Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.


Introduction
I n the literature there are numerous integral transforms [1] that are widely used in physics, astronomy as well as in engineering. In order to solve the differential equations, the integral transform was extensively used and thus there are several works on the theory and application of integral transform such as the Laplace, Fourier, Mellin, Hankel, Fourier Transform, Sumudu Transform, Elzaki Transform and Aboodh Transform. Aboodh Transform [2,3] was introduced by Khalid Aboodh in 2013, to facilitate the process of solving ordinary and partial differential equations in the time domain. This transformation has deeper connection with the Laplace and Elzaki Transform [4][5][6]. New integral transform, named as ZZ Transformation [7][8][9][10] introduce by Zain Ul Abadin Zafar. ZZ transform was successfully applied to integral equations and ordinary differential equations. The main objective of this article is to solve nonlinear ordinary differential equation using ZZ transform.

ZZ transform
Let f (t) be a function defined for all t ≥ 0. The ZZ transform of f (t) is the function Z(u, s) defined by or Equation (1) equivalent to

The ZZ decomposition method
Consider the general nonlinear ordinary differential equation of the form: where v is the unknown function, L is the linear differential operator of highest derivative, R is the reminder of the differential operator, g(t) is nonhomogeneous term and N(v) is the nonlinear term.
Suppose L is a differential operator of the first order, then by taking the ZZ transform of Equation (3), we have Substituting the given initial condition from Equation (4), we get Since, the solution can be written in the form of v(t) and also V(u, s) is the ZZ transform of v(t). Therefore, Taking the inverse ZZ transform of Equation (6) to obtain the solution in the form of v(t).
We now assume an infinite series solution of the unknown function v(t) of the form The nonlinear operator Nv = Ψ(v) is decomposed as where, A n is called Adomian's polynomials. This can be calculated for various classes of nonlinearity according to By using Equation (8), the Equation (7), can be rewritten as; Now, if we compare both sides of Equation (9), we can get the following recurrence relation Finally, we have the following general recurrence relation; Therefore, the exact or approximate solution is given by Example 1. Consider the non-linear system of initial value problems given by with exact solution y (t) = 1 1−t . Applying ZZ Transform of Equation (12), we have Substitute the given initial from Equation (13) s After simple calculation from Equation (14), we have By taking the inverse ZZ transform of Equation (15), we have We now assume an infinite series solution of the unknown function y (t) of the form By using Equation (17), we can write Equation (16) in the form where, A n is called Adomian's polynomials of the nonlinear term y 2 (t). Now, by comparing both sides of Equation (18), we can get the following recurrence relation; Finally, we have the following general recurrence relation; Now, by using the recursive relation in Equation (19), we can easily compute the remaining components of the unknown function y (t) in the following manner Similarly, we can find y 3 (t) After some calculation, we obtain t 3 and so on. Hence, the approximate solution is given by; The Octave Code is; plot(t, f , r , t, g, o ); ylabel( y(t) ); xlabel( t = 0 : 0.9 ); legend('Exact , 'Approximate ).
Hence, the exact solution is in closed agreement with the result obtained by ZZ decomposition Example 2. Consider the non-linear system of initial value problems given by Using the method of separation of variables, the exact solution is x (t) = e 2t −1 e 2t +1 . Applying ZZ Transform of Equation (20), we have Substitute the given initial condition from Equation (21), we have After simple calculation from Equation (22), we have By taking the inverse ZZ transform of Equation (23), we have We now assume an infinite series solution of the unknown function x (t) of the form By using Equation (25), we can write Equation (24) in the form where, A n is called Adomian's polynomials of the nonlinear term x 2 (t). Now, by comparing both sides of Equation (26), we can get the following recurrence relation: x 0 (t) = t, Finally, we have the following general recurrence relation Then by using the recursive relation in Equation (27), we can easily compute the remaining components of the unknown function x (t) in the following manner Similarly, we can find x 3 (t) After, some calculation step, we obtain x 3 (t) = −17 315 t 7 and so on. Hence, the approximate solution is given by; We obtain the following graph, that is the comparison of approximate and exact solution of the given differential equation depend on the order of expansion using Octave. The line (graph) in the red color indicates the actual solution, while the ring line (o) indicates the approximate solution.
Substitute the given initial condition from Equation (29), we have