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1.
Hermann Brunner 《计算数学(英文版)》1992,10(4):348-357
It is shown that the error corresponding to certain spline collocation approximations for nonlinear Volterra integral equations of the second kind is the solution of a nonlinearly perturbed linear Volterra integral equation. On the basis of this result it is possible to derive general estimates for the order of convergence of the spline solution at the underlying mesh points. Extensions of these techniques to other types of Volterra equations are indicated. 相似文献
2.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(3):527-535
In this study, an efficient method is presented for solving nonlinear two-dimensional Volterra integral equations (VIEs). Using piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrix of integration, two-dimensional first kind integral equations reduce to a lower triangular system. The rate of convergence and error analysis are given and numerical examples illustrate efficiency and accuracy of the proposed method. 相似文献
3.
我们在参考了相关文献的基础上,考察了一类非线性Volterra积分方程的Chebyshev谱配置法.方法中,我们将该类非线性方程转化为两个方程进行数值逼近.我们选择N阶Chebyshev Gauss-Lobatto点作为配置点,对积分项用N阶高斯数值积分公式逼近.收敛性分析结果表明数值误差的收敛阶为N(1/2)-m,其中m是已知函数最高连续导数的阶数.我们也开展数值实验证实这一理论分析结果. 相似文献
4.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(9):3512-3519
Few numerical methods such as projection methods, time collocation method, trapezoidal Nystrom method, Adomian decomposition method and some else are used for mixed Volterra–Fredholm integral equations. The main purpose of this paper is to use the piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrices for solving mixed nonlinear Volterra–Fredholm integral equations of the first kind (VFIE). This method leads to a linear system of equations by expanding unknown function as 2D-BPFs with unknown coefficients. The properties of 2D-BPFs are then utilized to evaluate the unknown coefficients. The error analysis and rate of convergence are given. Finally, some numerical examples show the implementation and accuracy of this method. 相似文献
5.
E. V. Frolova 《Journal of Mathematical Sciences》2007,147(1):6538-6542
It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial
integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra
partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions
for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator
are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial
integral equations under examination.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal
Conference-2004, Part 3, 2006. 相似文献
6.
Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method
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Jafar Eshaghi Hojatollah Adibi Saeed Kazem 《Mathematical Methods in the Applied Sciences》2016,39(12):3411-3425
In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
7.
Wilhelm Hock 《Numerische Mathematik》1979,33(1):77-100
Summary This paper deals with linear multistep methods applied to nonlinear, nonsingular Volterra integral equations of the second kind. Analogously to the theory of W.B. Gragg, the existence of asymptotic expansions in the stepsizeh is proved. Under certain conditions only even powers ofh occur. As a special case, the midpoint rule is treated, a short numerical example for the applicability to extrapolation techniques is given. 相似文献
8.
A. S. Apartsin 《Computational Mathematics and Mathematical Physics》2016,56(5):810-825
A nonclassical Volterra linear integral equation of the first kind describing the dynamics of an developing system with allowance for its age structure is considered. The connection of this equation with the classical Volterra linear integral equation of the first kind with a piecewise-smooth kernel is studied. For solving such equations, the quadrature method is applied. 相似文献
9.
Banach空间中不连续非线性Volterra型积分方程的唯一解 总被引:12,自引:0,他引:12
本文在一般序Banach空间中研究了不连续非线性Volterra型积分方程的唯一解.在非常弱的条件下证明了非线性Volterra型积分方程的唯一解可以由迭代序列的一致极限得到,并给出了逼近解的迭代序列的误差估计式,然后应用到一阶微分方程的初值问题,本质改进并推广了最近的一些结果. 相似文献
10.
Hao Chen Chengjian Zhang 《Applied mathematics and computation》2011,218(6):2619-2630
New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods. 相似文献
11.
M.I. Berenguer D. Gámez A.I. Garralda-Guillem M. Ruiz Galán M.C. Serrano Pérez 《Journal of Computational and Applied Mathematics》2011,235(7):1875-1883
With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically. 相似文献
12.
A. N. Artyushin 《Differential Equations》2017,53(7):841-854
We suggest a simple method for reducing problems with an integral condition for evolution equations to a Volterra integral equation of the first kind. For Volterra equations of the convolution type, we indicate necessary and sufficient solvability conditions for the case in which the right-hand side lies in some classes of functions of finite smoothness. We use these conditions to construct examples of nonexistence of a local solution for the heat equation with an integral condition. 相似文献
13.
Gladwin [4] proved that Newton-Gregory formulas of order larger than 2 produce unstable algorithms when applied to nonlinear Volterra integral equations of the first kind. It is shown that similar results are true for all interpolatory quadrature rules using equidistant nodes. Upper bounds for the error order of quadrature rules, which lead to stable methods are given. Some higher order stable methods are indicated. 相似文献
14.
Spline Collocation-Interpolation Method for Linear and Nonlinear Cordial Volterra Integral Equations
Gennadi Vainikko 《Numerical Functional Analysis & Optimization》2013,34(1):83-109
We study the convergence and convergence speed of the discontinuous spline collocation and collocation-interpolation methods on uniform grids for linear and nonlinear Volterra integral equations of the second kind with noncompact operators. 相似文献
15.
S. Amini 《BIT Numerical Mathematics》1985,25(1):281-284
The concept of (A
0,S)-stability for Volterra integral equations of the second kind will be extended to that of the first kind equations. We will show that stability polynomials for methods employing reducible quadrature rules, as applied to the first kind equations, can be trivially obtained from the results for the second kind equations. 相似文献
16.
17.
Summary Two classes of high order finite difference methods for first kind Volterra integral equations are constructed. The methods are shown to be convergent and numerically stable. 相似文献
18.
Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations
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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. 相似文献
19.
《Numerical Functional Analysis & Optimization》2013,34(2):249-269
ABSTRACT Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented. 相似文献
20.
In this survey paper, the author examines nonlinear Volterra integral equations of the second kind with solutions that blow-up or quench. The focus is on analytical results, although a few words about numerical solutions for such equations are provided. The integral equations arise in the mathematical modeling of thermal processes within a reactive–diffusive medium. The scope of this review is on the published literature between 1997 and 2005, serving as an update to a previous review by the same author. 相似文献