The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

     

A Haar wavelet collocation approach for solving one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one- and two-dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one- and two-dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate.     

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

Application of wavelet transforms to the solution of boundary value problems for linear parabolic equations

A method based on wavelet transforms is proposed for finding weak solutions to initial-boundary value problems for linear parabolic equations with discontinuous coefficients and inexact data. In the framework of multiresolution analysis, the general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. An error estimate is obtained for the stable approximate solution obtained by solving a set of linear algebraic equations for the wavelet coefficients of the desired solution.     

New approach to the Lax‐Wendroff modified differential equation for linear and nonlinear advection

The paper presents an enhanced analysis of the Lax‐Wendroff difference scheme—up to the eighth‐order with respect to time and space derivatives—of the modified‐partial differential equation (MDE) of the constant‐wind‐speed advection equation. The modified equation has been so far derived mainly as a fourth‐order equation. The Π ‐form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth‐order of the analyzed modified differential equation for the second‐order Lax‐Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 – 8) in the modified differential equation for the Lax‐Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two‐step variants of the Lax‐Wendroff type difference schemes and the MacCormack predictor–corrector scheme (see MacCormack's study) constructed for the scalar hyperbolic conservation laws are also presented in this paper. The analysis of the inviscid Burgers equation solution with the initial condition in a form of a shock wave has been discussed on their basis. The inviscid Burgers equation with the source is also presented. The theory of MDE started to develop after the paper of C. W. Hirt was published in 1968.     

On sinc discretization and banded preconditioning for linear third‐order ordinary differential equations

Some draining or coating fluid‐flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third‐order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd.     

On the existence of a stable limit cycle to a piecewise linear system in R2

The present paper is devoted to the existence of limit cycles of planar piecewise linear (PWL) systems with two zones separated by a straight line and singularity of type “focus-focus” and “focus-center.” Our investigation is a supplement to the classification of Freire et al concerning the existence and number of the limit cycles depending on certain parameters. To prove existence of a stable limit cycle in the case “focus-center,” we use a pure geometric approach. In the case “focus-focus,” we prove existence of a special configuration of five parameters leading to the existence of a unique stable limit cycle, whose period can be found by solving a transcendent equation. An estimate of this period is obtained. We apply this theory on a two-dimensional system describing the qualitative behavior of a two-dimensional excitable membrane model.     

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.     

On the stability of the spline collocation method for a class of integral equations of the first kind

This paper is concerned with the stability of the spline collocation method for a class of integral equations of the first kind with logarithmic kernels. It is shown that a proper choice of the mesh size can be made in the numerical computation so that one will obtain an optimal rate of convergence for the approximate solutions.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

某类线性微分方程亚纯解的增长性

研究了一类亚纯函数系数的线性微分方程的解的增长性问题,得到了齐次和非齐线性微分方程亚纯解的增长级、超级、二级不同零点收敛指数的精确估计．     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions

The paper deals with the impulsive nonlinear boundary value problem

     

On the kernel space for an L0‐linear function

A random normed module is a random generalization of an ordinary normed space, and it is the randomization that makes a random normed module possess rich stratification structures. On the basis of these stratification structures, this paper shows that either the kernel space N(f) for an L⁰‐linear function f from a random normed module S to the algebra is a closed submodule or N(f) on some specifical stratification is a dense proper submodule of S, which generalizes the classical case. In the meantime, a characterization for the kernel space N(f) to be closed is also given.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

The global solvability and asymptotic behavior of a transmission problem for Kirchhoff‐type wave equations with memory source on the boundary

A transmission problem for Kirchhoff‐type wave equations with memory source term on one part of the boundary feedback is considered. By using the Faedo‐Galerkin approximation technique, the method of Lyapunov functional and the energy perturbation technique, we establish well‐posedness of global solution and derive a general decay estimate of the energy.     

Application of collocation method for solving a parabolic‐hyperbolic free boundary problem which models the growth of tumor with drug application

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first‐order hyperbolic equations are given that describe the evolution of cells and other two second‐order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.     

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane–Emden equations

The purpose of this paper is to present a numerical algorithm for solving the Lane–Emden equations as singular initial value problems. The proposed algorithm is based on an operational Tau method (OTM). The main idea behind the OTM is to convert the desired problem to some operational matrices. Firstly, we use a special integral operator and convert the Lane–Emden equations to integral equations. Then, we use OTM to linearize the integral equations to some operational matrices and convert the problem to an algebraic system. The concepts, properties, and advantages of OTM and its application for solving Lane–Emden equations are presented. Some orthogonal polynomials are also used to reduce the volume of computations. Finally, several experiments of Lane–Emden equations including linear and nonlinear terms are given to illustrate the validity and efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.