scholarly journals Wang-Ball Polynomials for the Numerical Solution of Singular Ordinary Differential Equations

2021 ◽  
pp. 941-949
Author(s):  
Ahmed Kherd ◽  
Azizan Saaban ◽  
Ibrahim Eskander Ibrahim Fadhel
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Operational Matrix ◽  
Second Order ◽  
Operational Matrices ◽  
Second Order Equations

This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions.

scholarly journals A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 806
Author(s):  
Ali Shokri ◽  
Beny Neta ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Mohammad Mehdi Rashidi ◽  
Hamid Mohammad-Sedighi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability Property ◽  
Stability Region ◽  
Mathieu Equation ◽  
Variable Coefficients ◽  
Second Order ◽  
Implicit Schemes ◽  
The Stability

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

scholarly journals An efficient integrator that uses Gauss-Radau spacings

1985 ◽  
Vol 83 ◽  
pp. 185-202 ◽  
Author(s):  
Edgar Everhart
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Integration ◽  
Second Order ◽  
Double Precision ◽  
The Past ◽  
Fortran Code ◽  
Order Case ◽  
Speed And Accuracy ◽  
Second Order Equations

AbstractThis describes our integrator RADAU, which has been used by several groups in the U.S.A., in Italy, and in the U.S.S.R. over the past 10 years in the numerical integration of orbits and other problems involving numerical solution of systems of ordinary differential equations. First- and second-order equations are solved directly, including the general second-order case. A self-starting integrator, RADAU proceeds by sequences within which the substeps are taken at Gauss-Radau spacings. This allows rather high orders of accuracy with relatively few function evaluations. After the first sequence the information from previous sequences is used to improve the accuracy. The integrator itself chooses the next sequence size. When a 64-bit double word is available in double precision, a 15th-order version is often appropriate, and the FORTRAN code for this case is included here. RADAU is at least comparable with the best of other integrators in speed and accuracy, and it is often superior, particularly at high accuracies.

scholarly journals Linearizations of ordinary differential equations by area preserving maps

1999 ◽  
Vol 156 ◽  
pp. 109-122 ◽  
Author(s):  
Tetsuya Ozawa ◽  
Hajime Sato
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Contact Form ◽  
Third Order ◽  
Area Preserving Maps ◽  
Geometric Explanation ◽  
Coordinate Changes ◽  
Area Preserving ◽  
Second Order Equations

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations Y″ = 0 and Y‴ = 0. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

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