共查询到20条相似文献,搜索用时 31 毫秒
1.
This article proposes a simple efficient direct method for solving Volterra integral equation of the first kind. By using block-pulse functions and their operational matrix of integration, first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. Some examples are presented to illustrate efficiency and accuracy of the proposed method. 相似文献
2.
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.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal
Conference-2004, Part 3, 2006. 相似文献
3.
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
4.
A. F. Voronin 《Journal of Applied and Industrial Mathematics》2014,8(3):428-435
We study a Volterra convolution integral equation of the first kind on a semi-infinite interval. Under some rather natural constraints on the kernel and the right-hand side of the Volterra integral equation (the kernel has bounded support, while the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (i.e., the solution and the kernel of the integral operator) from the right-hand side of the equation. The uniqueness theorem is proved, the necessary and sufficient conditions for solvability are found, and the explicit formulas for the solution and the kernel are obtained. 相似文献
5.
Zhong ChenWei Jiang 《Applied mathematics and computation》2011,217(19):7790-7798
The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective. 相似文献
6.
Lothar von Wolfersdorf 《Journal of Mathematical Analysis and Applications》2007,331(2):1314-1336
The autoconvolution equation of the third kind with coefficient of general power type is dealt with by the method of weighted norms developed for equations with coefficients of linear and integer power type in recent joint work of the author with L. Berg, J. Janno, and B. Hofmann. For this equation two existence theorems and a uniqueness theorem are proved. Further, as an auxiliary equation a linear singular integral equation of Abel is treated anew and the existence of solutions to a related class of linear Volterra equations of the third kind is derived. 相似文献
7.
Mohamed A. Ramadan Heba S. Osheba Adel R. Hadhoud 《Numerical Methods for Partial Differential Equations》2023,39(1):268-280
This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block-pulse functions method is introduced and used where the key point is to transform linear second-type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process. 相似文献
8.
We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional
problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on
the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the
problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite
well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is
reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra
integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does
not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled
elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra
equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions
at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces,
normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 96–103. Original article submitted March 15, 1997. 相似文献
9.
We consider source identification problems for the wave equation on an interval and on trees. The main advantage of our approach is its locality. Our algorithm reduces essentially to the resolution of a linear integral Volterra equation of the second kind and is new even for an interval. 相似文献
10.
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
11.
The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given. 相似文献
12.
Information about the density distribution of a gas flowingin a pipe is obtained by measuring the mean densities alongstraight lines across the pipe. The density distribution isrepresented by a Fourier series with variable coefficients whichsatisfy a Volterra type integral equation of the first kind.This equation is solved and a numerical technique for calculatingthe solution is given. 相似文献
13.
Peter Linz 《BIT Numerical Mathematics》1971,11(4):413-421
The numerical solution of Volterra integral equations of the first kind can be accomplished if the integral is replaced by certain simple quadrature rules, such as the midpoint or the trapezoidal methods. When the kernel of the integral equation oscillates more rapidly than the solution one can use product integration techniques to increase the accuracy. Such an approach is investigated in this paper. 相似文献
14.
This paper presents a computational technique for Fredholm integral equation of the second kind and Volterra integral equation of the second kind. The method is based upon Haar functions approximation. Properties of Rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix and Newton–Cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples. 相似文献
15.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(6):2469-2477
In this paper a modification of Block Pulse Functions is introduced and used to solve Volterra integral equation of the first kind. Some theorems are included to show convergence and advantage of the method. Some examples show accuracy of the method. 相似文献
16.
I. V. Sapronov 《Russian Mathematics (Iz VUZ)》2011,55(1):50-61
We construct a multiparametric set of solutions to a singular Volterra integral equation of the first kind with a sufficiently smooth kernel in the space of integrable functions whose values belong to a Banach space. 相似文献
17.
18.
This paper presents a new and an efficient method for determining solutions of the linear second kind Volterra integral equations system. In this method, the linear Volterra integral equations system using the Taylor series expansion of the unknown functions transformed to a linear system of ordinary differential equations. For determining boundary conditions we use a new method. This method is effective to approximate solutions of integral equations system with a smooth kernel, and a convolution kernel. An error analysis for the proposed method is provided. And illustrative examples are given to represent the efficiency and the accuracy of the proposed method. 相似文献
19.
We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution. 相似文献
20.
We consider two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problems are reduced to systems of linear Volterra integral equations of the second kind for the unknown functions. These systems are used to prove the existence and uniqueness of the solution of the inverse problems and numerically solve them. The applicability of the methods developed here to the approximate solution of the problem on an unknown source in the heat equation is studied numerically. 相似文献