Solution of Nonlinear Partial Differential Equations by Mixture Adomian Decomposition Method and Sumudu Transform

This chapter is fundamentally centering on the application of the Adomian decomposition method and Sumudu transform for solving the nonlinear partial differential equations. It has instituted some theorems, definitions, and properties of Adomian decomposition and Sumudu transform. This chapter is an elegant combination of the Adomian decomposition method and Sumudu transform. Consequently, it provides the solution in the form of convergent series, then, it is applied to solve nonlinear partial differential equations.


Introduction
Many of nonlinear phenomena are a necessary part in applied science and engineering fields [1]. The wide use of nonlinear partial differential equations is the most important reason why they have drawn mathematician's attention. Despite this, they are not easy to find an answer, either numerically or theoretically. In the past, active study attempts were given a large amount of attention to the study of getting exact or approximate solutions of this kind of equations.
In this chapter, our presentation will be based on applying the new method, namely the Adomian Decomposition Sumudu Transform Method (ADSTM) for solving the nonlinear partial differential equations. This method is an elegant combination of the Sumudu transform method and decomposition method. 1

Sumudu Transform
A long time ago, differential equations wared a necessary part in all aspects of applied sciences and engineering fields. In this chapter we need to develop a new technique for help us to obtain the exact and approximate solutions of these differential equations.
Watugula [13] introduced a new integral transform and called it as Sumudu transform, which is defined as: Watugula [13] applied this transforms to the solution of ordinary differential equations. Because of its useful properties, the Sumudu transforms helps in solving complex problems in applied sciences and engineering mathematics. Henceforward, is the definition of the Sumudu transforms and properties describing the simplicity of the transform.

The relation between Sumudu and Laplace transform
The Sumudu transform F s u ðÞ of a function ft ðÞdefined for all real numberst ≥ 0. The Sumudu transform is essentially identical with the Laplace transform.
Given an initial ft ðÞits Laplace transform Gu ðÞcan be translated into the Sumudu transform F s u ðÞof f by means of the relation;  ii.
Using integration by parts;

Adomian decomposition method
Many of nonlinear phenomena are a necessary part in applied science and engineering fields. Nonlinear equations are noticed in a different type of physical problems [1], such as fluid dynamics, plasma physics, solid mechanics, and quantum field theory.
The wide use of these equations is the most important reason why they have drawn mathematician's attention. Despite this, they are not easy to find an answer, either numerically or theoretically.
In the past, active study attempts were given a large amount of attention to the study of getting exact or approximate solutions of this kind of equations. It is significant to note that several powerful methods have been advanced for this purpose.
The Adomian decomposition method will be used in this chapter and in other chapters to deal with nonlinear equations. The Adomian decomposition method proves to be powerful, effective and successfully used to handle most types of linear or nonlinear ordinary or partial differential equations, and linear or nonlinear integral equations.
In the following, the Adomian scheme for calculating a wide variety of forms of nonlinearity.

Calculation of Adomian polynomials
It is well known that the Adomian decomposition method suggests the unknown linear function u may be represented by the decomposition series; Where the components u n , n ≥ 0 can be elegantly computed in a recursive way. However, the nonlinear term Fu ðÞ , such as u 2 , u 3 , u 4 , sin u, e u , uu x , u x 2 , etc., can be expressed by an infinite series of the so-called Adomian polynomials A n given in the form; Fu ðÞ¼ X ∞ n¼0 A n u 0 , u 1 , u 2 , … , u n ðÞ : The Adomian polynomials A n for the nonlinear term Fu ðÞcan be evaluated by using the following expression; Assuming that the nonlinear function is Fu ðÞ , therefore, by using (6), Adomian polynomials are given by; , Other polynomials can be generated in a similar manner. Substituting (7) into (5) gives; The last expansion confirms a fact that the series in A n polynomials is a Taylor series about a function u 0 and not about a point as is usually used.
In the following, we will calculate Adomian polynomials for several forms of nonlinearity.

Nonlinear polynomials
If Fu ðÞ¼u 2 The polynomials can be found as follows: And so on. Proceeding as before, we find u 3 , u 4 , u 5 , … , etc.

Nonlinear derivatives
And so on. In a similar, we get u x 3 , u x 4 , u x 5 , : … , etc.
The A n polynomials in this case given by; And so on.

Trigonometric nonlinearity
If Fu ðÞ¼sin u The Adomian polynomials for this form nonlinearity are given by; And so on. In a similar way, we find Fu ðÞ¼cos u.

Hyperbolic nonlinearity
If Fu ðÞ¼sinh u The A n polynomials in this case are given by; And so on. In a parallel manner, Adomian polynomials can be calculated for Fu ðÞ¼cosh u.

Exponential nonlinearity
If Fu ðÞ¼e u The Adomian polynomials in this form of nonlinearity are given by; And so on. Proceeding as a before, we find Fu ðÞ¼e Àu .

Logarithmic nonlinearity
If Fu ðÞ¼ln u, u > 0 The A n polynomials for logarithmic nonlinearity are given by; And so on. In a similar way, we find Fu ðÞ¼ln 1 þ u ðÞ , À 1 < u ≤ 1.

A New algorithm for calculating Adomian polynomials (The alternative algorithm for calculating Adomian polynomials)
It is well known about the main disadvantage of the calculating Adomian polynomials A n , that it is a difficult method to perform calculation so called nonlinear terms. There is an alternative algorithm to reduce the demerits of formula introduced before, which depends mainly on algebraic, trigonometric identities and on Taylor expansions.
In the alternative algorithm which is a simple and reliable may be employed to calculate Adomian Polynomials A n .
The new algorithm will be clarified by discussing the following suitable forms of nonlinearity.

Nonlinear polynomials
If Fu ðÞ¼u 2 We first set, Substituting (8) into Fu ðÞ¼u 2 gives; Expanding the expression at the right-hand side gives; The expansion in (10) can be rearranged by grouping all terms with the sum of the subscripts of the components is the same. This means that we can rewrite (10) as; This gives Adomian polynomials for Fu ðÞ¼u 2 by; And so on. Proceeding as before, we get u 3 , u 4 , u 5 , : … , etc.

Nonlinear derivatives
Substituting (12) into Fu ðÞ¼u x 2 giving; Squaring the righthand side gives; Grouping the terms as discussed above, we find; Adomian polynomials are given by; Fu ðÞ¼uu x Note that this form of nonlinearity appears in advection problems and in nonlinear Burgers equations. We first set; Substituting (16) into Fu ðÞ¼uu x yields; Multiplying the two factors gives; Proceeding with grouping the terms are obtained; Consequently, the Adomian polynomials are given by; Proceeding as before, we find Fu ðÞ¼u 2 u x .

Trigonometric nonlinearity
If Fu ðÞ¼sin u First, we should be separate A 0 ¼ Fu 0 ðÞ from other terms. To achieve this goal, we first substitute; Into Fu ðÞ¼sin u to obtain; 8

Advances in the Solution of Nonlinear Differential Equations
Fu ðÞ¼sin u 0 þ u 1 þ u 2 þ u 3 þ u 4 þ … ðÞ ½ : To separate A 0 , recall the trigonometric identity; sin θ þ ϕ ðÞ ¼ sin θ cos ϕ þ cos θ sin ϕ: This means that; Separating Fu 0 ðÞ ¼ sin u 0 from other factors and using Taylor expansion for, cos u 1 þ u 2 þ u 3 þ u 4 þ : … ðÞ and, sin u 1 þ u 2 þ u 3 þ u 4 þ : … ðÞ gives; So that; The last expansion can be rearranged by grouping all terms with the same sum of subscripts. This leads to; This completes the calculation of the Adomian polynomials for nonlinear operator Fu ðÞ¼sin u, therefore we write; And so on. In a similar way, we find Fu ðÞ¼cos u.

Hyperbolic nonlinearity
If Fu ðÞ¼sinh u then, we first substitute Into Fu ðÞ¼sinh u to obtain; To calculate A 0 , recall the hyperbolic identity; Accordingly, Eq. (29) becomes; Separating Fu 0 ðÞ ¼ sinh u 0 from other factors and using Taylor expansion for cosh u 1 þ u 2 þ u 3 þ u 4 þ : … ðÞ and sinh u 1 þ u 2 þ u 3 þ u 4 þ : … ðÞ gives; By grouping all terms with the same sum of subscripts we find Consequently, the Adomian polynomials for Fu ðÞ¼sinh u are given by; Similarly as before, we find Fu ðÞ¼cosh u.

Exponential nonlinearity
If Fu ðÞ¼e u : Into Fu ðÞ¼e u gives; Fu ðÞ¼e u 0 þu 1 þu 2 þu 3 þu 4 þ … ðÞ : Or equivalently; Fu ðÞ¼e u 0 Â e u 1 þu 2 þu 3 þu 4 þ … ðÞ : Keeping the term Fu 0 ðÞ ¼ e u 0 and using Taylor expansion for the other factors we obtain; By grouping all terms with an identical sum of subscripts we find It then follows that; And so on. Proceeding as a before, we find Fu ðÞ¼e Àu .

Adomian decomposition Sumudu transform method for solving nonlinear partial differential equations
In this section, we will concentrate our study on the nonlinear PDEs. There are many nonlinear partial differential equations which are quite useful and applicable in engineering and physics.
The nonlinear phenomena that appear in the many scientific fields' such as solid state physics, plasma physics, fluid mechanics and quantum field theory can be modeled by nonlinear differential equations. The significance of obtaining exact or approximate solutions of nonlinear partial differential equations in physics and mathematics is yet an important problem that needs new methods to develop new techniques for obtaining analytical solutions. Several powerful mathematical methods are used for this purpose. We, propose a new method, namely Adomian Decomposition Sumudu Transform Method (ADSTM) for solving nonlinear equations. This method is a combination of Sumudu transform and decomposition method which was introduced by D. Kumar, J. Singh and S. Rathore.
(ADSTM) provides the solution for nonlinear equations in the form of convergent series. This forms the motivation for us to apply (ADSTM) for solving nonlinear equations in understanding different physical phenomena.
To illustrate the basic idea of this method, we consider a general nonhomogeneous partial differential equation with the initial conditions of the form: Where D is the second order linear differential operator D ¼ ∂ 2 ∂ t 2 , R is linear differential operator of less order than D, N represent the general nonlinear operator and gx , t ðÞ is the source term. Taking the Sumudu transform of both sides of Eq. (42), we get: Using the differentiation property of the Sumudu transform and given initial conditions, we have: If we apply the inverse operator S À1 to both sides of Eq. (44), we obtain: Where Gx , t ðÞ represents the term arising from the source term and the prescribed initial conditions. Now, apply the Adomain decomposition method: The nonlinear term can be decomposed as: For some Adomain polynomials A n U ðÞ that are given by: Substituting Eq. (46) and Eq. (47) in Eq. (45), we get: Accordingly, the formal recursive relation is defined by: The Adomian decomposition Sumudu transform method will be illustrated by discussing the following examples.
Example 1: Consider the following nonlinear partial differential equation: With the initial condition: Taking the Sumudu transform of both sides of Eq. (50) and using the initial condition, we have: Applying S À1 to both sides of Eq. (52) implies that: Following the technique, if we assume an infinite series of the form (53), we obtain: Where A n U ðÞ are Adomian polynomials that represent the nonlinear terms. The first few components of the Adomian polynomials are given by; :: : : This gives the recursive relation: The first few components are given by: And so on. The solution in a series form is given by: And in a closed form of: Example 2: Consider the following nonlinear partial differential equation: With the initial condition: Ux ,0 ðÞ ¼ 0: Proceeding as in Example 1, Eq. (60) becomes: The modified decomposition method admits the of a modified recursive relation given by: U kþ1 x, t ðÞ ¼ À L t À1 A k þ U k À B k ðÞ , k ≥ 1: (73) The first few of the components are given by; U 0 x, t ðÞ ¼ te Àx , The solution in a closed form is given by; Example 4: Consider the following nonlinear partial differential equation, With the initial conditions By taking Sumudu transform for (76) and using (77) we obtain: By applying the inverse Sumudu transform of (78), we get: This assumes a series solution of the function Ux , t ðÞ is given by: U n x, t ðÞ ; Using (80) into (79), we get: Where A n U ðÞ and B n U ðÞ are Adomian polynomials that represents nonlinear terms.
The few components of the Adomian polynomials are given as follows: And so on. From the above equations we obtain: