共查询到20条相似文献,搜索用时 15 毫秒
1.
Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which generalized some known weakly singular inequalities and can be used in the analysis of various problems in the theory of certain classes of differential equations, integral equations and evolution equations. Some applications to fractional differential and integral equations are also indicated. 相似文献
2.
Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.
3.
A fast/quasifast solver for weakly singular periodic Fredholm integral equations is constructed in the situation where the
information about the kernel and the free term of the equation is restricted to a finite number of sample values. Hence the
complexity of weakly singular integral equations is the same as that for equations without singularities or close to that.
AMS subject classification (2000) 65Y20, 65T05, 45B05 相似文献
4.
Bijaya Laxmi Panigrahi Gnaneshwar Nelakanti 《Journal of Applied Mathematics and Computing》2013,43(1-2):175-197
In this paper, we consider Galerkin method for weakly singular Fredholm integral equations of the second kind and its corresponding eigenvalue problem using Legendre polynomial basis functions of degree ≤n. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind and also obtain the error bounds for the approximated eigenelements in the corresponding eigenvalue problem. We illustrate our results with numerical examples. 相似文献
5.
Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method 下载免费PDF全文
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
6.
在L~1空间中讨论弱奇异积分方程的特征值问题.利用弱奇异积分算子为紧算子,便可以对其直接进行离散化算法求出特征值.并将直接离散的方法与以往的迭代后离散的方法用实例通过Matlab作图进行对比,说明直接进行离散的方法更佳. 相似文献
7.
We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and it is used to derive properties of the volume integral equation. 相似文献
8.
For classes of weakly singular integral equations of the second kind whose kernels have a power singularity, we find the optimal order of the rate of convergence of projection-iterative methods. Moreover, iterative methods of the Sokolov type are considered and, for weakly singular equations with differentiable coefficients, we present estimates of the rate of convergence of such methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 498–505, April, 1995. 相似文献
9.
Zhongying Chen Charles A. Micchelli Yuesheng Xu 《Advances in Computational Mathematics》2002,16(1):1-28
In this paper, we develop a discrete wavelet Petrov–Galerkin method for integral equations of the second kind with weakly singular kernels suitable for solving boundary integral equations. A compression strategy for the design of a fast algorithm is suggested. Estimates for the rate of convergence and computational complexity of the method are provided. 相似文献
10.
S. R. Enikeeva 《Russian Mathematics (Iz VUZ)》2010,54(2):11-15
In this paper we theoretically justify the Bogolyubov-Krylov method for weakly singular integral equations. 相似文献
11.
Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels 总被引:1,自引:0,他引:1
We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods. 相似文献
12.
High-Accuracy Numerical Approximations to Several Singularly Perturbed Problems and Singular Integral Equations by Enriched Spectral Galerkin Methods 下载免费PDF全文
Sheng Chen 《数学研究》2020,53(2):143-158
Usual spectral methods are not effective for singularly perturbed problems
and singular integral equations due to the boundary layer functions or weakly singular solutions. To overcome this difficulty, the enriched spectral-Galerkin methods
(ESG) are applied to deal with a class of singularly perturbed problems and singular
integral equations for which the form of leading singular solutions can be determined.
In particular, for easily understanding the technique of ESG, the detail of the process
are provided in solving singularly perturbed problems. Ample numerical examples
verify the efficiency and accuracy of the enriched spectral Galerkin methods. 相似文献
13.
《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
14.
In this paper, the piecewise polynomial collocation methods are used for solving the fractional integro-differential equations with weakly singular kernels. We present that a suitable transformation can convert fractional integro-differential equations to one type of second kind Volterra integral equations (VIEs) with weakly singular kernels. Then we solve the VIEs by standard piecewise polynomial collocation methods. It is shown that such kinds of methods are able to yield optimal convergence rate. Finally, some numerical experiments are given to show that the numerical results are consistent with the theoretical results. 相似文献
15.
M. Azizov 《Ukrainian Mathematical Journal》1996,48(10):1473-1485
We find the exact order of the ε-complexity of weakly singular integral equations with periodic and analytic coefficients of logarithmic singularities. This class of equations includes boundary equations for outer boundary-value problems for the two-dimensional Helmholtz equation. 相似文献
16.
Hermann Brunner 《Numerical Functional Analysis & Optimization》2013,34(7-8):717-736
The open problems (and conjectures) discussed in this paper arise in the discretization of Volterra integral equations, including equations with weakly singular kernels and delay arguments, by collocation methods in piecewise polynomial spaces. They focus on questions of stability versus accuracy; extrapolation on regular and graded meshes; and equations with certain variable delays 相似文献
17.
Lü Tao 《Journal of Mathematical Analysis and Applications》2006,324(1):225-237
Based on a new generalization of discrete Gronwall inequality in [L. Tao, H. Yong, A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equality of the second kind, J. Math. Anal. Appl. 282 (2003) 56-62], Navot's quadrature rule for computing integrals with the end point singularity in [I. Navot, A further extension of Euler-Maclaurin summation formula, J. Math. Phys. 41 (1962) 155-184] and a transformation in [P. Baratella, A. Palamara Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004) 401-418], a new quadrature method for solving nonlinear weakly singular Volterra integral equations of the second kind is presented. The convergence of the approximation solution and the asymptotic expansion of the error are proved, so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error. 相似文献
18.
19.
Numerical Algorithms - We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal... 相似文献