In the first part of this paper we study the regularity properties of solutions to initial or boundary-value problems of Fredholm integro-differential equationswithweakly singular or other nonsmooth kernels.We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. Presented numerical examples display that theoretical results ar...

We analyze the optimal superconvergence properties of piecewise polynomial collocation solutions on uniform meshes for Volterra integral and integrodifferential equations with multiple (vanishing) proportional delays θj(t) = qjt (0 < q1 < · · · < qr < 1). It is shown that for delay integro-differential equations the recently obtained optimal order is also attainable. For integral equations with...

A parameter-uniform numerical method based on Shishkin mesh is constructed and analyzed for a weakly coupled system of singularly perturbed second order reaction-diffusion equations. A B-spline collocation method is combined with a piecewise-uniform Shishkin mesh. An asymptotic error bound (independent of the perturbation parameter ε) in the maximum norm is established theoretically. To illustr...

In this paper we consider hyperbolic initial value problems subject to periodic boundary conditions with nonsmooth data. We show that if we filter the data and solve the problem by the Galerkin-Collocation method, recently proposed by us, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smooth...

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a prior...

In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind Fredholm equations. In a certain sense this projection is an interpolation projection. By introducing this technique of analysis, we particularly consider the case of polygonal boundaries....

In [14], the authors developed a new approach to computation of Hausdorff dimension invariant set an iterated function system or IFS. this paper, we extend incorporate high order approximation methods. We again rely on fact that can associate IFS parametrized family positive, linear, Perron-Frobenius operators $L_s$, idea known in varying degrees generality for many years. Although $L_s$ is not...

We study a fine hierarchy of Borel-piecewise continuous functions, especially, between closed-piecewise continuity and Gδ-piecewise continuity. Our aim is to understand how a priority argument in computability theory is connected to the notion of Gδ-piecewise continuity, and then we utilize this connection to obtain separation results on subclasses of Gδ-piecewise continuous reductions for unif...

We propose and analyze a C spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is establ...

In this study, the Fredholm hypersingular integral equation of first kind with a singular right-hand function on interval − 1 , is solved. The discontinuous solution domain id="M2"> approximated by piecewise polynomial, and collocation method introd...