P A ERROR ANALYSIS OF EXTENDED EXPONENTIAL GENERAL LINEAR METHODS OF TYPE 524 (original) (raw)

Construction of extended exponential general linear methods 524 for solving semi-linear problems

2014

This paper introduces a new approach for constructing higher order of EEGLM which have become very popular and novel due to its enviable stability properties. This paper also shows that methods 524 is stable with its characteristics root lies in a unit circle. Numerical experiments indicate that Extended Exponential General Linear Methods perform better than existing Methods. Key words : General Linear Methods, Exponential methods.

CONVERGENCE AND STABILITY ANALYSIS OF EXTENDED EXPONENTIAL GENERAL LINEAR METHODS

This paper dwells on the stability analysis of extended exponential general linear methods. Like the paper of Butcher [1], we are able to show using the root locus method that the various methods constructed posses favorable stability properties as they are zero stable as their parasitic roots lie in a unit circle. It is also shown that for positive stepsizes, the error estimate holds with a positive constant, independent of n and the stepsizes (h). Experimental experience reveals that our scheme converges.

Construction of Extended Exponential General Linear Methods

International Journal of Apllied Mathematics, 2014

This paper is concerned with the construction and numerical analysis of Extended Exponential General Linear Methods (EEGLM). These methods are related to the methods of Butcher [1] and Calvo and Palencia [3] but, in contrast to the latter, we make use of higher terms of the exponential and related matrix functions. This feature enables us to derive the order conditions which in turn aided in the construction of family of methods of higher order. The numerical experiments indicate that Extended Exponential General Linear Methods perform better than the existing methods.

Stability and Accuracy of Exponential Extrapolation Method

International Journal on Advanced Science, Engineering and Information Technology, 2012

The stability of a numerical methods is important consideration that must be considered when solving ODEs, where Since the extrapolation methods are best away to get a high-accuracy solution of ODEs. In this paper we studied the stability and accuracy of the exponential extrapolation method which is in the form of y=Ae αh where A and α are constant to be determined using least square fitting. We gave example to explain the accuracy of this method and we explained that the stability region of exponential extrapolation method always inside the | 1+ z | ≤ 1.

A New Numerical Algorithm to Compute a Root of Non-linear Equations using Exponential Method

International Symposium on Mathematical Analysis of Fractalsand Dynamical Systems (ISMAFDS--2023), 2023

This paper presents a three-step numerical root-finding algorithm using the exponential method to compute an approximate root of the given non-linear equations in one variable. The main idea of the proposed algorithm is based on the predictor-corrector method, which is developed by Traub. The proposed algorithm is efficient, easy to understand and convergent faster than other existing methods. Several numerical examples are presented to illustrate the method. Maple implementation of the proposed algorithm is discussed with sample computations. Using the proposed algorithm, one can extend the implementation in other mathematical software tools such as MATLAB, Mathematica, SCILab etc.

A Family of Nested General Linear Methods for Solving Ordinary Differential Equations

Asian Research Journal of Mathematics

General linear methods (GLMs) was introduced as a generalization of Runge{Kutta methods (RKMs) and linear multistep methods (LMMs). The discovery of general linear method gave insight into the discovery of new methods that are neither RKMs or LMMs. Here, new classes of GLMs that are nested in their stages and mono-implicit in the output are presented, these methods are referred to as nested general linear methods (NGLMs). Procedures for deriving members that are algebraically stable are discussed herein and algebraically stable NGLMs have been derived up to order p = 5. Implementation procedure of these nested general linear methods which include the solution of non-linear systems of equations by simplified Newton iterations and step size changing strategy are discussed. The order p = 3 NGLM has been implemented on two test problems by variable step size, and the results compared with the results of MATLAB ode15s and RADAU IIA.

The local modified extrapolated Gauss–Seidel (LMEGS) method

Computers & Structures, 2004

In this paper we present the convergence analysis of the Local Modified Extrapolated Gauss-Seidel (LMEGS) method. The related theory of convergence is developed. Convergence ranges and optimum values for the involved parameters of the LMEGS method are obtained. It is proved that even if l, the smallest in absolute value eigenvalue of the iteration matrix of the Jacobi method, becomes larger than unity LMEGS will converge. In fact, the larger l the faster the convergence of LMEGS.

An analysis of the properties of the variants of Newton's method with third order convergence

Applied Mathematics and Computation, 2006

This paper proposes a new numerical method for the solution of the Initial Value Problems (IVPs) of first order ordinary differential equations. The new scheme has been derived via the transcendental function of exponential type. The analysis of the properties of the method such as local truncation error, order of accuracy, consistency, stability and convergence were investigated. Two illustrative examples/test problems were solved successfully to test the accuracy, performance and suitability of the method in terms of the absolute relative errors computed at the final nodal point of the associated integration interval via MATLAB codes. It is observed that the method is found to be of third order convergence, consistent and stable. The numerical results obtained via the method agree with the exact solution. Moreover, it is also observed that the method is an improvement on Fadugba-Falodun scheme. Hence, the proposed numerical method is a good approach for solving the IVPs of various nature and characteristics in diverse areas of Ordinary Differential Equations (ODEs).

Hybrid Method for Solving Special Fourth Order Ordinary Differential Equations | Ibrahim Badamasi Babangida University Open Education Resources (OER)

2019

In recent time, Runge-Kutta methods that integrate special fourth order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Although, the methods require approximation of y′, y′′ and y′′′ of the solution at every step. In this paper, a hybrid type method is proposed, which can directly integrate special fourth order ODEs. The method does not require the approximation of any derivatives of the solution. Algebraic order conditions of the methods are derived via Taylor series technique. Using the order conditions, eight algebraic order method is presented. Absolute stability of the method is analyzed and the stability region presented. Numerical experiment is conducted on some test problems. Results from the experiment show that the new method is more efficient and accurate than the existing Runge-Kutta and hybrid methods with similar number of function evaluation.

The modified method of successive linear problems by Taylor expansion

This paper is due to modifying the Method of Successive Linear Problems (MSLP) in solving non-linear eigenvalue problems. The MSLPis a basic method for acquiring roots of non-linear equations with the second rank convergence. In this paper, both the second and the third Taylor expansion are applied for improving the speed of convergence in order to acquire the exact roots of non-linear equations. Numericalexamples are used to illustrate the method results.