Abstract
An analytical solution of the time-fractional Fisher equation with small delay is established by means of residual the residual power series method (RPSM) where the fractional derivative is taken in the Caputo sense. Taking advantage of small delay, the time-fractional Fisher equation is expanded in powers series of delay term . By using RPSM analytical solution of time-fractional of Fisher equation is constructed. The final results and graphical consequences illustrate that the proposed method in this study is very efficient, effective, and reliable for the solution of the time-fractional Fisher equation with small delay.
1. Introduction
In last few decades, fractional calculus growing considerable interest is used in bioengineering, thermodynamics, viscoelasticity, control theory, aerodynamics, electromagnetics, signal processing, chemistry, and finance [1–7]. Many numerical methods have been applied and analyzed for differential equations with fractional order derivative of Riemann-Liouville or Caputo sense [6–12]. Delay differential equations (DDEs) can be considered as the generalization of the ordinary differential equations which are appropriate for modelling physical systems with memory. The delay is an important fact of the physical systems such as kinetics, controllers, signal processing, and damping behavior of viscoelastic materials.
The RPSM was established as a powerful method for fuzzy differential equations [13]. It has been successfully put into practice in various fields [14–20]. The solution of problems by RPSM is obtained in the form of Maclaurin series. It is an efficient method to find out the coefficients of the series solutions. Construction of multidimensional and multiple solutions for fractional differential equations in the form of power series is an important advantage of RPSM. RPSM is effective and easy to use for solving linear and nonlinear FPDEs without linearization, perturbation, or discretization.
In this study, we extend the application of the RPSM in order to establish an approximate solution to time-fractional Fisher equation with small delaysubject to initial condition which was originally proposed by Fisher [20] which describes the spatial and temporal propagation of a virile gene in an infinite medium. The time-fractional Fisher equation can be readily solved by many methods [21–25]. Moreover, there are various studies on nonlocal of action of fractional differential equations including space fractional derivative [26–30]. In this article, by making use of the residual power series method (RPSM), we find series solution for the time-fractional Fisher equation with small delay. By taking advantage of the small delay we expand the term including small delay to Taylor series which reduced the problem to a perturbation problem. Applying RPSM we determine the solution of equations which are the obtained from the coefficients of . Replacing these solutions in the Taylor series the solution of the original problem is obtained.
2. Preliminaries
In this section, the basic definitions and various features for fractional calculus theory are shown [6, 31–35].
Definition 1. The Riemann-Liouville fractional integral of order is given as [14]
Definition 2. The Caputo fractional derivative with order is given as [11, 12] where is the classic differential operator with order .
By the Caputo derivative, we get
Definition 3. Caputo’s time-fractional derivative of order of is defined as [11, 12]
Definition 4. A power series expansion of the form is called fractional power series about [25].
Definition 5. A power series expansion of the form is called multiple fractional power series about [11, 12].
Theorem 6. Suppose that has a multiple fractional power series representation at of the formIf , , are continuous on , then .
3. RPSM of the Time-Fractional Fisher Equation with Small Delay
Consider the time-fractional Fisher equation (1) with small delay. Since the delay term is very small, we replace (1) by the following Taylor series expansion: where we ignore the higher order terms, which leads to and assuming that , we obtain In this study we obtain the solution of time-fractional Fisher equation with small delay for ; hence we get the following equation: Using (12) in (1) we reduce the problem to a perturbation problem for which we use the following series solution: to obtain the solution of (14). Substituting (15) into (14) leads to the following equation: Hence we get the following equations: and so on. We apply the RPSM to find out series solution for these equation subject to given initial conditions by replacing its fractional power series expansion with its truncated residual function. From each equation, a repetition formula for the calculation of coefficients is supplied, while coefficients in fractional power series expansion can be calculated by repeatedly fractional differentiation of the truncated residual function [13–20]. The RPSM propose the solutions for (18)-(21) as a fractional power series about the initial point [13] To obtain the numerical values from this series, let denote the -th truncated series of . That is, By the initial condition, the 0 residual power series approximate solution of can be written as follows: Equation (22) can be written as Define the residual function for (18) [35] and the th residual function can be expressed as From [13–23], by making use of some results such as for each and and are used to obtain the solution.
Substituting the m truncated series of into (18), calculating the fractional derivative of , at and solving the following obtained algebraic system the required coefficients in (24) are determined.
In order to determine , the 1 residual function in (26) can be written as follows: where . Therefore, From (27), we deduce that , which leads to Similarly, to obtain , the 2 residual function in (26) can be written in the following form: where . Therefore, The operator is applied on both sides of (32) as follows: From (27) and (33), The same manner is repeated as above; the following recurrence results is obtained and so on.
Thus, we have Define the residual function for (19) Suppose that By the initial condition, the 0 residual power series approximate solution of can be written as follows: We apply the RPSM to find out , , in (19). Thus, we have Define the residual function for (20) Suppose that By the initial condition, the 0 residual power series approximate solution of can be written as follows: We apply the RPSM to find out in (20).
Thus, we have Define the residual function for (11) Suppose that By the initial condition, the 0 residual power series approximate solution of can be written as follows: We apply the RPSM to find out , , in (21).
Thus, we have
For , we obtain , and as follows: and and
and For , we have , and as follows: and and and
4. Numerical Results
Consider the following time-fractional Fisher equation with small delay subject to initial condition Then, the exact solution of (1) without delay when is given by It is clear from Figures 1 and 2 and Tables 1-2 that the graph of the solution of time-fractional Fisher equation with small delay with respect to space in some neighborhood of solution acts as a critically damped oscillator for whereas it acts as an over damped oscillator for . In Figures 3 and 4, as it is seen from the graph of the solution of time-fractional Fisher equation with small delay with respect to time , the effect of the small time delay is limited for the fractional derivative of order ; i.e., the solution of time-fractional Fisher equation with small delay keeps close to the solution of Fisher equation without delay. As a result, it is concluded that the solution is stable under small time delay for . On the other hand, the effect of the small time delay increases as time increases for ; i.e., the solution of time-fractional Fisher equation with small delay gets away from the solution of Fisher equation without delay. As a result, it is concluded that the solution is unstable under small time delay for .
5. Conclusion
Based on the analysis of the solution of time-fractional Fisher equation with small delay, it is deduced that, at the fractional derivative of order , there is a transition of the solution with respect to both variables and .
Nomenclature
| Gamma function | |
| Riemann-Liouville fractional integral | |
| Caputo fractional derivative | |
| Caputo time-fractional derivative. |
Data Availability
The data in our manuscript is obtained by our calculations.
Disclosure
The previous version of the manuscript is presented in ICOMAA-2018 under the name of “On the Time-Fractional Fisher Equation with Small Delay”.
Conflicts of Interest
The authors declare that they have no conflicts of interest.