The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

Numerical analysis of a high order method for state-dependent delay integral equations

In this paper we study the piecewise collocation method for a class of functional integral equations with state-dependent delays that is, where the delays depend on the solution. It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called ”singular points”, which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. Here, we present a numerical algorithm based on the piecewise collocation method and an approach for tracking the singular points relying on the recent strategy for implicit delay differential equations which has been proposed by Guglielmi and Hairer in 2008. The convergence analysis of the method is investigated and some numerical experiments are presented for clarifying the robustness of the method.     

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using weighted ENO local Lax-Friedrichs methods as developed recently by Jiang and Peng. We verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent Hokkaido University preprint. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.

     

椭圆-抛物偏微分方程奇异摄动问题的差分解法

本文讨论了含有小参数在高阶导数项的椭圆型方程奇异摄动问题的差分解法.当ε=0时椭圆型方程退化为抛物型方程.作者根据此问题解的边界层性质,构造了特殊的差分格式:研究了它的收敛性和解的渐近性态.最后给出一个数值例题.     

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

In this paper, a fast numerical algorithm based on the Taylor wavelets is proposed for finding the numerical solutions of the fractional integro‐differential equations with weakly singular kernels. The properties of Taylor wavelets are given, and the operational matrix of fractional integration is constructed. These wavelets are utilized to reduce the solution of the given fractional integro‐differential equation to the solution of a linear system of algebraic equations. Also, convergence of the proposed method is studied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

Bifurcation in the neighbourhood of a non-isolated singular point

The Lyapunov-Schmidt method for bifurcation problems has, until recently, been applied only to operator equations whose singular points are isolated in the solution set of the equation. For bifurcation at a multiple eigenvalue involving several parameters, however, singular points are often non-isolated. In this paper, the case of intersecting curves of singular points is considered. Under natural hypotheses on these curves, and assuming suitable transversality conditions on the first order nonlinearity of the operator, it is shown that the solution set of the equation may be completely determined locally in terms of the solutions of associated finite dimensional polynomial equations.     

Weighted least squares solutions to general coupled Sylvester matrix equations

This paper is concerned with weighted least squares solutions to general coupled Sylvester matrix equations. Gradient based iterative algorithms are proposed to solve this problem. This type of iterative algorithm includes a wide class of iterative algorithms, and two special cases of them are studied in detail in this paper. Necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented. Sufficient conditions that are easy to compute are also given. The optimal step sizes such that the convergence rates of the algorithms, which are properly defined in this paper, are maximized and established. Several special cases of the weighted least squares problem, such as a least squares solution to the coupled Sylvester matrix equations problem, solutions to the general coupled Sylvester matrix equations problem, and a weighted least squares solution to the linear matrix equation problem are simultaneously solved. Several numerical examples are given to illustrate the effectiveness of the proposed algorithms.     

A Twofold Spline Chebyshev Linearization Approach for a Class of Singular Second-Order Nonlinear Differential Equations

The aim of this paper is to present a twofold approach for the numerical solution of a class of singular second-order nonlinear differential equations. The first is based on a modified version of an adaptive spline collocation method (ASCM). The second is a patching approach (PASCM) that splits the problem domain into two subintervals: Chebyshev economization procedure is implemented in the vicinity of the singular point and outside this domain the resulting initial or boundary value problem is handled by the (ASCM). The second strategy is based on the linearization of the nonlinear term about the given initial condition at the singular point. The choice of either technique relies on the specified boundary or initial conditions. Performance of the approach is investigated numerically through a number of application examples that demonstrate the efficiency of the approach and that it has O(h ⁴) rate of convergence. Results confirm that the scheme yields highly accurate results when compared with the exact and/or numerical solutions that exist in the literature.     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

Discretization Methods for Three-Dimensional Singular Integral Equations of Electromagnetism

Theorems providing the convergence of approximate solutions of linear operator equations to the solution of the original equation are proved. The obtained theorems are used to rigorously mathematically justify the possibility of numerical solution of the 3D singular integral equations of electromagnetism by the Galerkin method and the collocation method.     

Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels

In this paper, we introduce efficient methods for the approximation of solutions to weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels. Based on the asymptotic analysis of the solution, we derive corresponding convergence rates in terms of the frequency for the Filon method, and for piecewise constant and linear collocation methods. We also present asymptotic schemes for large values of the frequency. These schemes possess the property that the numerical solutions become more accurate as the frequency increases.     

Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities

In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution.     

Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem

In this paper, we consider Galerkin method for weakly singular Fredholm integral equations of the second kind and its corresponding eigenvalue problem using Legendre polynomial basis functions of degree ≤n. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind and also obtain the error bounds for the approximated eigenelements in the corresponding eigenvalue problem. We illustrate our results with numerical examples.     

大范围求解非线性方程组的指数同伦法

为了解决关于奇异的非线性方程组求根问题,提出了一种由同伦算法推出大范围收敛的连续型方法-指数同伦法,构造了一类指数同伦方程,克服了Jacobi矩阵的奇异,分析了指数同伦方     

Approximate solution for a class of two-dimensional nonlinear singular integral equations by the contraction mapping method

We obtain solutions for a class of two-dimensional nonlinear singular integral equations with Hilbert kernel using the contraction mapping method and find the rate of convergence of successive approximations to the exact solution.     

Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

A two-point boundary value problem is considered on the interval [0, 1], where the leading term in the differential operator is a Riemann-Liouville fractional derivative of order 2 ? δ with 0 < δ < 1. It is shown that any solution of such a problem can be expressed in terms of solutions to two associated weakly singular Volterra integral equations of the second kind. As a consequence, existence and uniqueness of a solution to the boundary value problem are proved, the structure of this solution is elucidated, and sharp bounds on its derivatives (in terms of the parameter δ) are derived. These results show that in general the first-order derivative of the solution will blow up at x = 0, so accurate numerical solution of this class of problems is not straightforward. The reformulation of the boundary problem in terms of Volterra integral equations enables the construction of two distinct numerical methods for its solution, both based on piecewise-polynomial collocation. Convergence rates for these methods are proved and numerical results are presented to demonstrate their performance.