Extrapolation method for Fredholm integral equations with non-smooth kernels

Summary This note analyses the methods of extrapolation from certain approximate solutions of integral equations whose kernels have lower degree smoothness. We show that in order to generate a global superconvergent approximation the extrapolation procedure must be applied to the iterated collocation solution rather than to the usual Nyström solution.     

An exponential approximation for solutions of generalized pantograph-delay differential equations

In this paper, a new matrix method based on exponential polynomials and collocation points is proposed for solutions of pantograph equations with linear functional arguments under the mixed conditions. Also, an error analysis technique based on residual function is developed for the suggested method. Some examples are given to demonstrate the validity and applicability of the method and the comparisons are made with existing results.     

SUPERCONVERGENT APPROXIMATIONS FOR WIENER-HOPF EQUATIONS

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An algorithm for solving Fredholm integro-differential equations

The aim of this paper is to present an efficient numerical procedure for solving linear second order Fredholm integro-differential equations. The scheme is based on B-spline collocation and cubature formulas. The analysis is accompanied by numerical examples. The results demonstrate reliability and efficiency of the proposed algorithm.      

Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order

In this paper we present a computational method for solving a class of nonlinear Fredholm integro-differential equations of fractional order which is based on CAS (Cosine And Sine) wavelets. The CAS wavelet operational matrix of fractional integration is derived and used to transform the equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.     

Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.     

Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind

Periodic harmonic wavelets (PHW) were applied as basis functions in solution of the Fredholm integral equations of the second kind. Two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions. It is proved that PHW satisfy the properties of the multiresolution analysis.     

PQWs in complex plane: Application to Fredholm integral equations

In this paper, we use a numerical procedure for solving Fredholm integral equations of the second kind in complex plane. The periodic quasi-wavelets (PQWs) constructed on [0,2π]$[0\text{,}2\pi ]$ are utilized as a basis in collocation method to reduce the solution of linear integral equations to a system of algebraic equations. Convergence analysis is derived and we used some numerical examples to illustrate the accuracy and the implementation of the method.     

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

An approximate Newton method for non-smooth equations with finite max functions

A new version of finite difference approximation of the generalized Jacobian for a finite max function is constructed. Numerical results are reported for the generalized Newton methods using this approximation.     

A spline collocation method for parabolic pseudodifferential equations

The purpose of this paper is to examine a boundary element collocation method for some parabolic pseudodifferential equations. The basic model problem for our investigation is the two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition. Certain choices of the representation formula for the heat potential yield boundary integral equations of the first kind, namely the single layer and the hypersingular heat operator equations. Both of these operators, in particular, are covered by the class of parabolic pseudodifferential operators under consideration. Moreover, the spatial domain is allowed to have a general smooth boundary curve. As trial functions the tensor products of the smoothest spline functions of odd degree (space) and continuous piecewise linear splines (time) are used. Stability and convergence of the method is proved in some appropriate anisotropic Sobolev spaces.     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

A new defect correction method for the Navier-Stokes equations at high Reynolds numbers

In this paper, a new defect correction method for the Navier-Stokes equations is presented. With solving an artificial viscosity stabilized nonlinear problem in the defect step, and correcting the residual by linearized equations in the correction step for a few steps, this combination is particularly efficient for the Navier-Stokes equations at high Reynolds numbers. In both the defect and correction steps, we use the Oseen iterative scheme to solve the discrete nonlinear equations. Furthermore, the stability and convergence of this new method are deduced, which are better than that of the classical ones. Finally, some numerical experiments are performed to verify the theoretical predictions and show the efficiency of the new combination.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels

A fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Some of our theoretical results are illustrated by numerical experiments.     

A Legendre spectral-element method for the one-dimensional fourth-order equations

A Legendre-Galerkin spectral-element method is proposed to solve the one-dimensional fourth-order equations. C¹-continuity between the elemental-faces is imposed by constructing appropriate basis functions. The method leads to linear systems with sparse matrices for the discrete variational formulations. Rigorous error analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results.