Adaptive power series solution for second order ordinary differential equations with initial conditions

2015 ◽  
Author(s):  
Tarek I. Haweel ◽  
AM. Alhasan
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Series Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Power Series Solution ◽  
Adaptive Power

Analytic Solution to a Virus Infection Model

2013 ◽  
Vol 367 ◽  
pp. 503-507
Author(s):  
Liang Xu ◽  
Sha Xu
Keyword(s):  
Virus Infection ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Series Solution ◽  
Infection Model ◽  
Power Series Solution ◽  
Analytic Method ◽  
Linear Ordinary Differential Equations ◽  
Virus Infection Model

A functional analytic method was developed by E.K.Ifantis in 1987 to prove that certain non-linear ordinary differential equations (ODEs) have a unique power series solution which converges absolutely in a specified disc of the complex plane. In this paper, we first applied this method to certain systems of two non-linear ordinary differential equations. We proved that the power series solutions can be determined by some recurrence relations which depend on the parameters of the equations and the initial conditions. Then, we found a method to extend the range of the converge bound. At last, we applied the functional analytic method to the resistant virus infection model to obtain a power series solution and compared our solution with the numerical solution obtained by the Runge-Kutta method using the software Matlab (Version 7.0.1).

scholarly journals Modified nonlinearities distribution Homotopy Perturbation method as a tool to find power series solutions to ordinary differential equations

Nova Scientia ◽  
2014 ◽  
Vol 6 (12) ◽  
pp. 13 ◽  
Author(s):  
Umberto Filobello-Nino ◽  
Héctor Vázquez-Leal ◽  
Yasir Khan ◽  
D. Pereyra-Díaz ◽  
A. Pérez-Sesma ◽  
...  
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Series Solutions ◽  
Homotopy Perturbation ◽  
Polynomial Coefficients ◽  
Power Series Solutions

In this article, modified non-linearities distribution homotopy perturbation method (MNDHPM) is used in order to find power series solutions to ordinary differential equations with initial conditions, both linear and nonlinear. We will see that the method is particularly relevant in some cases of equations with non-polynomial coefficients and inhomogeneous non-polynomial terms

DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 4 (4) ◽  
pp. 477-483
Author(s):  
O. E. Abolarin ◽  
B. G. Ogunware ◽  
A. F. Adebisi ◽  
S. O. Ayinde
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Multistep Method ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Order Error ◽  
Block Algorithm ◽  
Grid Points

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

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