Numerical solution of linear Volterra integral equations of the second kind with sharp gradients

Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.     

Numerical solution of integral equations

Summary A new method for the solution of integral equations is presented. The method is based on direct approximation of Dirac's delta operator by linear combination of integral operators. This avoids some pitfalls which arise in more conventional numerical procedures for integral equations.Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. the Union Carbide Corporation.     

Nonnegative integral solution of linear equations

A method to obtain a nonnegative integral solution of a system of linear equations, if such a solution exists is given. The method writes linear equations as an integer programming problem and then solves the problem using a combination of artificial basis technique and a method of integer forms.     

Numerical solution of Volterra integral equations with singularities

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.     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

Numerical implementations of Cauchy-type integral equations

In this paper, a good interpolation formulae are applied to the numerical solution of Cauchy integral equations of the first kind with using some Chebyshev quadrature rules. To demonstrate the effectiveness of the Radau-Chebyshev with respect to the olds, [6], [7], [8] and [12], some examples are given.     

Numerical solution of Volterra integral equations

Summary The numerical method discussed in this paper is based on quadrature formulae. With some assumptions on the coefficients of the quadrature formula and on the integrand, convergence properties of the method for both linear and non-linear equations are established.This article is a part of the author's D. Sc. Thesis.     

Noncommutative integration of linear differential equations

A method of noncommutative integration of linear partial differential equations that is analogous to noncommutative integration of finite-dimensional Hamiltonian systems is proposed. The method is based on the concept, introduced in the paper, of a representation of Lie algebras. The method can be applied to the integration of the Klein-Gordon equation in Riemannian spaces of non-Stäckel type (i.e., in spaces that do not admit complete separation of the variables).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 195–213, August, 1995.     

Asymptotic behaviour of linear evolutionary integral equations

The asymptotic behaviour of bounded solutions of evolutionary integral equations in a Banach spaceX

Fast Fourier-Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

We develop a fast fully discrete Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the number of multiplications used in generating the compressed matrix is O(nlog³n), and the solution of the proposed method preserves the optimal convergence order O(n^−t), where n is the order of the Fourier basis functions used in the method and t denotes the degree of regularity of the exact solution. Moreover, we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm.     

Representations of solutions to linear integral equations with difference kernels

For linear integral equations with difference kernels, we give a formula for the inverse in terms of the solutions to two specific problems. If the equation is self-adjoint, these solutions are simply related.     

Numerical solutions for functional integral equations

Convergence criteria are given for a family of numerical methodsfor functional integral equations with delay. Several numericalmethods are compared.     

Some results on linear delay integral equations

Recently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.     

Three theorems on third-kind linear integral equations

Consider a solution space $\text{P}$_τ consisting essentially of linear combinations of continuous functions and poles with integration over the poles defined by Cauchy's principal value. In $\text{P}$_τ the linear integral equation of the third kind is shown to have properties similar to those of the Fredholm equation of the second kind. The space $\text{P}$_τ is a subspace of a solution space P_τ previously studied where no unique solution to the non-homogeneous third-kind equation exists. Solutions in $\text{P}$_τ may be explicitly constructed via Fredholm theory. For simplicity full details are presented only for the case involving a single pole. Generalization to the case of several poles is, however, briefly discussed.     

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.