Sumudu transform and the solution of integral equations of convolution type

The convolution theorem for the Sumudu transform of a function which can be expressed as a polynomial or a convergent infinite series is proved and its applicability demonstrated in solving convolution type integral equations.     

Some algorithms for approximate solution of convolution integral equations

We study algorithms for approximate solution of convolution integral equations of first kind in which an approximate solution of the original equations can be constructed using the solution of equations having simpler structure or certain necessary properties, for example regularized equations. We obtain a number of algorithms for recursive refinement of the solution, where if the a priori information is given by linear algorithms, such algorithms are pseudo-recursive. The results may have application in problems of secondary processing of the data of indirect measurement. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Duality estimates for the numerical solution of integral equations

Summary We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.     

Error estimates and extrapolation for the numerical solution of Mellin convolution equations

In this paper we consider a quadrature method for the numericalsolution of a second-kind integral equation over the interval,where the integral operator is a compact perturbation of a Mellinconvolution operator. This quadrature method relies upon a singularitysubtraction and transformation technique. Stability and convergenceorder of the approximate solution are well known. We shall derivethe first term in the asymptotics of the error which shows that,in the interior of the interval, the approximate solution convergeswith higher order than over the whole interval. This implieshigher orders of convergence for the numerical calculation ofsmooth functionals to the exact solution. Moreover, the asymptoticsallows us to define a new approximate solution extrapolatedfrom the dilated solutions of the quadrature method over mesheswith different mesh sizes. This extrapolated solution is designedto improve the low convergence order caused by the non-smoothnessof the exact solution even when the transformation techniquecorresponds to slightly graded meshes. Finally, we discuss theapplication to the double-layer integral equation over the boundaryof polygonal domains and report numerical results.     

The numerical solution of Fredholm integral equations using product type quadrature formulas

Product type quadrature formulas are applied to obtain approximate solutions of Fredholm integral equations. A convergence theorem, and several numerical examples which demonstrate the efficacy of the technique, are presented.     

On multi-dimensional integral equations of convolution type with shift

The criterion of invertibility or Fredholmness of some multi-dimensional integral equations with Carleman type shifts are given. The investigation is based on some Banach space approach to equations with an involutive operator. A modified version of this approach is also presented in the paper.This approach is applied to multi-dimensional convolution type equations when the kernels may be integrable or of singular Calderon-Zygmund-Mikhlin type and shift generated by a linear transformation in the Euclidean space satisfying the generalized Carleman condition. The convolution type equations are also specially considered in the two-dimensional case in a sector on the plane symmetric with respect to one of the axes and the corresponding reflection shift. Another application deals with multi-dimensional equations with homogeneous kernels and the shift .     

On the eigenproblem for convolution integral equations

Summary This paper concerns the eigenproblem for convolution integral equations whose kernels can be expressed as finite or infinite Fourier transforms of integrable functions. A procedure which closely parallels previous work on displacement integral equations is derived and the problem of existence is treated. Approximations are obtained for both the eigenvalues and the eigenfunctions.The results of this paper are taken from the author's doctoral dissertation at the University of New Mexico. The research was supported by the United States Atomic Energy Commission.     

The general solution of a non-linear integral equation of convolution type

All continuous, non-negative solutions of a non-linear convolution equation are explicitly determined.

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Zusammenfassung Alle stetigen nichtnegativen Lösungen einer nichtlinearen Konvolutionsgleichung werden explizit bestimmt.

     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

On solving a system of singular Volterra integral equations of convolution type

This paper presents approximate analytical solutions for a system of singular Volterra integral equations of convolution type by using the fractional differential transform method. The solutions are calculated in the form of convergent series with easily computable terms and also the exact solutions can be achieved by well-known series solutions. Several examples are given to demonstrate reliability and performance of the presented method.     

A consistency analysis for the numerical solution of boundary integral equations

A method is presented for assessing the nature of the error incurred in the boundary integral equation (BIE) solution of both harmonic and biharmonic boundary value problems (BVPs). It is shown to what order of accuracy the governing partial differential equation is actually represented by the approximating numerical scheme, and how raising the order of the boundary ‘shape functions’ affects this representation. The effect of varying both the magnitude and the aspect ratio of the solution domain is investigated; it is found that the present technique may suggest an optimum nondimensional scaling for the BIE solution of a particular harmonic or biharmonic BVP.     

A numerical solution of nonlinear Volterra-Fredholm integral equations

In this paper, a numerical procedure for solving a class ofnonlinear Volterra-Fredholm integral equations is presented. Themethod is based upon the globally defined sinc basis functions.Properties of the sinc procedure are utilized to reduce thecomputation of the nonlinear integral equations to some algebraicequations. Illustrative examples are included to demonstrate thevalidity and applicability of the method.     

The numerical solution of nonlinear Volterra integral equations

A numerical method for solving nonlinear integral equations of Volterra type is given. Starting procedures together with predictor and corrector algorithms are discussed in detail.     

Approximate solution of integral equations and convolution integrals using Legendre polynomials

It has been argued that Chebyshev polynomials are ideal to use as approximating functions to obtain solutions of integral equations and convolution integrals on account of their fast convergence. Using the standard deviation as a measure of the accuracy of the approximation and the CPU time as a measure of the speed, we find that for reasonable accuracy Legendre polynomials are more efficient.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.