Essay: On Analytical Approximate Solutions of Nonlinear Problems

1. Working Title:
On Analytical Approximate Solutions of Nonlinear Problems
2. On Analytical Approximate Solutions of Nonlinear Problems:
The main purpose of this technique is to resolve linear and nonlinear problems by using of differential transform method. We get the approximate solution with the help of differential transformation method (DTM), and also catch the quickly convergent series. Comparisons with exact solution show that the DTM is a most dominant method to catch the exact solution of nonlinear Ordinary differential equations.
3. Introduction:
The differential equations which can show very complicated behavior over long time intervals, is the non-linear differential equation. Even the major questions of existence, uniqueness, and extendibility of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) are tough problems and their purpose in special cases is considered to be a forceful advance in the mathematical theory. The set of simultaneous equations in which the unknowns appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one is called the non-linear set of equations . The nature of a nonlinear system is defined by a nonlinear system of equations.
Nonlinear problems are of interest to engineers, physicists and mathematicians and many other scientists because most systems are characteristically nonlinear in nature. Non-linear differential equations are solved by few methods to find the exact solution. Many effective methods have been proposed for finding the analytical solutions. For example, Adomian decomposition method (ADM), homotopy perturbation method (HPM), variation iteration method (VIM), and differential transform method (DTM) .The Adomian decomposition method (ADM) proposed by George Adomian . The homotopy perturbation method (HPM) and variation iteration method (VIM) proposed by Ji-Huan He. These methods provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions of differential equations.
The analytical method for solving differential equations is the differential transformation method. The idea of differential transform was first introduced by Zhou (1986), who was the first one to use differential transform method (DTM) in engineering applications. This method is based on Taylor’s series method. He applied DTM in solution of initial boundary value problems for the analysis of electric circuit. In recent years, theory of DTM has extended to the problems involving Ordinary differential equations and systems of differential equations. This method makes an analytical solution in the form of polynomial. Traditional higher order Taylor’s series method is different from Differential Transformation method, which involves symbolic computation of the necessary derivatives of the data functions. The method for computationally taken long time for large orders is Taylor’s series method. This method avoids the large computation values and rounding off error, it gives exact analytical solution. The DTM is an iterative process for obtaining the analytic Taylor’s series solution of ordinary or partial differential equations.
4. Research Question:
Is it Differential transformation Method always effective and gives consistent solution for non-linear problems and systems?
5. Hypothesis:
It will be confirmed about the reliability and efficiency of proposed method.
6. Literature Review:
Since the beginning of 1986, Zhou and Pukhov have introduced a so-called differential transformation method (DTM) for electrical circuit’s problems. The DTM is a technique that uses Taylor’s series for the solution of differential equations in the form of a polynomial. It is different from the high-order Taylor’s series method, which involves symbolic computation of the necessary derivatives of the data functions. Computationally tedious method for high order equation is Taylor’s series method. The Differential transform method is the indications of an iterative process for obtaining an analytic series solutions of functional equations. In most recent years researchers have practically apply the method to various linear and nonlinear problems such as two point boundary value problems by Chen and Liu [3], partial differential equations by Jang et al. [4], Ayaz worked in differential-algebraic equations[5], the KdV and mKdV equations by Kangalgil and Ayaz[6,7], Arikoglu and Ozkol worked in integro-differential [8],fractional differential equations by Arikoglu and Ozkol [9],the Schrodinger equations by Ravi Kanth and Aruna[10], Nonlinear gas dynamic equation by Hossein Jafari[11],Working in Systems of Volterra Integral Equations of the First Kind by Biazar and Eslami [12],Jaulent??Miodek (JM) equation, the equation Hirota-Satsuma by Raslan and zain[13],nonlinear ordinary differential equations by Yinwei Lin[14,15].
7. Method Approach:
This method provides analytical convergent solution.
8. Chapter & Dissertation Breakdown:

8.1. Chapter 1.
In the first chapter, we shall give some preliminaries and analysis of the DTM and literature review of application of proposed method
8.2. Chapter 2.
In second chapter is dedicated to the applications of Differential Transformation Method (DTM) for solving nonlinear Ordinary differential equations (ODEs) and graphically checks the efficiency of the Differential transformation method (DTM).
8.3. Chapters 3.
The third chapter is dedicated to the applications of Differential Transformation Method (DTM) for solving system of Ordinary differential equations (ODEs) and graphically checks the efficiency of the Differential transformation method (DTM).

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