الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
Nonlinear differential equations are the backbone in modeling many applications in physics and engineering. However, analytical methods can’t find exact solutions for most cases. Different approximate analytical methods have been introduced like perturbation expansion method, Lindstedt-Poincaré method, homotopy perturbation method, homotopy analysis method, variational iteration method, harmonic balance method and energy balance methods. The importance of the semi-analytical solutions appears in systems optimization. Power series method which considers a Taylor series solution is a promising method for solving nonlinear differential equations. Though for a long time the method has been used for linear problems, not long ago Parker and Sochacki (1996) expanded the power series method to solve nonlinear ordinary differential equations. This method does not need a small parameter like the perturbation methods. It does not need a trial function like the homotopy methods. However, power series solutions have some drawbacks. In general, they suffer from the problem of limited convergence intervals and outside these intervals the solutions are not valid. In particular, power series solutions do not demonstrate the periodic behavior which is the main feature of oscillators.
However, Since Parker and Sochacki (1996) presented their method; no researcher presented any study to improve the solution of Parker-Sochacki Method. Exclusively, in this work, four new approaches have been proposed to improve the solution of Parker-Sochacki Method and to overcome the drawbacks of the method. The proposed approaches are called Parker- Sochacki Padé method (PSPM), modified Parker-Sochacki method (MPSM), improved Parker-Sochacki method (PSM-AT), and extended Parker-Sochacki method (EPSM). These approaches depend on extending Parker-Sochacki method by combining it with four resummation techniques, namely: Padé resummation method, Laplace-Padé resummation method, periodic aftertreatment techniques (cosine aftertreatment technique and sine aftertreatment technique) and for the first time a new proposed resummation method called Sumudu-Padé resummation method.
The proposed approaches have been applied for solving different applications. The first one is solving different types of nonlinear heat transfer equations, namely: cooling of a combined convection-radiation system, cooling of a variable specific heat system, cooling of a variable heat transfer coefficient system, and non-Fourier conduction heat transfer equation with variable specific heat. The second application is solving Michaelis–Menten biochemical reaction model. The third application is solving several types of nonlinear oscillatory systems which have their widespread usages in mechanical vibrations, solid and structural mechanics, as well as fluid mechanics. The solved oscillatory systems are Duffing oscillator with cubic nonlinearity, Duffing oscillator with quintic nonlinearity, Helmholtz oscillator, relativistic oscillator, oscillation of a mass attached to elastic wire, exact Duffing oscillator equation, Duffing harmonic oscillator and Van der Pole oscillator.
To test the performance and correctness of the presented approaches, the obtained solutions are compared with the exact solutions if exist and with Runge-Kutta numerical solutions if there are no exact solutions. Also comparison between the obtained solutions and some solutions using