Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results

A Shifted Jacobi-Gauss Collocation Scheme for Solving Fractional Neutral Functional-Differential Equations

The shifted Jacobi-Gauss collocation (SJGC) scheme is proposed and implemented to solve the fractional neutral functional-differential equations with proportional delays. The technique we have proposed is based upon shifted Jacobi polynomials with the Gauss quadrature integration technique. The main advantage of the shifted Jacobi-Gauss scheme is to reduce solving the generalized fractional neutral functional-differential equations to a system of algebraic equations in the unknown expansion. Reasonable numerical results are achieved by choosing few shifted Jacobi-Gauss collocation nodes. Numerical results demonstrate the accuracy, and versatility of the proposed algorithm

A new numerical scheme for solving system of Volterra integro-differential equation

In this article, we apply Genocchi polynomials to solve numerically a system of Volterra integro-differential equations. This is done by approximating functions using Genocchi polynomials and derivatives using Genocchi polynomials operational matrix of integer order derivative. Com-bining approximation with collocation method, the problem is reduced to a system of algebraic equations in terms of Genocchi coefficients of the unknown functions. By solving the Genocchi coefficients, we obtain good approximate functions of the exact solutions of the system. A few numerical examples show that our proposed Genocchi polynomials method achieves better accu-racy compared to some other existing methods

Numerical Computation of Travelling Wave Solutions for the Nonlinear Ito System

The Ito equation (a coupled nonlinear wave equation which generalizes the KdV equation) has previously been shown to admit a reduction to a single nonlinear Casimir equation governing the wave solutions. Some analytical properties of the solutions to this equation in certain parameter regimes have been studied recently. However, for general parameter regimes where the analytical approach is not so useful, a numerical method would be desirable. Therefore, in this paper, we proceed to show that the Casimir equation for the Ito system can be solved numerically by use of the shifted Jacobi-Gauss collocation (SJC) spectral method. First, we present the general solution method, which is follows by implementation of the method for specific parameter values. The presented results in this article demonstrate the accuracy and efficiency of the method. In particular, we demonstrate that relatively few notes permit very low residual errors in the approximate numerical solutions. We are also able to show that the coefficients of the higher order terms in the shifted Jacobi polynomials decrease exponentially, meaning that accurate solutions can be obtained after relatively few terms are used. With this, we have a numerical method which can accurately and efficiently capture the behavior of nonlinear waves in the Ito equation

A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation

A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x)PL,m(y), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. 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

A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line

The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line

On a generalized Laguerre operational matrix of fractional integration

A new operationalmatrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived.The fractional integration is described in the Riemann-Liouville sense.This operational matrix is applied together with generalized Laguerre tau method for solving general linearmultitermfractional differential equations (FDEs).Themethod has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposedmethod is very effective and convenient for linear multiterm FDEs on a semi-infinite interval

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms

An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of powerlaw nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials J(n)((alpha,beta))(r) with alpha,beta epsilon (-1, infinity), r epsilon (0,1) and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity

Fractional and time-scales differential equations

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