Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.     

Discrete Legendre spectral Galerkin method for Urysohn integral equations

In this paper, we consider the discrete Legendre spectral Galerkin method to approximate the solution of Urysohn integral equation with smooth kernel. The convergence of the approximate and iterated approximate solutions to the actual solution is discussed and the rates of convergence are obtained. In particular we have shown that, when the quadrature rule is of certain degree of precision, the superconvergence rates for the iterated Legendre spectral Galerkin method are maintained in the discrete case.     

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

Legendre wavelets method for approximate solution of fractional-order differential equations under multi-point boundary conditions

In this paper, Legendre wavelet collocation method is applied for numerical solutions of the fractional-order differential equations subject to multi-point boundary conditions. The explicit formula of fractional integral of a single Legendre wavelet is derived from the definition by means of the shifted Legendre polynomial. The proposed method is very convenient for solving fractional-order multi-point boundary conditions, since the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. Several numerical examples are solved to demonstrate the validity and applicability of the presented method.     

Legendre spectral method for solving integral and integro-differential equations

A new spectral approximation of an integral based on Legendre approximation at the zeros of the first term of the residual is presented. The method is used to solve integral and integro-differential equations. The method generates approximations to the lower order derivatives of the function through successive integrations of the Legendre polynomial approximation to the highest order derivative. Numerical results are included to confirm the efficiency and accuracy of the method.     

An iterative integration scheme for solving second-order ordinary differential equations

This paper presents an iterative solution method for the numerical integration of second-order ordinary differential equations using a simple program for microcomputers (PC). The method of integration proposed is based on the geometrical considerations in the phase plane. The numerical results are compared to those obtained by the fourth-order Runge-Kutta method and to the closed form solutions when possible. Tests show good accuracy and, in some cases, computer time saving with respect to the Runge-Kutta's method for th same accuracy. The method of integration in the phase plane seems very good for treating every kind of nonlinear second-order differential equation whatever the degree of nonlinearity.     

A new solution method for stochastic differential equations via collocation approach

The objective of this paper is to propose a novel solution method for Itô stochastic differential equations (SDEs). It is discussed that how the SDEs could numerically be solved as matrix problems. To improve the accuracy of this technique in contrast to the existing solvers, some non-uniform grids of points for discretizations along the time direction are applied. Finally, the high accuracy of approximated solutions in this way are illustrated by several experiments.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

Study of general Taylor-like explicit methods in solving stiff ordinary differential equations

In this paper, we present a further study of Taylor-like explicit methods in solving stiff ordinary differential equations. We derive the general form for Taylor-like explicit methods in solving stiff differential equations. We also analyse the order of convergence and stability property for the general form. Moreover, we give its corresponding vector form via introducing a new definition of vector product and quotient in another article. The convergence and stability of the vector form are considered as well.     

Stability of 2h-step spline method for delay differential equations

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.     

On solving systems of ordinary differential equations on MIMD-computers

The error of solution of Cauchy problems for systems of ordinary differential equations is estimated in the case where the input data are approximate. It is shown how to prepare a program for computing the right-hand sides of the system automatically and simultaneously. Diagrams are presented to illustrate the efficiency of parallelization. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 175–182, March–April 2007.     

A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument

In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre–Gauss–Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method.     

Series solution of nonlinear differential equations by a novel extension of the Laplace transform method

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration.     

Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis

In this paper, a spectral Tau method based on Legendre Wavelet basis is proposed. For this purpose we present a stable operational Tau method based on Legendre Wavelet basis. This method provides an efficient approximate solution for weakly singular Volterra integral equations by using reduced set of matrix operations. An error estimation of the Tau method is also introduced. Finally we demonstrate the validity and applicability of the method by numerical examples.     

The weighted continuous galerkin scheme for ordinary differential equations

We introduce the Weighted Continuous Galerkin Scheme for initial value ordinary differential equations. This is an extension of the Continuous Galerkin Scheme, having an extra parameter for the purpose of error reduction. We prove convergence in the L ₂ norm in the time variable in a new way, similar to (elliptic) finite element techniques. Using the optimal L ₂ estimates, we then prove max norm convergence. Numerical evidence for the effectiveness of the proposed scheme is presented.     

Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.     

Numerical study of Fisher's equation by wavelet Galerkin method

Fisher's equation, which describes the logistic growth–diffusion process and occurs in many biological and chemical processes, has been studied numerically by the wavelet Galerkin method. Wavelets are functions which can provide local finer details. The solution of Fisher's equation has a compact support property and therefore Daubechies' compactly supported wavelet basis has been used in this study. The results obtained by the present method are highly encouraging and can be computed for a large value of the linear growth rate.