Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.     

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.     

On a method for solving a two-dimensional nonlinear integral equation of the second kind

In this article, the existence of at least one solution of a nonlinear integral equation of the second kind is proved. The degenerate method is used to obtain a nonlinear algebraic system, where the existence of at least one solution of this system is discussed. Finally, computational results with error estimates are obtained using Maple software.     

Toeplitz matrix method and nonlinear integral equation of Hammerstein type

The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated.     

Convergence theorems for HL integrable vector-valued functions with applications

In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities.     

The h-p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains

Summary. This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain and with constant coefficients in the exterior domain . This procedure uses the variational formulation of the differential equation in and involves integral operators on the interface between and . The finite elements are used to obtain approximate solutions of the differential equation in and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements. Received February 10, 1996 / Revised version received April 4, 1997     

Nyström type methods for Fredholm integral equations with weak singularities

Nyström type methods are constructed and justified for a class of Fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities. The proposed approach is based on a suitable smoothing change of variables and product integration techniques. Global convergence estimates are derived and a collection of numerical results is given.     

The Hukuhara–Kneser property for a nonlinear integral equation

Applying a structure theorem of Krasnosel’skii and Perov, we show that the solution set of a nonlinear integral equation satisfies the classical Hukuhara–Kneser property.     

System of nonlinear Volterra’s integral equations with polar kernel and singularities

We study an existence-uniqueness theorem for a system of nonlinear Volterra-type integral equations with polar kernel and singularities. To do so, we use a method of regularization of fractional derivatives with a delta sequence to avoid singularities over diagonal {y|x−y=0}$\left\{y\right|x-y=0\}$. The other singularities we regularize with delta sequences with different growth. As a framework, we use Colombeau special algebra.     

Integro-differential Burgers equation. Solvability and homogenization

We consider the integro-differential Burgers equation which appears in nonlinear acoustics. The integral term in the right-hand side describes the relaxation (memory) effects. The global existence and uniqueness of solution is proved. The smoothness of the solution is studied.In the case when the coefficients of the equation rapidly oscillate, we replace it by the homogenized equation with constant coefficients and prove the error estimates.     

Numerical analysis of the weighted particle method applied to the semiconductor Boltzmann equation

Summary The semiconductor Boltzmann equation involves an integral operator, the kernel of which is a measure supported by a surface. This feature introduces some singularities of the exact solution, which makes the numerical approximation of this equation difficult. This paper is devoted to the error analysis of the weighted particle method (introduced by Mas-Gallic and Raviart [14]) applied to the space homogeneous semiconductor Boltzmann equation. The results are commented in view of the practical use of the method. This paper is closely related to [12], where results of numerical simulations on both test and real problems are given.     

A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.     

Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

Finite sections for singular integral operators by weighted Chebyshev polynomials

A finite section method for the approximate solution of singular integral equations with piecewise continuous coefficients on intervals is considered. The problem is transformed in such a way that results which were previously obtained for singular integral equations on the unit circle using localization methods in Banach algebras are applicable to it. Thus, necessary and sufficient conditions for the stability of the approximation method can be proved.     

Solutions of some singular integral equation

A singular integral equation with a Holderian second member function on [a,b] is considered and solved for four different type of kernels in the class of functions that are unbounded at the end points of the interval.     

Collocation Methods and Their Modifications for Cauchy Singular Integral Equations on the Interval

In this paper we present polynomial collocation methods and their modi.cations for the numerical solution of Cauchy singular integral equations over the interval [-1, 1]. More precisely, the operators of the integral equations have the form with piecewise continuous coefficients a and b, and with a Jacobi weight . Using the splitting property of the singular values of the collocation methods, we obtain enough stable approximate methods to .nd the least square solution of our integral equation. Moreover, the modifications of the collocation methods enable us to compute kernel and cokernel dimensions of operators from a C*-algebra, which is generated by operators of the Cauchy singular integral equations.     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

Regularization of an Abel equation

We consider one class of Abel's equation with ill-posed natures. The stable approximate solution is obtained by using the well-known Tikhonov's regularization approach. We also conduct numerical computations to realize the approximate solutions for a concrete equation to demonstrate the applicability of our method.     

Integral equations and initial value problems for nonlinear differential equations of fractional order

We discuss the solvability of integral equations associated with initial value problems for a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative and the inhomogeneous term depends on the fractional derivative of lower orders. We obtain the existence of at least one solution for integral equations using the Leray–Schauder Nonlinear Alternative for several types of initial value problems. In addition, using the Banach contraction principle, we establish sufficient conditions for unique solutions. Our approach in obtaining integral equations is the “reduction” of the fractional order of the integro-differential equations based on certain semigroup properties of the Caputo operator.