Abstract

This paper considers the approximation of solution for a fractional order biological population model. The fractional derivative is considered in the Caputo sense. By using Laplace Adomian decomposition method (LADM), we construct a base function and provide deformation equation of higher order in a simple equation. The considered scheme gives us a solution in the form of rapidly convergent infinite series. Some examples are used to show the efficiency of the method. The results show that LADM is efficient and accurate for solving such types of nonlinear problems.

1. Introduction

Nowadays in most of the research areas, the importance of fractional differential equations has been increased due to its wide range of applications in real world problems. In different scientific and engineering categories such as chemistry, mechanics, and physics applications of fractional calculus can be found. It can be also used in control theory, optimization theory, image processing, economics, and so on [1–5]. Mathematical models of fractional order are very important to study natural problems. As is known, the nature of the trajectory of the fractional order derivatives is nonlocal, describing that the fractional order derivative possesses memory effect features and any dynamical or physical system related to fractional order differential operators has a memory effect, which shows that the future states depend on the present and past states. The limitations of Caputo fractional derivative are removed while introducing Caputo-Fabrizio derivative; see [6]. This new approach was used by Singh et al. to model the dynamics of computer viruses, particularly in those cases where the physical processes do not bear plasticity, fatigue, damage, and electromagnetic hysteresis effects. In view of the great significance of new fractional derivative, a fractional order model of chemical kinetic system with full memory effect has been studied very well. The authors examined the existence and uniqueness of the solutions for chemical kinetic system of arbitrary order by using the fixed-point theorem.

System of nonlinear differential equations are very important for mathematicians, engineers, and physicists, because in most of physical systems the input is not proportional to output in nature. In addition to the study of simple nonlinear differential equations, exact solution of the nonlinear evolution equation also plays an important role in the study of some nonlinear problems. For the exact solution there are many approaches such as Hirota’s method, Darboux transformation, and Painleve expansions.

In last few years, more alternative numerical methods have been used for solving both linear and nonlinear problems of physical interest including homotopy perturbation method (HPM) [7, 8], Adomian decomposition method (ADM)[9, 10], and homotopy analysis method (HAM) [11]. Due to various applications of Lienard’s equation, Kumar et al. proposed a numerical algorithm based on fractional homotopy analysis transform method to study fractional order Lienard’s equation, while recently in 2017 the authors used -homotopy analysis method and Laplace transform approach to explore some aspects of the FitzHugh-Nagumo equation of fractional order. Laplace decomposition method was adopted in [12, 13] to deal with some other nonlinear problems while homotopy perturbation transform method is selected in [14] to explore the approximate solution. Very recently Laplace transform combined with homotopy analysis method is used to produce effective method called homotopy analysis transform method (HATM) [15, 16]. Its main purpose is to handle nonlinear physical problems. Baleanu et al. [17] have solved the fractional order optimal control problems. For the solution of system of Schrodinger-Korteweg-de Vries equation, Golmankhaneh has used the homotopy perturbation method [18]. In [19], the authors presented the following nonlinear fractional order biological population model using homotopy analysis transform method:with initial condition aswhere , , and Further, represents density of the population and denotes supply of the population due to birth and death rate. This nonlinear fractional order biological population model is obtained by replacing the first-order derivative term in the corresponding biological population model by fractional order , where . The derivative is considered in Caputo’s sense. The parameter involved in fractional derivative shows various responses. If , the fractional order biological model reduces to standard biological model. In [20, 21], the authors discussed such models while exploring their different aspects. Promoting the above work, we solve model (1) by using LADM.with the given initial condition Since the proposed model is highly nonlinear and contains nonlinear terms and for , therefore, we decompose and in terms of Adomian polynomials aswhere

2. Preliminaries

In order to assist the readers, here in this section we recall some fundamental definitions and results from fractional calculus.

Definition 1 (see [22]). The fractional order derivative of in the Caputo sense is defined asFor , In case of function , we defined Caputo derivative with respect to “” asFurther for , we have .

Definition 2 (see [22]). The Riemann-Liouville fractional order integral operator of order of function is given as In case of then the Riemann-Liouville fractional order integral operator of order is given by The Riemann-Liouville fractional order integral [22] is given by provided that integral on the right side is point wise defined on .

Lemma 3. Let and . Then hold. holds almost everywhere on

Definition 4 (see [19]). The Laplace transform of the Caputo derivative is given as

Definition 5 (see [23]). The Mittag-Leffler function in term of power series is defined as

3. LADM for Biological Model

This section is devoted to the general procedure of the LADM for solving (3) with given initial conditions. To apply Laplace transform to model (3) we proceed aswith the given initial condition From definition of Laplace transform on both sides of (3), we haveUsing given initial conditions yields that Therefore, after decomposing nonlinear terms in terms of Adomian polynomials and considering the unknown solutions , (18) can be written as Comparing terms on both sides, we getAfter taking Laplace inverse transform of system (21), we get

4. Applications

In this section we present some application of LADM to solve biological population model.

Example 6. Consider the following fractional order biological model [21]:with given initial condition Upon using the proposed method on (23) and comparing terms of both sides and then taking Laplace inverse transform, we get The series solution is provided as In closed form, the solution is given bywhich is the exact solution. Putting in (27), we get solution asThis is the classical solution.

Example 7. Consider the following fractional order biological model [21]:with given initial condition Applying Laplace transform on both sides, we haveAfter using Laplace inverse transform on both sides of (31), we have and so no. Now the series solution of problem (29) is given by The closed form is given by Considering , we get classical solution as

Example 8. Consider the following fractional order biological model.corresponding to the initial condition Applying Laplace transform on both sides, we have After using Laplace transform on both sides, we have and so on. The series solution of problem (36) is given by Hence closed form of the solution is given by Considering , we get classical solution as