Approximate solution for a class of hypersingular integral equations

A simple approximate method for solving a general hypersingular integral equation of the first kind with its kernel consisting of a hypersingular part and a regular part is developed here. The method is illustrated by considering some simple examples.     

Hierarchical basis methods for hypersingular integral equations

In this paper we present hierarchical basis methods for theGalerkin approximation of hypersingular integral equations onthe interval = (–1,1). The condition number of the stiffnessmatrix with respect to the hierarchical basis is shown to behavelike O(|logh|²). The implementations are based on the preconditionedconjugate gradient method using a hierarchical basis (HB) preconditioner.The numerical results are presented with a comparison betweenthe HB preconditioner and the BPX (Bramble, Pasciak and Xu)preconditioner.     

Adaptive meshless Galerkin boundary node methods for hypersingular integral equations

Adaptive refinement techniques are developed in this paper for the meshless Galerkin boundary node method for hypersingular boundary integral equations. Two types of error estimators are derived. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two consecutive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the numerical solution itself and its projection. These error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization scheme is presented to accomodate the non-local property of hypersingular integral operators for the needed computable local error indicators. The convergence of the adaptive meshless techniques is verified theoretically. To confirm the theoretical results and to show the efficiency of the adaptive techniques, numerical examples in 2D and 3D with high singularities are provided.     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].     

A solution method for hypersingular integral equations

The Neumann problem for the Helmholtz equation is considered. The double-layer potential is used to reduce the problem to a hypersingular integral equation. The properties of the hypersingular operator in a neighborhood lead to a method for approximate solution of the hypersingular equation with an arbitrary contour. Some numerical results are reported.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 130–136, 1993.     

On a class of nonlinear stochastic integral equations

We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution.     

Block methods for nonlinear Volterra integral equations

A method used to solve linear Fredholm equations is adapted to obtain a generalization of block methods for solving nonlinear Volterra integral equations. Conditions for such methods to be convergent andA-stable are given along with some numerical examples.     

Additive schwarz methods for indefinite hypersingular integral equations in R3 - the p-version

We propose and analyze preconditioners for the p-version of the boundary element method in three dimensions. We consider indefinite hypersingular integral equations on surfaces and use quadrilateral elements for the boundary discretization. We use the GMRES method as iterative solver for the linear systems and prove for an overlapping additive Schwarz method that the number of iterations is bounded. This bound is independent of the polynomial degree of the ansatz functions and of the size of the underlying mesh. For a modified diagonal scaling, which uses special basis functions, we prove that the number of iterations grows only polylogarithmically in the polynomial degree. Here, a sufficiently fine mesh is required. Numerical results supporting the theory are presented.     

On a class of nonlinear integral equations with a noncompact operator

The paper is devoted to investigation of the solvability of a class of nonlinear integral equations on the semiaxis in a critical case, which possess a noncompact operator of almost Hammerstein type. Under some conditions on the equation kernel, the existence of a bounded, monotone increasing, positive solution is proved. The asymptotic behavior of the solution near infinity is studied.     

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

On a class of Newton-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of Newton-like methods considered also in [I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Computational theory of iterative methods, in: C.K. Chui, L. Wuytack (Eds.), Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co, New York, USA, 2007; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971], in order to approximate a locally unique solution of an equation in a Banach space.     

A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

On a class of nonlinear integral equations arising in neutron transport

We apply the method of continuation to study the structure of the solutions of quadratic integral equations for a certain class of kernel functions.     

Preconditined semilinear stochastic singular and hypersingular integral equations

In this paper, three classes of preconditioners are proposed for solving some stochastic integral equations with the weakly singular kernel and the hypersingular kernel. The first and the second class of preconditioners are based on circulant operators, but the third class of preconditioners is based on iterative substructuring. It is proved that substructuring preconditioners can be better than other preconditioners. Also, the spaces of solutions are discussed such that the solutions of these equations are smooth, therefore, we give special Banach spaces for these integral equations. Finally, numerical results which support our theories are presented