scholarly journals A Jacobi Dual-Petrov-Galerkin Method for Solving Some Odd-Order Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
E. H. Doha ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
Fully Integrated ◽  
Polynomial Coefficients ◽  
Odd Order ◽  
Constant Coefficients ◽  
Fifth Order

A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for theJth order ODE involvesn-fold indefinite integrals forn=1,…,J. Extension of the JDPG for ODEs with polynomial coefficients is treated using the Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.

scholarly journals Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kasim Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Order Conditions ◽  
Numerical Results ◽  
Type Method ◽  
First Order ◽  
New Methods ◽  
Runge Kutta ◽  
Fifth Order

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

scholarly journals Solving Second-Order Delay Differential Equations by Direct Adams-Moulton Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hoo Yann Seong ◽  
Zanariah Abdul Majid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Direct Method ◽  
Second Order ◽  
Numerical Results ◽  
Step Size ◽  
First Order ◽  
Delay Differential ◽  
Fifth Order

This paper will consider the implementation of fifth-order direct method in the form of Adams-Moulton method for solving directly second-order delay differential equations (DDEs). The proposed direct method approximates the solutions using constant step size. The delay differential equations will be treated in their original forms without being reduced to systems of first-order ordinary differential equations (ODEs). Numerical results are presented to show that the proposed direct method is suitable for solving second-order delay differential equations.

scholarly journals Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
M. El-Kady ◽  
S. M. El-Sayed ◽  
H. E. Fathy
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Quadrature Formula ◽  
Numerical Results ◽  
Basis Functions ◽  
Gauss Quadrature ◽  
Gauss Quadrature Formula ◽  
Huxley Equation ◽  
Numerical Treatments

Numerical treatments for the generalized Burger's—Huxley GBH equation are presented. The treatments are based on cardinal Chebyshev and Legendre basis functions with Galerkin method. Gauss quadrature formula and El-gendi method are used to convert the problem into a system of ordinary differential equations. The numerical results are compared with the literatures to show efficiency of the proposed methods.

scholarly journals A NEW SUPERCLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

2014 ◽  
Vol 07 (01) ◽  
pp. 1350034 ◽  
Author(s):  
M. B. Suleiman ◽  
H. Musa ◽  
F. Ismail ◽  
N. Senu ◽  
Z. B. Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Results ◽  
New Method ◽  
Differentiation Formula ◽  
Stiff Ordinary Differential Equations ◽  
Backward Differentiation Formula ◽  
Error Constant

A superclass of block backward differentiation formula (BBDF) suitable for solving stiff ordinary differential equations is developed. The method is of order 3, with smaller error constant than the conventional BBDF. It is A-stable and generates two points at each step of the integration. A comparison is made between the new method, the 2-point block backward differentiation formula (2BBDF) and 1-point backward differentiation formula (1BDF). The numerical results show that the method developed outperformed the 2BBDF and 1BDF methods in terms of accuracy. It also reduces the integration steps when compared with the 1BDF method.

Sign in / Sign up

Export Citation Format

Share Document