions. Moreover, conditions for the solution to converge asymptotically to a determined limit are obtained.     

Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration,

(0.1)     

A new numerical approximation for Volterra integral equations combining two quadrature rules

This paper describes a numerical approximation to the solution of Volterra integral equations of the second kind. This algorithm combines trapezoidal and Simpson rules. We prove the convergence of the method. Numerical examples are provided to illustrate the accuracy of the method.     

The method of Kantorovich majorants to nonlinear singular integral equation with shift

The paper is concerned with the applicability of the method of Kantorovich majorants to nonlinear singular integral equation with shift. The result is illustrated in the generalized Hölder space.     

A note on blow-up solutions to some nonlinear Volterra integral equations

In this paper we investigate certain nonlinear Volterra integral equations with the power nonlinearity. The basic results provide a criterion involving the kernel and the nonlinearity for the existence of the nontrivial and blow-up solution.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

Approximate solutions of linear Volterra integral equation systems with variable coefficients

In this paper, a new approximate method has been presented to solve the linear Volterra integral equation systems (VIEs). This method transforms the integral system into the matrix equation with the help of Taylor series. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to show this capability and robustness, some systems of VIEs are solved by the presented method in order to obtain their approximate solutions.