Adomian polynomials method for dynamic equations on time scales

Öz

In a recent paper, a series solution method based on combining the Laplace transform and Adomian polynomial expansion was proposed to find an approximate solution of nonlinear differential equations \cite{FA2016}. It uses the expansion in Adomian polynomials defined in \cite {A1,A2}. An important drawback of the Laplace transform method is the fact that it cannot be applied in the case of nonlinear differential equation in general. In order to cope with this problem, the authors of \cite{FA2016} suggested the use of Adomian polynomial expansion of the nonlinear function of the dependent variable involved in the differential equation. In this work, we propose a counterpart of this method on an arbitrary time scale and derive its general formulation for a dynamic equation of any order. We confirm that when the time scale is the set of real numbers, our method reduces to that in \cite{FA2016}. Our presentation is organized as follows. First, we recollect some preliminary information on time scales in Secton 2. In Section 3, we derive the method for an $n$-th order nonlinear dynamic equation. The next section contains the application of the method to specific examples of first order nonlinear dynamic equations. The last section is devoted to conclusion and some further directions for study.

Anahtar Kelimeler

• time scale
• Adomian polynomials
• Laplace transform
• dynamic equation

Kaynakça

1. G. Adomian, A new approach to nonlinear partial differential equations, J. Math. Anal. Appl., 102 (1984), 420--434.
2. G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comp. Math. Appl. 21(1991), 101--127. M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh\"auser, Boston, 2001.
3. M. Bohner, S.Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, 2016.
4. S. Georgiev, Integral Equations on Time Scales. Atlantis Press 2016.
5. S. Georgiev. Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, 2017.
6. S. Georgiev, I. Erhan, Nonlinear Integral Equations on Time Scales. Nova Science Publishers, 2019.
7. H. Fatoorehchi, H. Abolghasemi, Series solution of nonlinear differential equations by a novel extension of the Laplace transform method, International Journal of Computer Mathematics, 93(8) 1299-1319, 2016.