On Volterra-Fredholm integral equations

The existence and uniqueness of solutions of more general Volterra-Fredholm integral equations are investigated. The successive approximations method based on the general idea of T. Wazewski is the main tool.     

Extermal solutions of terminal value problems for nonlinear impulsive integro-differential equations in Banach spaces

Abstract. The comparison principle is first established ,and then the lower and upper solutionmethod and the monotone iterative technique are employed to the study of terminal value prob-lems for the first order nonlinear impulsive integro-differential equations in Banach spaces. Fi-nally ,the existence theorem on the maximal and minimal solutions is obtained.     

Integral operators of Sturm-Liouville type

This paper extends the class of integral equations whose solutions can be generated from a finite number of particular cases to include those of Sturm-Liouville type, including the case where the associated operators are not self-adjoint. An explicit expression for the resolvent operator is generated from two particular solutions, in a form amenable to the use of approximation techniques. A usable estimate of the norm of the inverse operator is obtained even in cases where approximate solutions have to be used.     

Existence theorems of solutions for nonlinear impulsive Volterra integral equations in Banach spaces

In this paper,the Monch fixed point theorem and an impulsive integral inequality is used to prove some existence theorems of solutions for nonlinear impulsive Volterra integral equations in Banach spaces that improve and extend the previous results.     

Recent results on blow-up and quenching for nonlinear Volterra equations

In this survey paper, the author examines nonlinear Volterra integral equations of the second kind with solutions that blow-up or quench. The focus is on analytical results, although a few words about numerical solutions for such equations are provided. The integral equations arise in the mathematical modeling of thermal processes within a reactive–diffusive medium. The scope of this review is on the published literature between 1997 and 2005, serving as an update to a previous review by the same author.     

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.     

Global solutions of abstract semilinear parabolic equations with memory terms

The main purpose of this paper is to obtain the existence of global solutions to semilinear integro-differential equations in Hilbert spaces for rather general convolution kernels and nonlinear terms with superlinear growth at infinity. The included application to a nonlinear model of heat flow in materials of fading memory type provides motivations for the abstract theory.     

Hölder continuous solutions for integro-differential equations and maximal regularity

We characterize existence and uniqueness of solutions for a linear integro-differential equation in Hölder spaces. Our method is based on operator-valued Fourier multipliers. The solutions we consider may be unbounded. Concrete equations of the type we study arise in the modeling of heat conduction in materials with memory.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

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     

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.     

Nyström method for Cauchy singular integral equations with negative index

In this paper, the authors propose a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates.     

Numerical method for solving diffusion-wave phenomena

We find solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0,1),(1,2) and (0,2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. This method transfers the diffusion-wave problem into the corresponding infinite system of linear algebraic equations through the coefficients, which are uniquely solvable under some relations between the coefficients with index zero.The method is applicable for nonlinear problems too.     

Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.Received: March 25, 2002     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.     

Conditions for blow-up of solutions of some nonlinear Volterra integral equations

In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

On the existence of nonnegative solutions to an integral equation with applications to boundary value problems

By the use of a Guo-Krasnoselskii theorem in cones, existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied. The results are applied to a variety of boundary value problems concerning ordinary differential equations.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.