An <Emphasis Type="Italic">hp</Emphasis>-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels

In this paper, we present an hp-version Legendre–Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the \(H^1\)-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method.     

Smoothing Transformation and Piecewise Polynomial Collocation for Weakly Singular Volterra Integral Equations

We discuss a possibility to construct high-order numerical algorithms on uniform or mildly graded grids for solving linear Volterra integral equations of the second kind with weakly singular or other nonsmooth kernels. We first regularize the solution of integral equation by introducing a suitable new independent variable and then solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

     

Collocation Methods for General Caputo Two-Point Boundary Value Problems

A general class of two-point boundary value problems involving Caputo fractional-order derivatives is considered. Such problems have been solved numerically in recent papers by Pedas and Tamme, and by Kopteva and Stynes, by transforming them to integral equations then solving these by piecewise-polynomial collocation. Here a general theory for this approach is developed, which encompasses the use of a variety of transformations to Volterra integral equations of the second kind. These integral equations have kernels comprising a sum of weakly singular terms; the general structure of solutions to such problems is analysed fully. Then a piecewise-polynomial collocation method for their solution is investigated and its convergence properties are derived, for both the basic collocation method and its iterated variant. From these results, an optimal choice can be made for the transformation to use in any given problem. Numerical results show that our theoretical convergence bounds are often sharp.     

L1 Fourier spectral methods for a class of generalized two-dimensional time fractional nonlinear anomalous diffusion equations

In this paper, L1 Fourier spectral methods are derived to obtain the numerical solutions for a class of generalized two-dimensional time-fractional nonlinear anomalous diffusion equations involving Caputo fractional derivative. Firstly, we establish the L1 Fourier Galerkin full discrete and L1 Fourier collocation schemes with Fourier spectral discretization in spatial direction and L1 difference method in temporal direction. Secondly, stability and convergence for both Galerkin and collocation approximations are proved. It is shown that the proposed methods are convergent with spectral accuracy in space and $(2?\alpha )$ order accuracy in time. For implementation, the equivalence between pseudospectral method and collocation method is discussed. Furthermore, a numerical algorithm of Fourier pseudospectral method is developed based on two-dimensional fast Fourier transform (FFT2) technique. Finally, numerical examples are provided to test the theoretical claims. As is shown in the numerical experiments, Fourier spectral methods are powerful enough with excellent efficiency and accuracy.     

Least squares support vector regression for solving Volterra integral equations

In this paper, a numerical approach is proposed based on least squares support vector regression for solving Volterra integral equations of the first and second kind. The proposed method is based on using a hybrid of support vector regression with an orthogonal kernel and Galerkin and collocation spectral methods. An optimization problem is derived and transformed to solving a system of algebraic equations. The resulting system is discussed in terms of the structure of the involving matrices and the error propagation. Numerical results are presented to show the sparsity of resulting system as well as the efficiency of the method.

     

A Fractional Order Collocation Method for Second Kind Volterra Integral Equations with Weakly Singular Kernels

In this paper, we develop a fractional order spectral collocation method for solving second kind Volterra integral equations with weakly singular kernels. It is well known that the original solution of second kind Volterra integral equations with weakly singular kernels usually can be split into two parts, the first is the singular part and the second is the smooth part with the assumption that the integer m being its smooth order. On the basis of this characteristic of the solution, we first choose the fractional order Lagrange interpolation function of Chebyshev type as the basis of the approximate space in the collocation method, and then construct a simple quadrature rule to obtain a fully discrete linear system. Consequently, with the help of the Lagrange interpolation approximate theory we establish that the fully discrete approximate equation has a unique solution for sufficiently large n, where \(n+1\) denotes the dimension of the approximate space. Moreover, we prove that the approximate solution arrives at an optimal convergence order \(\mathcal{O}(n^{-m}\log n)\) in the infinite norm and \(\mathcal{O}(n^{-m})\) in the weighted square norm. In addition, we prove that for sufficiently large n, the infinity-norm condition number of the coefficient matrix corresponding to the linear system is \(\mathcal{O}(\log ^2 n)\) and its spectral condition number is \(\mathcal{O}(1)\). Numerical examples are presented to demonstrate the effectiveness of the proposed method.     

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

     

A collocation and Cartesian grid methods using new radial basis function to solve class of partial differential equations

In the present paper, a class of partial differential equation represented by Poisson's type problems are solved using a proposed Cartesian grid method and a collocation technique using a new radial basis function. The advantage of using this new radial basis function represented by overcoming singularity from the diagonal elements when thin plate radial basis function is used. The new function is a combination of both multiquadric and thin plate radial basis functions. The new radial basis function contains a control parameter ?, that takes one when evaluating the singular elements and equals zero elsewhere. Collocation of the approximate solution of the potential over the governing and boundary condition equations leads to a double linear system of equations. A proposed algebraic procedure is then developed to solve the double system. Examples of Poisson and Helmholtz equations are solved and the present results are compared with the their analytical solutions. A good agreement with analytical results is achieved.     

Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems

The classical collocation and Galerkin methods are used for the numerical solution of singular integral equations of the first kind involving a finite-part integral with a double pole singularity. Such equations appear in plane elasticity crack problems, where they were suggested by Bueckner, and the unknown function in them is proportional to the crack opening displacement function. An application of the proposed methods to the problem of a straight crack under an exponential normal loading distribution is also made and shows the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips to their theoretical values.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

Direct Integration of the Newton Potential over Cubes

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way. Received April 18, 2001; revised September 17, 2001 Published online April 25, 2002     

On kalandiya’ method for the numerical solution of singular integral equations

For the numerical solution of one-dimensional singular integral equations with Cauchy type kernels, one can use an appropriate quadrature rule and an appropriate set of collocation points for the reduction of this equation to a system of linear equations. In this short paper, we use as collocation points the nodes of the quadrature rule and we rederive, in a more direct manner, Kalandiya’ method for the numerical solution of the aforementioned class of equations, which was originally based on a trigonometric interpolation formula. Furthermore, we test this method in numerical applications. Finally, a discussion on the accuracy of the same method is made.     

On the selection of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems

A modification of the collocation method for the numerical solution of Cauchy type singular integral equations with generalized kernels is proposed. In accordance with this modification, although the abscissas and weights used in the numerical integration rule for the approximation of the integrals of the integral equation remain unaltered, yet the collocation points are selected in such a way that the poles of the integrands due not only to the Cauchy principal value part of the kernel, but also to the singularities of the generalized part of the kernel are taken into account. This modification assures the convergence of the method to the correct results since the error terms, usually neglected for the collocation points nearest to the end-points of the integration interval and generally tending to infinity, are now taken into consideration for the selection of the collocation points. The method was applied to the singular integral equations derived for the antiplane and plane elasticity problems of a crack terminating at a bimaterial interface.     

A Fast Collocation Method for Eigen-Problems of Weakly Singular Integral Operators

A multiscale collocation method is developed for solving the eigen-problem of weakly singular integral operators. We employ a matrix truncation strategy of Chen, Micchelli and Xu to compress the collocation matrix, which the compressed matrix has only O(NlogN)\mathcal{O}(N\log N) nonzero entries, where N denotes the order of the matrix. This truncation leads to a fast collocation method for solving the eigen-problem. We prove that the fast collocation method has the optimal convergence order for approximation of the eigenvalues and eigenvectors. The power iteration method is used for solving the corresponding discrete eigen-problem. We present a numerical example to demonstrate how the methods can be used to compute a nonzero eigenvalue rapidly and efficiently.     

CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel

In this paper, a computational method for numerical solution of a class of integro-differential equations with a weakly singular kernel of fractional order which is based on Cos and Sin (CAS) wavelets and block pulse functions is introduced. Approximation of the arbitrary order weakly singular integral is also obtained. The fractional integro-differential equations with weakly singular kernel are transformed into a system of algebraic equations by using the operational matrix of fractional integration of CAS wavelets. The error analysis of CAS wavelets is given. Finally, the results of some numerical examples support the validity and applicability of the approach.