Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations

doi:10.1016/j.amc.2005.12.049

Applied Mathematics and Computation

Volume 181, Issue 1, 1 October 2006, Pages 598-608

https://doi.org/10.1016/j.amc.2005.12.049 Get rights and content

Abstract

A modification of the Adomian decomposition method applied to systems of linear/nonlinear ordinary and fractional differential equations, which yields a series solution with accelerated convergence, has been presented. Illustrative examples have been given.

Introduction

Numerous problems in Physics, Chemistry, Biology and Engineering science are modeled mathematically by systems of ordinary and fractional differential equations, e.g. series circuits, mechanical systems with several springs attached in series lead to a system of differential equations. On the other hand motion of an elastic column fixed at one end and loaded at the other, can be formulated in terms of a system of fractional differential equations [4]. Since most realistic differential equations do not have exact analytic solutions approximation and numerical techniques, therefore, are used extensively. Recently introduced Adomian Decomposition Method (ADM) [2] has been used for solving a wide range of problems. This new iterative method has proven rather successful in dealing with both linear as well as nonlinear problems, as it yields analytical solutions and offers certain advantages over standard numerical methods. It is free from rounding off errors since it does not involve discretization, and is computationally inexpensive. Biazar et al. [6] have applied this method to a system of ordinary differential equations. Daftardar-Gejji and Jafari [7], [8] have explored this method to obtain solutions of a system of linear and nonlinear fractional differential equations. Further in [9] they have suggested a modification (termed as “revised ADM”) of this method and have applied revised ADM for solving a system of nonlinear algebric equations. In the present paper we use the revised Adomian decomposition method to obtain solutions of systems of linear/nonlinear ordinary and fractional differential equations. We demonstrate that the series solution thus obtained converges faster relative to the series obtained by standard ADM. Several illustrative examples have been presented.

The present paper has been organized as follows. In Section 2, we give basic definitions and preliminaries. Section 3 deals with the analysis of ADM applied to a system of ordinary differential equations. In Sections 4 Revised ADM for a system of ordinary differential equations, 5 Revised ADM for a system of fractional differential equations, we introduce revised ADM for systems of ordinary and fractional differential equations, respectively. Section 6 compares the revised ADM and standard ADM with illustrative examples. This is followed by the conclusions in Section 7.

Section snippets

Definitions and preliminaries

Definition 2.1

A real function f(x), x > 0 is said to be in the space C_α, $α \in R$ if there exists a real number p (>α), such that f(x) = x^pf₁(x) where f₁(x) ∈ C[0, ∞). Clearly C_α ⊂ C_β if β ⩽ α.

Definition 2.2

A function f(x), x > 0 is said to be in the space $C_{α}^{m}, m \in N ⋃ {0}$ , if f^(m) ∈ C_α.

Definition 2.3

The (left sided) Riemann–Liouville fractional integral of order μ ⩾ 0 [9], [10], [11], [12] of a function f ∈ C_α, α ⩾ −1 is defined as $I^{μ} f (x) = \frac{1}{Γ (μ)} \int_{0}^{x} \frac{f (t)}{(x - t)^{1 - μ}} d t, μ > 0, x > 0,$ $I^{0} f (x) = f (x) .$

Definition 2.4

The (left sided) Caputo fractional derivative of $f \in C_{- 1}^{m}, m \in N$ is defined as [9], [10] $D^{μ} f ($

System of ordinary differential equations and Adomian decomposition

Consider the following system of ordinary differential equations: $y_{i}^{'} (x) = \sum_{j = 1}^{n} b_{ij} (x) y_{j} + N_{i} (x, y_{1}, y_{2}, \dots, y_{n}) + g_{i} (x), y_{i} (0) = c_{i}, i = 1, 2, \dots, n,$ where b_ij(x), g_i(x) ∈ C[0, T] and N_i’s are nonlinear continuous functions of its argument. Integrating both side of Eq. (5) from 0 to x and the using initial conditions, we get $y_{i} (x) = c_{i} + \int_{0}^{x} g_{i} (x) d x + \int_{0}^{x} \sum_{j = 1}^{n} b_{ij} (x) y_{j} d x + \int_{0}^{x} N_{i} (x, y_{1}, y_{2}, \dots, y_{n}) d x, for i = 1, 2, \dots, n .$ The standard ADM [2] yields the solution y_i(x) by the series $y_{i} (x) = \sum_{m = 0}^{\infty} y_{im} (x)$ and the nonlinear terms by an infinite series

Revised ADM for a system of ordinary differential equations

In this section we propose a modification of the Adomian decomposition. We set $\begin{matrix} y_{10} (x) = c_{1} + \int_{0}^{x} g_{1} (x) d x, \\ y_{1, m + 1} (x) = \int_{0}^{x} \sum_{j = 1}^{n} b_{1 j} (x) y_{jm} d x + \int_{0}^{x} A_{1 m} d x, \\ y_{l 0} (x) = c_{l} + \int_{0}^{x} g_{l} (x) d x + \int_{0}^{x} \sum_{j = 1}^{l - 1} b_{lj} (x) y_{j 0} d x, l = 2, \dots, n, \\ y_{l, m + 1} (x) = \int_{0}^{x} \sum_{j = 1}^{l - 1} b_{lj} (x) y_{jm + 1} d x + \int_{0}^{x} \sum_{j = l}^{n} b_{lj} (x) y_{jm} d x + \int_{0}^{x} A_{lm}^{*} d x, \end{matrix}$ where $A_{lm}^{*}$ , is defined as $A_{lm}^{*} = \{\begin{matrix} A_{lm + 1}, & if N_{l} are independent of y_{l}, y_{l + 1}, \dots, y_{n}, \\ ^{1} A_{l, m + 1} +^{2} A_{l, m} & if N_{l} (y_{1}, \dots, y_{n}) =^{1} N_{l} (y_{1}, \dots, y_{l - 1}) +^{2} N_{l} (y_{1}, \dots, y_{n}), \\ A_{lm}, & otherwise, \end{matrix} l = 2, 3, \dots$ Here ¹A_l,m+1, ²A_l,m are Adomian polynomials corresponding to ¹N_l and ²N₂ as defined in Eq. (9).

Revised ADM for a system of fractional differential equations

We consider the following system of fractional differential equations: $D^{α_{i}} y_{i} (x) = \sum_{j = 1}^{n} (ϕ_{ij} (x) + γ_{ij} D^{α_{ij}}) y_{j} + N_{i} (x, y_{1}, \dots, y_{n}) + g_{i} (x), y_{i}^{(k)} (0) = c_{k}^{i}, 0 ⩽ k ⩽ m_{i}, α_{ij} ⩽ α_{i}, m_{i} < α_{i} ⩽ m_{i} + 1, 1 ⩽ i ⩽ n,$ where N_i’s are nonlinear functions of x, y₁, … , y_n and $α_{i}, α_{ij} \in R^{+}$ . Applying $I^{α_{i}}$ to both the sides of Eq. (14), we get [7], [8] $y_{i} = \sum_{k = 0}^{m_{i}} c_{k}^{i} \frac{x^{k}}{k!} + I^{α_{i}} \sum_{j = 1}^{n} (ϕ_{ij} (x) + γ_{ij} D^{α_{ij}}) y_{j} + I^{α_{i}} N_{i} (x, y_{1}, \dots, y_{n}), 1 ⩽ i ⩽ n,$ where m_i < α_i ⩽ m_i + 1 and m_ij < α_ij ⩽ m_ij + 1. Here 0 ⩽ α_ij < α_i, for 1 ⩽ i, j ⩽ n, γ_ij’s are constants and ϕ_ij(x), g_i(x) ∈ C[0, T].

Using standard ADM we get $\sum_{m = 0}^{\infty} y_{im}$

Illustrative examples

To give a clear overview of the revised method, we present the following examples. We apply the revised ADM and compare the results with the standard ADM.

(i) Consider the following system of linear equations: $\begin{matrix} y_{1}^{'} = y_{3} - \cos x, y_{1} (0) = 1, \\ y_{2}^{'} = y_{3} - e^{x}, y_{2} (0) = 0, \\ y_{3}^{'} = y_{1} - y_{2}, y_{3} (0) = 2 . \end{matrix}$ This system is equivalent to the following system of integral equations: $\begin{matrix} y_{1} = y_{1} (0) - \int_{0}^{x} \cos x d x + \int_{0}^{x} y_{3}, \\ y_{2} = y_{2} (0) - \int_{0}^{x} e^{x} d x + \int_{0}^{x} y_{3}, \\ y_{3} = y_{3} (0) + \int_{0}^{x} (y_{1} - y_{2}) d x . \end{matrix}$ The revised Adomian procedure would lead to: $\begin{matrix} y_{10} = 1 - \sin x, y_{1 m + 1} = \int_{0}^{x} y_{3 m} d x, \\ y_{20} = 1 - e^{x}, y_{2 m + 1} = \int_{0}^{x} y_{3 m} d x, \end{matrix}$

Conclusions

Adomian decomposition is a powerful method which yields a convergent series solution for linear/nonlinear problems. This method is better than numerical methods, as it is free from rounding off errors, and does not require large computer power. We [9] have suggested a modification of this method, termed as “revised ADM”. In the present paper we employ the revised ADM for solving a system of ordinary/fractional differential equations. The revised method yields a series solution which converges

Acknowledgements

Hossein Jafari thanks University Grants Commission, New Delhi, India for the award of Junior Research Fellowship and acknowledges Y. Talebi, University of Mazandaran, Babolsar, Iran for encouragement.

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