Numerical solution of the Goursat problem by a nonlinear trapezoidal formula

In this paper the solution of the Goursat problem is obtained by the use of a nonlinear Trapezoidal formula based on geometric means. The numerical results indicate the new strategy to be superior.     

Numerical solution of a singular integral equation arising in a cruciform crack problem

A singular integral equation arising in a cruciform crack problem is investigated in the present paper. Based on the convex technique, the piecewise Taylor-series expansion method is extended by introducing a weight parameter. An approximate solution of the singular integral equation is constructed and its convergence and error estimate are made. The variations of the approximate solutions associating with stress intensity factors are analyzed by considering internal pressures of power and sine functions, respectively. By comparing with the known methods, the observations reveal that a good approximation can be achieved using less derivative times, less discretization points, and a suitable weight parameter. The obtained results show that the crack growth is dependent on applied mechanical loadings.     

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.     

Numerical solution of nonlinear integral equations using alternative Legendre polynomials

In this paper, the operational matrices of integration and the product for the alternative Legendre polynomials (ALPs) are first derived. Then, using these operational matrices and the collocation method, the nonlinear Volterra–Fredholm–Hammerstein integral equations are reduced to a set of nonlinear algebraic equations with unknown ALP coefficients. Some error estimations are provided and the efficiency and accuracy is verified by applying the method to some examples chosen from other literature.     

Numerical solution of nonlinear volterra integral equations using the idea of quasilinearization

The iterations of the quasilinear technique, employed in nonlinear volterra integral equations, are expressed as linear integral equations. By using Collocation Method, the solutions of these linear equations are approximated. Combining this and iterations of the quasilinear technique yields an approximation solution for nonlinear integral equations. The convergence is considered and the examples confirm the accuracy of the solution.     

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem

In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the mean value theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.     

Numerical solution of the abel integral equation

A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.     

Numerical solution of the boundary-value problem for a nonlinear diffusion equation in image processing

Diffusion filtering methods involve solving an initial boundary-value problem for the diffusion equation in which the initial condition is specified by a function representing the filtered image. The output of this filter is the solution u(x, y, t) of the initial boundary-value problem at some fixed time t = T. In a previous study we have proposed a new version of the diffusion filtering method that ensures improved noise removal due to inclusion of a dependence of the diffusion coefficient on local image intensity. The present study analyzes the resulting finite-difference method for the initial boundary-value problem, examines its numerical implementation, and analyzes its efficiency on prototype and real images. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 35–43, 2006.     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

Numerical approximation of the solution in infinite dimensional global optimization using a representation formula

A non convex optimization problem, involving a regular functional J, on a closed and bounded subset S of a separable Hilbert space V is here considered. No convexity assumption is introduced. The solutions are represented by using a closed formula involving means of convenient random variables, analogous to Pincus (Oper Res 16(3):690–694, 1968). The representation suggests a numerical method based on the generation of samples in order to estimate the means. Three strategies for the implementation are examined, with the originality that they do not involve a priori finite dimensional approximation of the solution and consider a hilbertian basis or enumerable dense family of V. The results may be improved on a finite-dimensional subspace by an optimization procedure, in order to get higher-quality solutions. Numerical examples involving both classical situation and an engineering application issued from calculus of variations are presented and establish that the method is effective to calculate.     

Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Two-dimensional rationalized Haar (RH) functions are applied to the numerical solution of nonlinear second kind two-dimensional integral equations. Using bivariate collocation method and Newton–Cotes nodes, the numerical solution of these equations is reduced to solving a nonlinear system of algebraic equations. Also, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Numerical solution of Volterra integral equation

Summary This paper discusses the use of Gregory's formula for numerical integration of Volterra linear integral equations of the second type. The order of magnitude of the truncation error and the asymptotic behavior of this error are obtained by means of recursive inequalities.     

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.     

Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations

We consider the three-dimensional Dirichlet problem for equations of elliptic type in inhomogeneous media. The problem can be reduced to a system of loaded Fredholm integral equations of the second kind over the volume. We prove the uniqueness of a classical solution of the problem. We suggest a numerical solution algorithm of iterative type. An example of the numerical solution of the problem is considered, and the convergence of the iterative procedure is demonstrated numerically.     

Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function

In this paper, we introduce a numerical method for the solution of two-dimensional Fredholm integral equations. The method is based on interpolation by Gaussian radial basis function based on Legendre-Gauss-Lobatto nodes and weights. Numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method.