A new numerical technique for solving Caputo time-fractional biological population equation

Ali Khalouta; Abdelouahab Kadem; Ali Khalouta; Abdelouahab Kadem

doi:10.3934/math.2019.5.1307

AIMS Mathematics

2019, Volume 4, Issue 5: 1307-1319. doi: 10.3934/math.2019.5.1307

Previous Article Next Article

Research article Special Issues

A new numerical technique for solving Caputo time-fractional biological population equation

Ali Khalouta ^,,
Abdelouahab Kadem

Laboratory of Fundamental and Numerical Mathematics, Department of Mathematics, Faculty of Sciences, Ferhat Abbas University of Setif 1, Sétif 19000, Algeria

Received: 01 July 2019 Accepted: 16 August 2019 Published: 03 September 2019
MSC : Primary 35R11, 26A33; Secondary 74G10, 34K28

In this paper, we propose a new numerical technique called modified generalized Taylor fractional series method (MGTFSM) for solving Caputo time-fractional biological population equation.We present our obtained results in the form of a new theorem.This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without need of linearization, discretization, perturbation or unrealistic assumptions.The accuracy and efficiency of the method is tested by means of three numerical examples.The results prove that the proposed method is very effective and simple for solving fractional partial differential equations.

Keywords:

biological population equation,
Caputo fractional derivative,
modified generalized Taylor fractional series method

Citation: Ali Khalouta, Abdelouahab Kadem. A new numerical technique for solving Caputo time-fractional biological population equation[J]. AIMS Mathematics, 2019, 4(5): 1307-1319. doi: 10.3934/math.2019.5.1307

Related Papers:

[1]	Ali Khalouta, Abdelouahab Kadem . A new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients. AIMS Mathematics, 2020, 5(1): 1-14. doi: 10.3934/math.2020001
[2]	D. Priyadarsini, P. K. Sahu, M. Routaray, D. Chalishajar . Numerical treatment for time fractional order phytoplankton-toxic phytoplankton-zooplankton system. AIMS Mathematics, 2024, 9(2): 3349-3368. doi: 10.3934/math.2024164
[3]	Muhammad Imran Liaqat, Sina Etemad, Shahram Rezapour, Choonkil Park . A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients. AIMS Mathematics, 2022, 7(9): 16917-16948. doi: 10.3934/math.2022929
[4]	Sumbal Ahsan, Rashid Nawaz, Muhammad Akbar, Saleem Abdullah, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar . Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method. AIMS Mathematics, 2022, 7(7): 13169-13191. doi: 10.3934/math.2022726
[5]	M. Ali Akbar, Norhashidah Hj. Mohd. Ali, M. Tarikul Islam . Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Mathematics, 2019, 4(3): 397-411. doi: 10.3934/math.2019.3.397
[6]	Rahat Zarin, Abdur Raouf, Amir Khan, Aeshah A. Raezah, Usa Wannasingha Humphries . Computational modeling of financial crime population dynamics under different fractional operators. AIMS Mathematics, 2023, 8(9): 20755-20789. doi: 10.3934/math.20231058
[7]	Hayman Thabet, Subhash Kendre, James Peters . Travelling wave solutions for fractional Korteweg-de Vries equations via an approximate-analytical method. AIMS Mathematics, 2019, 4(4): 1203-1222. doi: 10.3934/math.2019.4.1203
[8]	Ahmad Qazza, Rania Saadeh, Emad Salah . Solving fractional partial differential equations via a new scheme. AIMS Mathematics, 2023, 8(3): 5318-5337. doi: 10.3934/math.2023267
[9]	Abdul Samad, Imran Siddique, Fahd Jarad . Meshfree numerical integration for some challenging multi-term fractional order PDEs. AIMS Mathematics, 2022, 7(8): 14249-14269. doi: 10.3934/math.2022785
[10]	Zeliha Korpinar, Mustafa Inc, Dumitru Baleanu . On the fractional model of Fokker-Planck equations with two different operator. AIMS Mathematics, 2020, 5(1): 236-248. doi: 10.3934/math.2020015

Abstract

1. Introduction

Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders ^[6,9,11,17]. Recently, fractional partial differential equations play an important role in interpretation and modeling of many of realism matters appear in applied mathematics and physics including fluid mechanics, electrical circuits, diffusion, damping laws, relaxation processes, optimal control theory, chemistry, biology, and so on ^{[7,13,14,15,16]}. Therefore, the search of the solutions for fractional partial differential equations is an important aspect of scientific research.

Many powerful and efficient methods have been proposed to obtain numerical solutions and analytical solutions of fractional partial differential equations. The most commonly used ones are: Adomian decomposition method (ADM) ^[5], variational iteration method (VIM) ^[18], new iterative method (NIM) ^[8], fractional difference method (FDM) ^[11], reduced differential transform method (RDTM) ^[1], homotopy analysis method (HAM) ^[3], homotopy perturbation method (HPM) ^[4].

The main objective of this paper is to present a new numerical technique called modified generalized Taylor fractional series method (MGTFSM) to obtain the approximate and exact solutions of Caputo time-fractional biological population equation. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. The main advantage of the proposed method compare with the existing methods is, that method solves the nonlinear problems without using linearization and any other restriction.

Consider the following Caputo time-fractional biological population equation

$\begin{equation} D_{t}^{\alpha }u = \frac{\partial ^{2}u^{2}}{\partial x^{2}}+\frac{\partial^{2}u^{2}}{\partial y^{2}}+F(u), \end{equation}$

(1.1)

with the initial condition

$\begin{equation} u(x,y,0) = u_{0}(x,y), \end{equation}$

(1.2)

where $D_{t}^{\alpha } = \dfrac{\partial ^{\alpha }}{\partial t^{\alpha }}$ is the Caputo fractional derivative operator of order $\alpha,$ $0 < \alpha \leq 1, u = u(x, y, t), (x, y)\in \mathbb{R}^{2}, t > 0$ denotes the population density and $F$ represents the population supply due to birth and death, $\alpha$ is a parameter describing the order of the fractional derivative.

The plan of our paper is as follows: In Section 2, we present some necessary definitions and properties of the fractional calculus theory. In Section 3, we will propose an analysis of the modified generalized Taylor fractional series method (MGTFSM) for solving the Caputo time-fractional biological population equation (1.1) subject to the initial condition (1.2). In Section 4, we present three numerical examples to show the efficiency and effectiveness of this method. In Section 5, we discuss our obtained results represented by figures and tables. These results were verified with Matlab (version R2016a). Section 6, is devoted to the conclusions on the work.

2. Basic definitions

In this section, we present some basic definitions and properties of the fractional calculus theory which are used further in this paper. For more details see, ^[9,11].

Definition 2.1. A real function $u(X, t),$ $X = (x_{1}, x_{2}, ..., x_{n})\in \mathbb{R}^{N}, N\in \mathbb{N}^{\ast }, t\in \mathbb{R}^{+}$ , is considered to be in the space $C_{\mu }(\mathbb{R}^{N}\times \mathbb{R}^{+}),$ $\mu \in$ $\mathbb{R},$ if there exists a real number $p > \mu,$ so that $u(X, t) = t^{p}v(X, t)$ , where $v$ $\in C\left(\mathbb{R}^{N}\times \mathbb{R}^{+}\right)$ , and it is said to be in the space $C_{\mu }^{n}$ if $u^{\left(n\right) }\in C_{\mu }(\mathbb{R}^{N}\times \mathbb{R}^{+}), n\in \mathbb{N}.$

Definition 2.2. The Riemann-Liouville fractional integral operator of order $\alpha \geq 0$ of $u\in C_{\mu }(\mathbb{R}^{N}\times \mathbb{R}^{+}), \mu \geq -1$ , is defined as follows

$\begin{equation} I_{t}^{\alpha }u(X,t) = \left\{ \begin{array}{c} \dfrac{1}{\Gamma (\alpha )}\int\limits_{0}^{t}\left( t-\xi \right) ^{\alpha -1}u(X,\xi )d\xi,\alpha \gt 0,t \gt \xi \gt 0, \\ u(X,t),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \alpha = 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array} \right. \end{equation}$

(2.1)

where $\Gamma (.)$ is the well-known Gamma function.

Definition 2.3. The Caputo time-fractional derivative operator of order $\alpha > 0$ of $\ u\in C_{-1}^{n}(\mathbb{R}^{N}\times \mathbb{R}^{+}), n\in \mathbb{N},$ is defined as follows

$\begin{equation} D_{t}^{\alpha }u(X,t) = \left\{ \begin{array}{c} \dfrac{1}{\Gamma (n-\alpha )}\int\limits_{0}^{t}\left( t-\xi \right) ^{n-\alpha -1}u^{(n)}(X,\xi )d\xi,n-1 \lt \alpha \lt n, \\ u^{\left( n\right) }(X,t), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \alpha = n. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array} \right. \end{equation}$

(2.2)

For this definition we have the following properties

(1)

$\begin{equation*} D_{t}^{\alpha }(c) = 0,\text{ where }c\text{ is a constant.} \end{equation*}$

(2)

$\begin{equation*} D_{t}^{\alpha }t^{\beta } = \left\{ \begin{array}{c} \frac{\Gamma (\beta +1)}{\Gamma (\beta -\alpha +1)}t^{\beta -\alpha }\text{ if }\beta \gt n-1, \\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if }\beta \leq n-1. \end{array} \right. \end{equation*}$

Definition 2.3. The Mittag-Leffler function is defined as follows

$\begin{equation} E_{\alpha }\left( z\right) = \sum\limits_{n = 0}^{\infty }\frac{z^{n}}{\Gamma(n\alpha +1)},\alpha \in \mathbb{C},Re(\alpha ) \gt 0. \end{equation}$

(2.3)

For $\alpha = 1,$ $E_{\alpha }\left(z\right)$ reduces to $e^{z}$ .

3. Analysis of modified generalized Taylor fractional series method (MGTFSM)

Theorem 3.1. Consider the Caputo time-fractional biological population equation of the form (1.1) with the initial condition (1.2).

Then, by MGTFSM the solution of equations (1.1)-(1.2) is given in the form of infinite series which converges rapidly to the exact solution as follows

$\begin{equation*} u(x,y,t) = \sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)},(x,y)\in \mathbb{R}^{2},t\in \left[ 0,R\right), \end{equation*}$

where $u_{i}(x, y)$ the coefficients of the series and $R$ is the radius of convergence.

Proof. In order to achieve our goal, we consider the following Caputo time-fractional biological population equation of the form (1.1) with the initial condition (1.2).

Assume that the solution takes the following infinite series form

$\begin{equation} u(x,y,t) = \sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}. \end{equation}$

(3.1)

Consequently, the approximate solution of equations (1.1)-(1.2), can be written in the form of

$\begin{equation} u_{n}(x,y,t) = \sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma(i\alpha +1)} = u_{0}(x,y)+\sum\limits_{i = 1}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}. \end{equation}$

(3.2)

By applying the operator $D_{t}^{\alpha }$ on equation (3.2), and using the properties (1) and (2), we obtain the formula

$\begin{equation} D_{t}^{\alpha }u_{n}(x,y,t) = \sum\limits_{i = 0}^{n-1}u_{i+1}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}. \end{equation}$

(3.3)

Next, we substitute both (3.2) and (3.3) in (1.1). Therefore, we have the following recurrence relations

$\begin{eqnarray*} 0 & = &\sum\limits_{i = 0}^{n-1}u_{i+1}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}-\frac{\partial ^{2}}{\partial x^{2}}\left(\sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)} \right) ^{2} \\ &&-\frac{\partial ^{2}}{\partial y^{2}}\left( \sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}\right) ^{2}-F\left(\sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)} \right). \end{eqnarray*}$

We follow the same analogue used in obtaining the Taylor series coefficients. In particular, to calculate the function $u_{n}(x, y), n = 1, 2, 3, ..,$ we have to solve the following

$\begin{equation*} D_{t}^{\left( n-1\right) \alpha }\left\{ G(x,y,t,\alpha ,n)\right\}\downarrow _{t = 0} = 0, \end{equation*}$

where

$\begin{eqnarray*} G(x,y,t,\alpha ,n) & = &\sum\limits_{i = 0}^{n-1}u_{i+1}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}-\frac{\partial ^{2}}{\partial x^{2}}\left(\sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}\right) ^{2} \\ &&-\frac{\partial ^{2}}{\partial y^{2}}\left( \sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}\right) ^{2}-F\left(\sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}\right). \end{eqnarray*}$

Now, we calculate the first terms of the sequence $\left\{u_{n}(x, y)\right\} _{1}^{N}$ .

For $n = 1$ we have

$\begin{eqnarray*} G(x,y,t,\alpha ,1) & = &u_{1}(x,y)-\frac{\partial ^{2}}{\partial x^{2}}\left(u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right) ^{2} \\ &&-\frac{\partial ^{2}}{\partial y^{2}}\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right) ^{2}-F\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right) . \end{eqnarray*}$

Solving $G(x, y, 0, \alpha, 1) = 0,$ yields

$\begin{equation*} u_{1}(x,y) = \frac{\partial ^{2}}{\partial x^{2}}u_{0}^{2}(x,y)+\frac{\partial ^{2}}{\partial y^{2}}u_{0}^{2}(x,y)+F\left( u_{0}(x,y)\right). \end{equation*}$

For $n = 2$ we have

$\begin{eqnarray} G(x,y,t,\alpha ,2) & = &u_{1}(x,y)+u_{2}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha+1)} \\ &&-\frac{\partial ^{2}}{\partial x^{2}}\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma(2\alpha +1)}\right) ^{2} \\ &&-\frac{\partial ^{2}}{\partial y^{2}}\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma(2\alpha +1)}\right) ^{2} \\ &&-F\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}\right). \end{eqnarray}$

(3.4)

Applying $D_{t}^{\alpha }$ on both sides of equation (3.4) gives

$\begin{eqnarray*} D_{t}^{\alpha }G(x,y,t,\alpha ,2) & = &u_{2}(x,y)-2\frac{\partial ^{2}}{\partial x^{2}}\left[ \left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma(\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}\right)\right. \\ &&\times \left. \left( u_{1}(x,y)+u_{2}(x,y)\frac{t^{\alpha }}{\Gamma(\alpha +1)}\right) \right] \\ &&-2\frac{\partial ^{2}}{\partial y^{2}}\left[ \left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma(2\alpha +1)}\right) \right. \\ &&\times \left. \left( u_{1}(x,y)+u_{2}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right) \right] -\left( u_{1}(x,y)+u_{2}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}\right) \\ &&\times F^{\prime }\left( u_{0}(x,y)+u_{1}(x,y)\frac{t^{\alpha }}{\Gamma (\alpha +1)}+u_{2}(x,y)\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}\right). \end{eqnarray*}$

Solving $D_{t}^{\alpha }\left\{ G(x, y, t, \alpha, 2)\right\} \downarrow_{t = 0} = 0,$ yields

$\begin{equation*} u_{2}(x,y) = 2\frac{\partial ^{2}}{\partial x^{2}}\left[ u_{0}(x,y)u_{1}(x,y)\right] +2\frac{\partial ^{2}}{\partial y^{2}}\left[ u_{0}(x,y)u_{1}(x,y)\right] +u_{1}(x,y)F^{\prime }\left( u_{0}(x,y)\right). \end{equation*}$

To calculate $u_{3}(x, y)$ , we consider $G(x, y, t, \alpha, 3)$ and we solve

$\begin{equation*} D_{t}^{2\alpha }\left\{ G(x,y,t,\alpha ,3)\right\} \downarrow _{t = 0} = 0, \end{equation*}$

we have

$\begin{eqnarray*} u_{3}(x,y) & = &2\frac{\partial ^{2}}{\partial x^{2}}\left[3u_{1}(x,y)u_{2}(x,y)+u_{0}(x,y)u_{3}(x,y)\right]\\ &&+2\frac{\partial ^{2}}{\partial y^{2}}\left[3u_{1}(x,y)u_{2}(x,y)+u_{0}(x,y)u_{3}(x,y)\right] \\ &&+u_{2}(x,y)F^{\prime }\left( u_{0}(x,y)\right) +u_{1}^{2}(x,y)F^{^{\prime \prime }}\left( u_{0}(x,y)\right), \end{eqnarray*}$

and so on.

In general, to obtain the coefficient function $u_{k}(x, y)$ we solve

$\begin{equation*} D_{t}^{(k-1)\alpha }\left\{ G(x,y,t,\alpha ,k)\right\} \downarrow _{t = 0} = 0. \end{equation*}$

Finally, the solution of equations (1.1)-(1.2), can be expressed by

$\begin{eqnarray*} u(x,y,t) & = &\underset{n\rightarrow \infty }{\lim }u_{n}(x,y,t) \\ & = &\underset{n\rightarrow \infty }{\lim }\sum\limits_{i = 0}^{n}u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)} \\ & = &\sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}. \end{eqnarray*}$

The proof is complete.

4. Test examples

In this section, we test the validity and efficiency of the proposed method to solve three numerical examples of Caputo time-fractional biological population equation.

We define $E_{n}$ to be the absolute error between the exact solution $u$ and the approximate solution $u_{n},$ as follows

$\begin{equation*} E_{n}(x,y,t) = \left\vert u(x,y,t)-u_{n}(x,y,t)\right\vert ,n = 0,1,2,3,... \end{equation*}$

Example 4.1. Consider the Caputo time-fractional biological population equation in the form

$\begin{equation} D_{t}^{\alpha }u = \frac{\partial ^{2}u^{2}}{\partial x^{2}}+\frac{\partial^{2}u^{2}}{\partial y^{2}}+hu, \end{equation}$

(4.1)

with the initial condition

$\begin{equation} u(x,y,0) = u_{0}(x,y) = \sqrt{xy}. \end{equation}$

(4.2)

By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.1)-(4.2) in the form

$\begin{equation*} u(x,y,t) = \sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma(i\alpha +1)}, \end{equation*}$

and

$\begin{equation*} u_{i}(x,y) = h^{i}\sqrt{xy},\text{ for }i = 0,1,2,3,... \end{equation*}$

So, the solution of equations (4.1)-(4.2), can be expressed by

$\begin{eqnarray} u(x,y,t) & = &\sqrt{xy}\left( 1+h\frac{t^{\alpha }}{\Gamma (\alpha +1)}+h^{2}\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}+h^{3}\frac{t^{3\alpha }}{\Gamma(3\alpha +1)}+...\right) \\ & = &\sqrt{xy}\sum\limits_{i = 0}^{\infty }\frac{\left( ht^{\alpha }\right) ^{i}}{\Gamma (i\alpha +1)} = \sqrt{xy}E_{\alpha }\left( ht^{\alpha }\right), \end{eqnarray}$

(4.3)

where $E_{\alpha }\left(ht^{\alpha }\right)$ is the Mittag-Leffler function, defined by (2.3).

Taking $\alpha = 1$ in (4.3), we have

$\begin{eqnarray*} u(x,y,t) & = &\sqrt{xy}\left( 1+ht+\frac{\left( ht\right) ^{2}}{2!}+\frac{\left( ht\right) ^{3}}{3!}+...\right) \\ & = &\sqrt{xy}\exp (ht), \end{eqnarray*}$

which is an exact solution to the standard form biological population equation ^[10].

Example 4.2. Consider the Caputo time-fractional biological population equation in the form

$\begin{equation} D_{t}^{\alpha }u = \frac{\partial ^{2}u^{2}}{\partial x^{2}}+\frac{\partial^{2}u^{2}}{\partial y^{2}}+u, \end{equation}$

(4.4)

with the initial condition

$\begin{equation} u(x,y,0) = u_{0}(x,y) = \sqrt{\sin x\sinh y}. \end{equation}$

(4.5)

By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.4)-(4.5) in the form

$\begin{equation*} u(x,y,t) = \sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma(i\alpha +1)}, \end{equation*}$

and

$\begin{equation*} u_{i}(x,y) = \sqrt{\sin x\sinh y},\text{ for }i = 0,1,2,3,... \end{equation*}$

So, the solution of equations (4.4)-(4.5), can be expressed by

$\begin{eqnarray} u(x,y,t) & = &\sqrt{\sin x\sinh y}\left( 1+\frac{t^{\alpha }}{\Gamma (\alpha +1)}+\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}+\frac{t^{3\alpha }}{\Gamma(3\alpha +1)}+...\right) \\ & = &\sqrt{\sin x\sinh y}\sum\limits_{i = 0}^{\infty }\frac{t^{i\alpha }}{\Gamma (i\alpha +1)} = \sqrt{\sin x\sinh y}E_{\alpha }\left( t^{\alpha}\right), \end{eqnarray}$

(4.6)

where $E_{\alpha }\left(t^{\alpha }\right)$ is the Mittag-Leffler function, defined by (2.3).

Taking $\alpha = 1$ in (4.6), we have

$\begin{eqnarray*} u(x,y,t) & = &\sqrt{\sin x\sinh y}\left( 1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+...\right) \\ & = &\left( \sqrt{\sin x\sinh y}\right) \exp(t), \end{eqnarray*}$

which is an exact solution to the standard form biological population equation ^[12].

Example 4.3 Consider the Caputo time-fractional biological population equation in the form

$\begin{equation} D_{t}^{\alpha }u = \frac{\partial ^{2}u^{2}}{\partial x^{2}}+\frac{\partial^{2}u^{2}}{\partial y^{2}}+hu(1-ru), \end{equation}$

(4.7)

with the initial condition

$\begin{equation} u(x,y,0) = u_{0}(x,y) = \exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) . \end{equation}$

(4.8)

By applying the steps involved in the MGTFSM as presented in Section 3, we have the solution of equations (4.7)-(4.8) in the form

$\begin{equation*} u(x,y,t) = \sum\limits_{i = 0}^{\infty }u_{i}(x,y)\frac{t^{i\alpha }}{\Gamma (i\alpha +1)}, \end{equation*}$

and

$\begin{equation*} u_{i}(x,y) = h^{i}\exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) ,\text{ for } i = 0,1,2,3,... \end{equation*}$

So, the solution of equations (4.7)-(4.8), can be expressed by

$\begin{eqnarray} u(x,y,t) & = &\exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) \left( 1+h\frac{t^{\alpha }}{\Gamma (\alpha +1)}+h^{2}\frac{t^{2\alpha }}{\Gamma (2\alpha +1)}+h^{3}\frac{t^{3\alpha }}{\Gamma (3\alpha +1)}+...\right) \\ & = &\exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) \sum\limits_{i = 0}^{\infty }\frac{\left( ht^{\alpha }\right) ^{i}}{\Gamma (i\alpha +1)} \\ & = &\exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) E_{\alpha }\left( ht^{\alpha}\right) , \end{eqnarray}$

(4.9)

where $E_{\alpha }\left(ht^{\alpha }\right)$ is the Mittag-Leffler function, defined by (2.3).

Taking $\alpha = 1$ in (4.9), we have

$\begin{eqnarray*} u(x,y,t) & = &\exp \left( \sqrt{\frac{hr}{8}}(x+y)\right) \left( 1+ht+\frac{\left( ht\right) ^{2}}{2!}+\frac{\left( ht\right) ^{3}}{3!}+...\right) \\ & = &\exp \left( \sqrt{\frac{hr}{8}}(x+y)+ht\right), \end{eqnarray*}$

which is an exact solution to the standard form biological population equation ^[2].

5. Numerical results and discussion

In this section the numerical results for Examples 4.1, 4.2 and 4.3 are presented. Figures 1, and represents the surface graph of the exact solution and the approximate solution $u_{6}(x, y, t)$ at $\alpha = 0.6, 0.8, 1.$ Figures 2, and represents the behavior of the exact solution and the approximate solution $u_{6}(x, y, t)$ at $\alpha = 0.7, 0.8, 0.9, 1$ . These figures affirm that when the order of the fractional derivative $\alpha$ tends to $1$ , the approximate solutions obtained by MGTFSM tends continuously to the exact solutions. – show the absolute errors between the exact solution and the approximate solution $u_{6}(x, y, t)$ at $\alpha = 1$ for different values of $x, y$ and $t$ . These tables clarifies the convergence of the approximate solutions to the exact solutions.

Figure 1. The surface graph of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.1 when

$h = 1$ and

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. The behavior of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.1 when

$h = y = 1$ and

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. The surface graph of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.2 when

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. The behavior of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.2 when

$y = 1$ and

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. The surface graph of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.3 when

$h = 1, r = 2$ and

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. The behavior of the exact solution

$u$ and the approximate solution

$u_{6}$ by MGTFSM for different values of

$\alpha$ for Example 4.3 when

$h = y = 1, r = 2$ and

$t = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Table 1. Comparison of the absolute errors for the obtained results and the exact solution for Example 4.1 when

$h = 1, n = 6$ and

$\alpha = 1$ .

$t/x, y$	$0.1$	$0.3$	$0.5$	$0.7$
$0.1$	$1.4090\times 10^{-10}$	$4.2269\times 10^{-10}$	$7.0449\times 10^{-10}$	$9.8629\times 10^{-10}$
$0.3$	$1.0576\times 10^{-7}$	$3.1727\times 10^{-7}$	$5.2879\times 10^{-7}$	$7.4030\times 10^{-7}$
$0.5$	$2.3354\times 10^{-6}$	$7.0062\times 10^{-6}$	$1.1677\times 10^{-5}$	$1.6348\times 10^{-5}$
$0.7$	$1.8129\times 10^{-5}$	$5.4387\times 10^{-5}$	$9.0645\times 10^{-5}$	$1.2690\times 10^{-4}$
$0.9$	$8.4486\times 10^{-5}$	$2.5346\times 10^{-4}$	$4.2243\times 10^{-4}$	$5.9140\times 10^{-4}$

| Show Table

DownLoad: CSV

Table 2. Comparison of the absolute errors for the obtained results and the exact solution for Example 4.2 when

$n = 6$ and

$\alpha = 1$ .

$t/x, y$	$0.1$	$0.3$	$0.5$	$0.7$
$0.1$	$1.4090\times 10^{-10}$	$4.2268\times 10^{-10}$	$7.0425\times 10^{-10}$	$9.8497\times 10^{-10}$
$0.3$	$1.0576\times 10^{-7}$	$3.1726\times 10^{-7}$	$5.2860\times 10^{-7}$	$7.3932\times 10^{-7}$
$0.5$	$2.3354\times 10^{-6}$	$7.0059\times 10^{-6}$	$1.1673\times 10^{-5}$	$1.6326\times 10^{-5}$
$0.7$	$1.8129\times 10^{-5}$	$5.4385\times 10^{-5}$	$9.0614\times 10^{-5}$	$1.2673\times 10^{-4}$
$0.9$	$8.4486\times 10^{-5}$	$2.5345\times 10^{-4}$	$4.2228\times 10^{-4}$	$5.9061\times 10^{-4}$

| Show Table

DownLoad: CSV

Table 3. Comparison of the absolute errors for the obtained results and the exact solution for Example 4.3 when

$h = 1, r = 2, n = 6$ and

$\alpha = 1$ .

$t/x, y$	$0.1$	$0.3$	$0.5$	$0.7$
$0.1$	$1.5572\times 10^{-9}$	$1.9019\times 10^{-9}$	$2.3230\times 10^{-9}$	$2.8373\times 10^{-9}$
$0.3$	$1.1688\times 10^{-6}$	$1.4276\times 10^{-6}$	$1.7436\times 10^{-6}$	$2.1297\times 10^{-6}$
$0.5$	$2.5810\times 10^{-5}$	$3.1525\times 10^{-5}$	$3.8504\times 10^{-5}$	$4.7029\times 10^{-5}$
$0.7$	$2.0036\times 10^{-4}$	$2.4472\times 10^{-4}$	$2.9890\times 10^{-4}$	$3.6507\times 10^{-4}$
$0.9$	$9.3372\times 10^{-4}$	$1.1404\times 10^{-3}$	$1.3929\times 10^{-3}$	$1.7013\times 10^{-3}$

| Show Table

DownLoad: CSV

In addition, numerical results have confirmed the theoretical results and high accuracy of the proposed scheme.

Remark 5.1. In this paper, we only apply Six terms to approximate the solutions, if we apply more terms of the approximate solutions, the accuracy of the approximate solutions will be greatly improved.

6. Conclusion

In this paper, a new numerical technique called modified generalized Taylor fractional series method (MGTFSM) has been successfully applied for solving the Caputo time-fractional biological population equation. The method was applied to three numerical examples. The results show that the MGTFSM is an efficient and easy to use technique for finding approximate and exact solutions for these problems. The obtained approximate solutions using the suggested method is in excellent agreement with the exact solutions. This confirms our belief that the effciency of our technique gives it much wider applicability for general classes of fractional problems.

Acknowledgments

The authors are very grateful to the guest editors of this special issue and would like to express their sincere thanks to the referees for the careful and noteworthy reading of the paper and for their constructive comments and suggestions which are improved the paper substantially.

Conflict of interest

The authors declare that there is no conflict of interest in this paper.

References

[1]	O. Acana, M. M. Al Qurashib and D. Baleanu, Reduced differential transform method for solving time and space local fractional partial differential equations, J. Nonlinear Sci. Appl., 10 (2017), 5230-5238. doi: 10.22436/jnsa.010.10.09
[2]	A. A. M. Arafa, S. Z. Rida and H. Mohamed, Homotopy Analysis Method for Solving Biological Population Model, Commun. Theor. Phys., 56 (2011), 797-800. doi: 10.1088/0253-6102/56/5/01
[3]	A. Elsaid, Homotopy analysis method for solving a class of fractional partial differential equations, Commun. Nonlinear Sci., 16 (2011), 3655-3664. doi: 10.1016/j.cnsns.2010.12.040
[4]	A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, et al. A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Appl. Math. Comput., 218 (2012), 8329-8340.
[5]	Z. Hammouch and T. Mekkaoui, Adomian decomposition method for solving a time-fractional Burger-Huxley's equation, Nonlinear Stud., 19 (2012), 489-496.
[6]	R. Hilfer, Applications of fractional Calculus in Physics, World Scientific, 2000.
[7]	M. M. Khader and K. M. Saad, A numerical study using Chebyshev collocation method for a problem of biological invasion: fractional Fisher equation, Int. J. Biomath., 11 (2018), 1850099.
[8]	A. Khalouta and A. Kadem, Comparison of New Iterative Method and Natural Homotopy Perturbation Method for Solving Nonlinear Time-Fractional Wave-Like Equations with Variable Coefficients, Nonlinear Dyn. Syst. Theory, 19 (2019), 160-169.
[9]	A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential equations, Elsevier, North-Holland, 2006.
[10]	Y. Liu, Z. Li and Y. Zhang, Homotopy perturbation method to fractional biological population equation, Fractional Differential Calculus, 1 (2011), 117-124.
[11]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[12]	P. Roul, Application of Homotopy Perturbation Method to Perturbation Biological Population Model, Appl. Appl. Math., 5 (2010), 272-281.
[13]	K. M. Saad, S. Deniz and D. Baleanu, On a new modified fractional analysis of Nagumo equation, Int. J. Biomath., 12 (2019), 1950034.
[14]	K. M. Saad, M. M. Khader, J. F. Gómez-Aguilar, et al. Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29 (2019), 023116.
[15]	K. M. Saad, D. Baleanu and A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger's equations, Comput. Appl. Math., 37 (2018), 5203-5216. doi: 10.1007/s40314-018-0627-1
[16]	K. M. Saad and E. H. Faissal AL-Sharif, Comparative study of a cubic autocatalytic reaction via different analysis methods, Discrete & Continuous Dynamical Systems - S, 12 (2019), 665-684.
[17]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[18]	V. Turut and N. Güzel, On solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Padé Approximations, Eur. J. Pure Appl. Math., 6 (2013), 147-171.

This article has been cited by:

1.	Gbenga O. Ojo, Nazim I. Mahmudov, Aboodh Transform Iterative Method for Spatial Diffusion of a Biological Population with Fractional-Order, 2021, 9, 2227-7390, 155, 10.3390/math9020155
2.	Mehmet Giyas Sakar, Onur Saldır, A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems, 2020, 4, 2504-3110, 53, 10.3390/fractalfract4040053
3.	Nasir Ali, Rashid Nawaz, Laiq Zada, Abir Mouldi, Souhail Mohamed Bouzgarrou, Ndolane Sene, Taza Gul, Analytical Approximate Solution of the Fractional Order Biological Population Model by Using Natural Transform, 2022, 2022, 1687-4129, 1, 10.1155/2022/6703086
4.	Ali Khalouta, Abdelouahab Kadem, Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations, 2020, 34, 2391-4238, 203, 10.2478/amsil-2020-0008
5.	Nazek A. Obeidat, Daniel E. Bentil, Convergence analysis of the fractional decomposition method with applications to time‐fractional biological population models, 2023, 39, 0749-159X, 696, 10.1002/num.22916
6.	Md Nur Alam, Onur Alp Ilhan, Jalil Manafian, Muhammad Imran Asjad, Hadi Rezazadeh, Haci Mehmet Baskonus, Jorge E. Macias-Diaz, New Results of Some of the Conformable Models Arising in Dynamical Systems, 2022, 2022, 1687-9139, 1, 10.1155/2022/7753879
7.	Yeliz Karaca, Dumitru Baleanu, 2022, 9780323900324, 149, 10.1016/B978-0-323-90032-4.00006-7
8.	Ayşe Ata, Mehmet Giyas Sakar, Onur Saldır, Mehmet Şenol, Finite and Infinite Dimensional Reproducing Kernel Hilbert Space Approach for Bagley–Torvik Equation, 2025, 11, 2349-5103, 10.1007/s40819-024-01828-z

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(4626) PDF downloads(712) Cited by(8)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(6) / Tables(3)

AIMS Mathematics

A new numerical technique for solving Caputo time-fractional biological population equation

Related Papers:

Abstract

1. Introduction

2. Basic definitions

3. Analysis of modified generalized Taylor fractional series method (MGTFSM)

4. Test examples

5. Numerical results and discussion

6. Conclusion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

A new numerical technique for solving Caputo time-fractional biological population equation

Related Papers:

Abstract

1. Introduction

2. Basic definitions

3. Analysis of modified generalized Taylor fractional series method (MGTFSM)

4. Test examples

5. Numerical results and discussion

6. Conclusion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog