Numerical solution of two dimensional time fractional-order biological population model

Amit Prakash; Manoj Kumar

doi:10.1515/phys-2016-0021

Article Open Access

Numerical solution of two dimensional time fractional-order biological population model

Amit Prakash and Manoj Kumar

Published/Copyright: June 24, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 14 Issue 1

Abstract

In this work, we provide an approximate solution of a parabolic fractional degenerate problem emerging in a spatial diffusion of biological population model using a fractional variational iteration method (FVIM). Four test illustrations are used to show the proficiency and accuracy of the projected scheme. Comparisons between exact solutions and numerical solutions are presented for different values of fractional order α.

Keywords: Biological population model; Mittag-leffler function; Caputo fractional derivative; Lagrange multiplier; parabolic equation

PACS: 02.30.Jr; 02.60.-x; 04.20.Fy; 04.20.Jb

Introduction

Fractional calculus is an integral branch of mathematics for arbitrary derivatives and integrals. Fractional differential equations are vitally important due to their proved and diverse uses in engineering and every part of science. Fractional differential equations are the best for modeling various processes in engineering and physical sciences. Since many standard models with integer-order derivatives involving nonlinear models, do not work sufficiently well in many cases. At the outset of this century, there has been an increase in the role of fractional calculus in countless arenas such as economics, mechanics, bioinformatics, chemistry, electricity, control analysis, signal and image processing, fluid flow, propagation of seismic waves etc. Many fields of academics in various branches involve unusual diffusion, control and vibration, random walk with continuous time, Brownian motion and fractional Brownian motion, fractional kinetic model with neutron point, power law, Riesz potential etc and hence fractional calculus serves an important role in every part of science and technology. Many researchers developed various numerical methods to solve nonlinear phenomenon since analytic solution do not exist in every situation. In order to maintain precise and consistent solutions, many techniques have been applied to find the solution of the differential equations with fractional order derivative. Some of the neoteric numerical methods are finite element method, finite difference method, fractional differential transform method and the adomain decomposition method etc.

The Fractional variational iteration method (FVIM) is one of these novel approaches to solve nonlinear phenomenon of various fields. He [1–3] first proposed and solved fractional differential equations [4] with the help of VIM. Odibat et al. [5] used VIM to solve problems in fluid mechanics. Yulita et al. [6] solved Zakharov--Kuznetsov equations of fractional order. Inc [7] solved the space- and time-fractional Burgers differential equations. Freeborn et al. [8] applied a nonlinear least square fitting to extract the double -dispersion cole bioimpedance model. Lu [9] applied a variational iteration method to obtain approximate solution of the Fornberg–Whitham equation. Sakar et al. [10] and Prakash et al. [11] have made use of the variational iteration method to find numerical solutions of time-fractional Fornberg–Whitham equation and fractional coupled Burger’s equation respectively. Shakeri et al. [12] solved the two dimensional biological population model for the standard case when α = 1 by a variational iteration method. Recently, Srivastava et al. [13] studied two dimensional time fractional-order biological population model by Fractional reduced differential transform method. The biological population model is also studied by Cheng et al. [14] for the standard case when α = 1 by using the element-free kp-Ritz method and Kumar et al. [15] applied the Homotopy Analysis method to solve the Biological Population model.

The basic goal of the present effort is to make use of the Fractional Variational Iteration Method (FVIM) to find the solution of a time-fractional degenerate partial differential equation of parabolic type arising in the diffusion of the spatial type of biological model:

ptα=pxx2+pyy2+f(p), t≥0, x, y∈IR,(1)

subject to condition at initial stage p(x, y, 0), where p is the density of the population and function f is the supply of population (births and deaths in that region).

Biologists have a sustained and strong belief that immigration or dispersal are important factors for the regulation of some specie’s population. Gurtin and Maccamy [16] described the scattering of a race in a biological model in zone B assuming the functions of spot X = (x, y) in B and time t; where p(x, y, t) is the density of population, v(x, y, t) is the velocity of diffusion and f(x, y, t) is the cause of population.

The function p(x, y, t) indicates counting of characters on the field at location X and time t, per unit volume and total number of individuals at the sub region R of the region B can be found out by integrating over sub region R at any time t. Function f(x, y, t) brings forth the ratio of fellows that are brought inly at the positions X per unit volume by births and deaths. v(x, y, t) denotes the velocity of diffusion from one position to other position in the flow of population of those fellows which at any time t occupy the place X. The functions p, v and f must satisfy the law of population balance: for each sub region R of B at any time t as:

dαdtα∫Rpdv+∫∂Rpv.n^dA=∫RfdV.(2)

In this relation n^ is the normal unit vector outward side of boundary of the region R. This law presents that the changing rate of population in the region R plus the rate at which animals leave the boundary of R should be equal to the rate at which animals are right away to region R. Gurtin and Maccamy [16] proved it by assuming the conditions:

f=f(p), v=−k(p)∇p,(3)

where k(p) > 0 for p > 0. The ensuing nonlinear fractional partial differential equation with the density of p is attained as:

ptα=ϕ(p)xx+ϕ(p)yy+f(p), t≥0 , x, y∈IR.(4)

Gurney et al. [17] applied a particular case of ϕ(p) for the field of population of animals. Movements in the field are due to either grown-up creatures compelled out by invaders or by youthful creatures just reaching maturity moving out of their ancestral homes to establish reproduction province of their own. In these two cases, it can be inferred that these will be in a neighboring unoccupied zone. In the model at hand, drive will take place almost merely “down” the population density gradient, and this model will be more applicable in a high density population place than a low population density place. In this model, they assumed a rectangular place, in which a cattle may either dwell at its existing place or may change its position from high density population to low density population. The probability distribution in these two situations can be categorised with the aid of the magnitude of the population density gradient at the grid site concerned. This model leads to (1) with ϕ(p) = p², i.e. the following equation:

ptα=pxx2+pyy2+f(p), t≥0, x, y∈IR,(5)

subject to the initial condition p(x, y, 0). Some properties of equation (2) such as Holder estimates of its solutions are studied by Y.G. Lu [18]. Two models of constitutive equations for f are the Malthusian law

f=μp,(μ=constant),(6)

and the Verhulst law

f=μp−γp2,(μ,γ=constant).(7)

We study a more general form of f as f(p) = hp^a(1 − rp^b) which yields

ptα=pxx2+pyy2+hpa(1−rpb), t≥0, x, y∈IR,(8)

where α, β, h and r are real numbers. It can be noted that the Malthusian law and Verhulst law are particular situations obtained when h = μ, α = 1, r = 0 and h = μ, α = β = 1, r=γμ respectively.

Preliminaries

Definition

A real function f(t), t > 0 is said to be in the space C_α, α ∈ R if there exists a real number p(> α), such that f(t) = t^pf₁(t), where f₁ ∈ C[0, ∞] clearly C_α ⊂ C_β if β ≤ α[19–21].

Definition

A function f(t), t > 0 is said to be in the space Cαm, m ∈ N ∪ {0}, if f^(m) ∈ C_α[19–21].

Definition

The left sided Riemann-Liouville fractional integral of order μ > 0, of a function f ∈ C_α, α ≥ −1 is defined as [22,23]

Iμf(t)=1Γ(μ)∫0tf(τ)(t−τ)1−μ dτ=1Γ(1+μ)∫0tf(τ) (dτ)μ

where I⁰f(t) = f(t).

Definition

The (left sided) Caputo fractional derivative of f,f∈C−1m,m∈IN∪{0}[22,23],

Dtμf(t)={Im−μf(m)(t),m−1<μ<m, m∈Ndm(f(t))dtm,μ=m.

Note that [20–24]

Itαf(x,t)=1Γ(α)∫0t(t−s)α−1f(x,s)ds, α, t > 0.
Dtαu(x,t)=Itm−α∂mu(x,t)∂tmf(t),m − 1 < α < m.
Iμtγ=Γ(1+γ)Γ(1+γ+μ)tγ+μ.

Definition

The Mittag-Leffler function E_α(z) with α > 0 is defined by the following series representation, valid in the whole complex plane [24,25]: Eα(z)=∑n=0∞znΓ(1+αn),α>0,z∈C.

Fractional variational iteration method for fractional order biological population model

Consider the following two dimensional time fractional order biological population model:

ptα=pxx2+pyy2+hpa(1−rpb), t≥0,x,y∈R.(9)

According to the FVIM, the correction functional for this equation is as follows [3]:

pn+1(x,y,t)=pn+1Γ(1+α)∫0tλ(∂αpn∂τα−∂2p˜n2∂x2 −∂2p˜n2∂y2−h(p˜na(1−rp˜nb)))(dτ)α.(10)

Now by variational theory λ, must satisfy ∂αλ∂τα=0 and λ|_{τ = t} = 0. From these equations, we obtain that λ = −1 and a new correction functional emerges:

pn+1(x,y,t)=pn(x,y,t)−1Γ(1+α)∫0t(∂αpn(x,y,τ)∂τα −∂2pn2∂x2−∂2pn2∂y2−h(pna(1−rpnb)))(dτ)α.(11)

We can build consecutive approximations p_n, n ≥ 0 by using λ, a common Lagrange’s multiplier, that can be obtained by variational theory. The functions p˜n are restricted variation i.e. δp˜n=0 Consequently, first we elect the Lagrange multiplier λ, which can be obtained using integration by parts. In this way we can obtain sequences p_{n + 1}(x, y, t), n ≥ 0 of the solution and finally we can obtain the solution as p(x, y, t) = lim_{n → ∞}p_n(x, y, t).

Test Examples

In the present section, we use the projected technique on some test examples.

Example 1

Consider the ensuing time fractional- order biological population model ∂αp∂tα=∂2p2∂x2+∂2p2∂y2+hp, which can be obtained by putting a = 1, r = 0, b = 1 in (8), subject to the given condition p(x,y,0)=xy.

Now by the given initial condition:

p0(x,y,t)=xyp1(x,y,t)=xy(1+htαΓ(1+α))p2(x,y,t)=xy(1+htαΓ(1+α)+h2t2αΓ(1+2α))p3(x,y,t)=xy(1+htαΓ(1+α)+h2t2αΓ(1+2α)+h3t3αΓ(1+3α))pn(x,y,t)=xy(1+htαΓ(1+α)+h2t2αΓ(1+2α)+h3t3αΓ(1+3α) +…+hntαnΓ(1+nα))p(x,y,t)=limn→∞pn(x,y,t)=(Eα(htα))xy,

which is the exact solution.

Fig. 1 demonstrates the comparison between exact solution and approximate solution for different values of fractional order α=1,23,12,13 when h=15. Fig. 1(a) shows the comparison when t = 0.1, Fig. 1(b) when t = 0.2 and Fig. 1(c) when t = 0.3. Fig. 2 represents the graphical solution for α =1 when t = 0.1, 0.2 and 0.3 respectively. It can be observed from Fig. 1 that the exact and numerical solutions are almost identical for different values of fractional order α. Table 1 shows the comparison of absolute error between the exact solution and the tenth approximate solution for different value of fractional order α when t = 0.1, 0.2 and 0.3 respectively.

$Figure 1 Comparison between exact and approximate solution for different values of fractional order α for example 1.$

Figure 1

Comparison between exact and approximate solution for different values of fractional order α for example 1.

Figure 2

3D-plot for different values of t = 0.1, 0.2, 0.3 when α = 1 for example 1.

Table 1

Absolute error |p − p₁₀|.

x	y	t = 0.1				t = 0.2				t = 0.3
α →		1	2/3	1/2	1/3	1	2/3	1/2	1/3	1	2/3	1/2	1/3
0.1	0.2	0	0	0	4.4 × 10⁻¹⁴	2.7 × 10⁻¹⁷	0	1.5 × 10⁻¹⁵	5.8 × 10⁻¹³	2.7 × 10⁻¹⁷	2.7 × 10⁻¹⁷	1.4 × 10⁻¹⁴	2.5 × 10⁻¹²
0.1	0.6	0	0	0	7.7 × 10⁻¹⁴	5.5 × 10⁻¹⁷	0	2.6 × 10⁻¹⁵	1.0 × 10⁻¹²	5.5 × 10⁻¹⁷	5.5 × 10⁻¹⁷	2.4 × 10⁻¹⁴	4.4 × 10⁻¹²
0.1	1.0	0	0	0	1.0 × 10⁻¹³	5.5 × 10⁻¹⁷	0	3.4 × 10⁻¹⁵	1.2 × 10⁻¹²	5.5 × 10⁻¹⁷	5.5 × 10⁻¹⁷	3.1 × 10⁻¹⁴	5.8 × 10⁻¹²
0.5	0.2	0	0	0	1.0 × 10⁻¹³	5.5 × 10⁻¹⁷	0	3.4 × 10⁻¹⁵	1.2 × 10⁻¹²	5.5 × 10⁻¹⁷	5.5 × 10⁻¹⁷	3.1 × 10⁻¹⁴	5.8 × 10⁻¹²
0.5	0.6	0	0	0	1.7 × 10⁻¹³	1.1 × 10⁻¹⁶	0	5.9 × 10⁻¹⁵	2.2 × 10⁻¹²	1.1 × 10⁻¹⁶	1.1 × 10⁻¹⁶	5.4 × 10⁻¹⁴	1.0 × 10⁻¹¹
0.5	1.0	0	0	0	2.2 × 10⁻¹³	2.2 × 10⁻¹⁶	0	7.6 × 10⁻¹⁵	2.9 × 10⁻¹²	1.1 × 10⁻¹⁶	1.1 × 10⁻¹⁶	6.9 × 10⁻¹⁴	1.3 × 10⁻¹¹
0.9	0.2	0	0	0	1.3 × 10⁻¹³	1.1 × 10⁻¹⁶	0	4.6 × 10⁻¹⁵	1.7 × 10⁻¹²	5.5 × 10⁻¹⁷	1.1 × 10⁻¹⁶	4.2 × 10⁻¹⁴	7.7 × 10⁻¹²
0.9	0.6	0	0	0	2.3 × 10⁻¹³	2.2 × 10⁻¹⁶	0	7.9 × 10⁻¹⁵	3.0 × 10⁻¹²	2.2 × 10⁻¹⁶	1.1 × 10⁻¹⁶	7.2 × 10⁻¹⁴	1.3 × 10⁻¹¹
0.9	1.0	0	0	0	3.0 × 10⁻¹³	2.2 × 10⁻¹⁶	0	1.0 × 10⁻¹⁴	3.8 × 10⁻¹²	2.2 × 10⁻¹⁶	2.2 × 10⁻¹⁶	9.3 × 10⁻¹⁴	1.7 × 10⁻¹¹

Example 2

Consider the ensuing time fractional order biological population model ∂αp∂tα=∂2p2∂x2+∂2p2∂y2+p, which can be obtained by putting h = 1, a = 1, r = 0, b = 1 in (8), subject to given condition p(x,y,0)=sin(x)sinh(y).

Now by the given initial condition:

p0(x,y,t)=sin(x)sinh(y)p1(x,y,t)=p0(x,y,t)−1Γ(1+α)∫0t(∂αp(x,y,τ)∂τα −∂2p2(x,y,τ)∂x2−∂2p2(x,y,τ)∂y2 −p(x,y,τ))(dτ)α =(1+tαΓ(1+α))(sin(x))(sinh(y))p2(x,y,t)=(1+tαΓ(1+α)+t2αΓ(1+2α))sin(x)sinh(y)p3(x,y,t)=(1+tαΓ(1+α)+t2αΓ(1+2α) +t3αΓ(1+3α))sin(x)sinh(y)pn(x,y,t)=(1+tαΓ(1+α)+t2αΓ(1+2α)+t3αΓ(1+3α) +…+tαnΓ(1+nα))sin(x)sinh(y)p(x,y,t)=limn→∞pn(x,y,t)=(Eα(tα))sin(x)sinh(y),

which is the exact solution.

Fig. 3 demonstrates the comparison between exact solution and approximate solution for different values of fractional order α=1,23,12,13Fig. 3(a) shows the comparison when t = 0.1, Fig. 3(b) when t = 0.2 and Fig. 3(c) when t = 0.3. Fig. 4 represents graphical solution for α = 1 when t = 0.1, 0.2 and 0.3 respectively. It can be observed from Fig. 3 that the exact and numerical solutions are almost identical for different values of fractional order α. Table 2. shows the comparison of absolute error between the exact solution and the tenth approximate solution for different value of fractional order α when t = 0.1, 0.2 and 0.3 respectively.

$Figure 3 Comparison between exact and approximate solution for different values of fractional order α for example 2.$

Figure 3

Comparison between exact and approximate solution for different values of fractional order α for example 2.

Figure 4

3D-plot for different values of t = 0.1, 0.2, 0.3 when α = 1 for example 2.

Table 2

Absolute error |p − p₁₀|.

x	y	t = 0.1				t = 0.2				t = 0.3
α →		1	2/3	1/2	1/3	1	2/3	1/2	1/3	1	2/3	1/2	1/3
0.1	0.2	2.8 × 10^-17	7.0 × 10^-13	1.8 × 10^-9	2.9 × 10^-6	5.6 × 10^-17	1.2 × 10^-10	8.6 × 10^-8	4.1 × 10^-5	6.4 × 10^-15	2.3 × 10^-9	8.4 × 10^-7	2.0 × 10^-4
0.1	0.6	5.6 × 10^-17	1.2 × 10^-12	3.2 × 10^-9	5.1 × 10^-6	1.1 × 10^-16	2.1 × 10^-10	1.5 × 10^-7	7.3 × 10^-5	1.1 × 10^-14	4.2 × 10^-9	1.5 × 10^-6	3.5 × 10^-4
0.1	1.0	5.6 × 10^-17	1.7 × 10^-12	4.3 × 10^-9	7.0 × 10^-6	1.1 × 10^-16	2.8 × 10^-10	2.1 × 10^-7	9.9 × 10^-5	1.5 × 10^-14	5.7 × 10^-9	2.0 × 10^-6	4.7 × 10^-4
0.5	0.2	5.6 × 10^-17	1.5 × 10^-12	3.9 × 10^-9	6.3 × 10^-6	1.7 × 10^-16	2.6 × 10^-10	1.9 × 10^-7	8.9 × 10^-5	1.4 × 10^-14	5.1 × 10^-9	1.8 × 10^-6	4.3 × 10^-4
0.5	0.6	1.1 × 10^-16	2.7 × 10^-12	6.9 × 10^-9	1.1 × 10^-5	2.2 × 10^-16	4.5 × 10^-10	3.3 × 10^-7	1.6 × 10^-4	2.5 × 10^-14	9.1 × 10^-9	3.3 × 10^-6	7.6 × 10^-4
0.5	1.0	1.1 × 10^-16	3.7 × 10^-12	9.4 × 10^-9	1.5 × 10^-5	3.3 × 10^-16	6.2 × 10^-10	4.5 × 10^-7	2.2 × 10^-4	3.4 × 10^-14	1.2 × 10^-8	4.4 × 10^-6	1.0 × 10^-3
0.9	0.2	5.6 × 10^-17	2.0 × 10^-12	5.0 × 10^-9	8.6 × 10^-6	1.7 × 10^-16	3.3 × 10^-10	2.4 × 10^-7	1.1 × 10^-4	1.8 × 10^-14	6.6 × 10^-9	2.3 × 10^-6	5.5 × 10^-4
0.9	0.6	1.1 × 10^-17	3.5 × 10^-12	8.9 × 10^-9	1.4 × 10^-5	3.3 × 10^-16	5.8 × 10^-10	4.3 × 10^-7	2.0 × 10^-4	3.2 × 10^-14	1.2 × 10^-8	4.2 × 10^-6	9.7 × 10^-4
0.9	1.0	2.2 × 10^-17	4.7 × 10^-12	1.2 × 10^-8	2.0 × 10^-5	4.4 × 10^-16	7.9 × 10^-10	5.8 × 10^-7	2.8 × 10^-4	4.4 × 10^-14	1.6 × 10^-8	5.7 × 10^-6	1.3 × 10^-3

Example 3

Consider the ensuing time fractional- order biological population model ∂αp∂tα=∂2p2∂x2+∂2p2∂y2+p(1−λp) which can be obtained by putting h = 1, a = 1, r = λ, b = 1 in (8) subject to given condition

p(x,y,0)=eλ(x+y)22. Now by the given initial condition:

p0(x,y,t)=eλ(x+y)22p1(x,y,t)=p0−1Γ(1+α)∫0t(∂αp0∂τα−∂2p02∂x2−∂2p02∂y2 −p0(1−λp0))(dτ)α =(1+tαΓ(1+α))eλ(x+y)22p2(x,y,t)=(1+tαΓ(1+α)+t2αΓ(1+2α))eλ(x+y)22p3(x,y,t)=(1+tαΓ(1+α)+t2αΓ(1+2α) +t3αΓ(1+3α))eλ(x+y)22pn(x,y,t)=(1+tαΓ(1+α)++t2α1+Γ(2α)+t3αΓ(1+3α) +…+tnαΓ(1+nα))eλ(x+y)22p(x,y,t)=limn→∞pn(x,y,t)=(Eα(tα))eλ(x+y)22,

which is the exact solution.

Fig. 5 demonstrates the comparison between exact solution and approximate solution for different values of fractional order α=1,23,12,13 when λ = 1 Fig. 5(a) shows the comparison when t = 0.1, Fig. 5(b) when t = 0.2 and Fig. 5(c) when t = 0.3. Fig. 6 represents graphical solution for α = 1 when t = 0.1, 0.2 and 0.3 respectively. It can be observed from Fig. 5 that the exact and numerical solutions are almost identical for different values of fractional order α. Table 3. shows the comparison of absolute error between the exact solution and the tenth approximate solution for different value of fractional order α when t = 0.1, 0.2 and 0.3 respectively.

$Figure 5 Comparison between exact and approximate solution for different values of fractional order α for example 3.$

Figure 5

Comparison between exact and approximate solution for different values of fractional order α for example 3.

Figure 6

3D-plot for different values of t = 0.1, 0.2, 0.3 when α = 1 for example 3.

Table 3

Absolute error |p − p₁₀|.

x	y	t = 0.1				t = 0.2				t = 0.3
α →		1	2/3	1/2	1/3	1	2/3	1/2	1/3	1	2/3	1/2	1/3
0.1	0.2	2.2 × 10⁻¹⁶	5.5 × 10⁻¹²	1.4 × 10⁻⁸	2.3 × 10⁻⁵	4.4 × 10⁻¹⁶	9.1 × 10⁻¹⁰	6.7 × 10⁻⁷	3.2 × 10⁻⁴	5.0 × 10⁻¹⁴	1.8 × 10⁻⁸	6.6 × 10⁻⁶	1.5 × 10⁻³
0.1	0.6	4.4 × 10⁻¹⁶	6.3 × 10⁻¹²	1.6 × 10⁻⁸	2.6 × 10⁻⁵	4.4 × 10⁻¹⁶	1.1 × 10⁻⁹	7.7 × 10⁻⁷	3.7 × 10⁻⁴	5.8 × 10⁻¹⁴	2.1 × 10⁻⁸	7.6 × 10⁻⁶	1.8 × 10⁻³
0.1	1.0	2.2 × 10⁻¹⁶	7.3 × 10⁻¹²	1.9 × 10⁻⁸	3.0 × 10⁻⁵	6.7 × 10⁻¹⁶	1.2 × 10⁻⁹	8.9 × 10⁻⁷	4.2 × 10⁻⁴	6.7 × 10⁻¹⁴	2.4 × 10⁻⁸	8.7 × 10⁻⁶	2.0 × 10⁻³
0.5	0.2	4.4 × 10⁻¹⁶	6.3 × 10⁻¹²	1.6 × 10⁻⁸	2.6 × 10⁻⁵	4.4 × 10⁻¹⁶	1.1 × 10⁻⁹	7.7 × 10⁻⁷	3.7 × 10⁻⁴	5.8 × 10⁻¹⁴	2.1 × 10⁻⁸	7.6 × 10⁻⁶	1.8 × 10⁻³
0.5	0.6	2.2 × 10⁻¹⁶	7.3 × 10⁻¹²	1.9 × 10⁻⁸	3.0 × 10⁻⁵	6.7 × 10⁻¹⁶	1.2 × 10⁻⁹	8.9 × 10⁻⁷	4.2 × 10⁻⁴	6.7 × 10⁻¹⁴	2.4 × 10⁻⁸	8.7 × 10⁻⁶	2.0 × 10⁻³
0.5	1.0	4.4 × 10⁻¹⁶	8.4 × 10⁻¹²	2.1 × 10⁻⁸	3.5 × 10⁻⁵	8.9 × 10⁻¹⁶	1.4 × 10⁻⁹	1.0 × 10⁻⁷	4.9 × 10⁻⁴	7.7 × 10⁻¹⁴	2.8 × 10⁻⁸	1.0 × 10⁻⁵	2.3 × 10⁻³
0.9	0.2	2.2 × 10⁻¹⁶	7.3 × 10⁻¹²	1.9 × 10⁻⁸	3.0 × 10⁻⁵	6.7 × 10⁻¹⁶	1.2 × 10⁻⁹	8.9 × 10⁻⁷	4.2 × 10⁻⁴	6.7 × 10⁻¹⁴	2.4 × 10⁻⁸	8.7 × 10⁻⁶	2.0 × 10⁻³
0.5	1.0	4.4 × 10⁻¹⁶	8.4 × 10⁻¹²	2.1 × 10⁻⁸	3.5 × 10⁻⁵	8.9 × 10⁻¹⁶	1.4 × 10⁻⁹	1.0 × 10⁻⁷	4.9 × 10⁻⁴	7.7 × 10⁻¹⁴	2.8 × 10⁻⁸	1.0 × 10⁻⁵	2.3 × 10⁻³
0.9	1.0	4.4 × 10⁻¹⁶	9.7 × 10⁻¹²	2.5 × 10⁻⁸	4.0 × 10⁻⁵	8.9 × 10⁻¹⁶	1.6 × 10⁻⁹	1.2 × 10⁻⁷	5.6 × 10⁻⁴	8.9 × 10⁻¹⁴	3.2 × 10⁻⁸	1.2 × 10⁻⁵	2.7 × 10⁻³

Example 4

Consider the ensuing linear time fractional- order biological population model ∂αp∂tα=∂2p2∂x2+∂2p2∂y2−p(8p9+1) which can be obtained by putting h = 1, a = 1, r=89, b = 1 (8) subject to given condition p(x,y,0)=ex+y3.

Now by the given initial condition:

p0(x,y,t)=ex+y3p1(x,y,t)=p0−1Γ(1+α)∫0t(∂αp0∂τα−∂2p02∂x2−∂2p02∂y2 +p0(8p09+1))(dτ)α =(1−tαΓ(1+α))ex+y3p2(x,y,t)=(1−tαΓ(1+α)+t2αΓ(1+2α))ex+y3p3(x,y,t)=(1−tαΓ(1+α)+t2αΓ(1+2α)−t3αΓ(1+3α))ex+y3pn(x,y,t)=(1−tαΓ(1+α)+t2αΓ(1+2α)−t3αΓ(1+3α) +…+(−1)ntnαΓ(1+nα))ex+y3p(x,y,t)=limn→∞pn(x,y,t)=(Eα(−tα))ex+y3,

which is the exact solution.

Fig. 7 demonstrates the comparison between the exact solution and the approximate solution for different values of fractional order α=1,23,12,13 when λ = 1 Fig. 7(a) shows the comparison when t = 0.1, Fig. 7(b) when t = 0.2 and Fig. 7(c) when t = 0.3. Fig. 8 represents graphical solution for α = 1 when t = 0.1, 0.2 and 0.3 respectively. It can be observed from Fig. 7 that the exact and numerical solutions are almost identical for different values of fractional order α. Table 4 shows the comparison of absolute error between the exact solution and the approximate solution for different value of fractional order α when t = 0.1, 0.2 and 0.3 respectively.

$Figure 7 Comparison between exact and approximate solution for different values of fractional order α for example 4.$

Figure 7

Comparison between exact and approximate solution for different values of fractional order α for example 4.

Figure 8

3D-plot for different values of t = 0.1, 0.2, 0.3 when α = 1 for example 4.

Table 4

Absolute error |p − p₁₀|.

x	y	t = 0.1				t = 0.2				t = 0.3
α →		1	2/3	1/2	1/3	1	2/3	1/2	1/3	1	2/3	1/2	1/3
0.1	0.2	0	4.9 × 10⁻¹²	1.1 × 10⁻⁸	1.3 × 10⁻⁵	5.6 × 10⁻¹⁶	7.7 × 10⁻¹⁰	4.7 × 10⁻⁷	1.5 × 10⁻⁴	4.8 × 10⁻¹⁴	1.5 × 10⁻⁸	4.2 × 10⁻⁶	6.4 × 10⁻⁴
0.1	0.6	0	5.6 × 10⁻¹²	1.2 × 10⁻⁸	1.4 × 10⁻⁵	6.7 × 10⁻¹⁶	8.8 × 10⁻¹⁰	5.3 × 10⁻⁷	1.7 × 10⁻⁴	5.5 × 10⁻¹⁴	1.7 × 10⁻⁸	4.8 × 10⁻⁶	7.3 × 10⁻⁴
0.1	1.0	0	6.4 × 10⁻¹²	1.4 × 10⁻⁸	1.6 × 10⁻⁵	6.7 × 10⁻¹⁶	1.0 × 10M⁻⁹	6.1 × 10⁻⁷	2.0 × 10⁻⁴	6.3 × 10⁻¹⁴	1.9 × 10⁻⁸	5.5 × 10⁻⁶	8.4 × 10⁻⁴
0.5	0.2	0	5.6 × 10⁻¹²	1.2 × 10⁻⁸	1.4 × 10⁻⁵	6.7 × 10⁻¹⁶	8.8 × 10⁻¹⁰	5.3 × 10⁻⁷	1.7 × 10⁻⁴	5.5 × 10⁻¹⁴	1.7 × 10⁻⁸	4.8 × 10⁻⁶	7.3 × 10⁻⁴
0.5	0.6	0	6.4 × 10⁻¹²	1.4 × 10⁻⁸	1.6 × 10⁻⁵	6.7 × 10⁻¹⁶	1.0 × 10⁻⁹	6.1 × 10⁻⁷	2.0 × 10⁻⁴	6.3 × 10⁻¹⁴	1.9 × 10⁻⁸	5.5 × 10⁻⁶	8.4 × 10⁻⁴
0.5	1.0	0	7.3 × 10⁻¹²	1.6 × 10⁻⁸	1.9 × 10⁻⁵	8.9 × 10⁻¹⁶	1.1 × 10⁻⁹	6.9 × 10⁻⁷	2.3 × 10⁻⁴	7.2 × 10⁻¹⁴	2.2 × 10⁻⁸	6.2 × 10⁻⁶	9.6 × 10⁻⁴
0.9	0.2	0	6.4 × 10⁻¹²	1.4 × 10⁻⁸	1.6 × 10⁻⁵	6.7 × 10⁻¹⁶	1.0 × 10⁻⁹	6.1 × 10⁻⁷	2.0 × 10⁻⁴	6.3 × 10⁻¹⁴	1.9 × 10⁻⁸	5.5 × 10⁻⁶	8.4 × 10⁻⁴
0.9	0.6	0	7.3 × 10⁻¹²	1.6 × 10⁻⁸	1.9 × 10⁻⁵	8.9 × 10⁻¹⁶	1.1 × 10⁻⁹	6.9 × 10⁻⁷	2.3 × 10⁻⁴	7.2 × 10⁻¹⁴	2.2 × 10⁻⁸	6.2 × 10⁻⁶	9.6 × 10⁻⁴
0.9	1.0	0	8.4 × 10⁻¹²	1.8 × 10⁻⁸	2.1 × 10⁻⁵	1.1 × 10⁻¹⁵	1.3 × 10⁻⁹	7.9 × 10⁻⁷	2.6 × 10⁻⁴	8.2 × 10⁻¹⁴	2.5 × 10⁻⁸	7.1 × 10⁻⁶	1.1 × 10⁻⁴

Conclusion

In this paper, two dimensional time fractional-order biological population model is investigated, using the Fractional variational iteration method (FVIM). The numerical results are in excellent agreement with exact solutions. Numerical results shows that FVIM is a powerful and reliable algorithm for solving the approximate solution of the two dimensional time fractional-order biological population model as sequences of the approximate solution converge to the exact solution rapidly. The main advantage of this technique over other methods is that this technique does not require discretization of variables, perturbation and any other restrictive assumptions. Therefore FVIM is easy to implement and computationally very attractive over other methods.

References

[1] He J. H., Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul., 2007, 2 (4), 235–236.10.1016/S1007-5704(97)90008-3Search in Google Scholar

[2] He J. H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 1998, 167, 69–73.10.1016/S0045-7825(98)00109-1Search in Google Scholar

[3] He J. H., Variational iteration method – a kind of non-linear analytical technique: some examples, Int. J. Nonlinear Mech., 1999, 34, 699–708.10.1016/S0020-7462(98)00048-1Search in Google Scholar

[4] He J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 1998, 167, 57–68.10.1016/S0045-7825(98)00108-XSearch in Google Scholar

[5] Odibat Z., Momani S., The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 2009, 58, 2199–2208.10.1016/j.camwa.2009.03.009Search in Google Scholar

[6] Yulita Molliq R., Noorani M. S. M., Hashim I., Ahmad R. R., Approximate solutions of fractional Zakharov–Kuznetsov equations by VIM, J. Comput. Appl. Math., 2009, 233 (2), 103–108.10.1016/j.cam.2009.03.010Search in Google Scholar

[7] Inc M., The approximate and exact solutions of the space- and time-fractional burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 2008, 345 (1), 476–484.10.1016/j.jmaa.2008.04.007Search in Google Scholar

[8] Freeborn T. J., Maundy B. J., Elwakil S., Extracting the parameters of the double-dispersion Cole bioimpedance model from magnitude response measurements, Med. Biol. Eng. Comput., 2014, 52, 749–758.10.1007/s11517-014-1175-5Search in Google Scholar PubMed

[9] Lu J., An analytical approach to the Fornberg–Whitham type equations by using the variational iteration method, Comput. Math. Appl., 2011, 61, 2010–2013.10.1016/j.camwa.2010.08.052Search in Google Scholar

[10] Sakar M. G., Erdogan F., Yildirim A., Variational iteration method for the time-fractional Fornberg–Whitham equation, Computers and Mathematics with Applications., 2012, 63, 1382–1388.10.1016/j.camwa.2012.01.031Search in Google Scholar

[11] Prakash A., Kumar M., Sharma K. K., Numerical method for solving fractional coupled Burgers equations, Applied Mathematics and Computation, 2015, 260, 314–320.10.1016/j.amc.2015.03.037Search in Google Scholar

[12] Shakeri F., Dehghan M., Numerical solution of a biological population model using He’s variational iteration method, Computers and Mathematics with Applications, 2007, 54, 1197–1209.10.1016/j.camwa.2006.12.076Search in Google Scholar

[13] Srivastava V. K., Kumar S., Awasthi M. K., Singh B. K., Two-dimensional time fractional-order biological population model and its analytical solution, Egyptian journal of basic and applied sciences, 2014, 1, 71–76.10.1016/j.ejbas.2014.03.001Search in Google Scholar

[14] Cheng R. J., Zhang L. W., Liew K. M., Modeling of biological population problems using the element-free kp-Ritz method, Applied Mathematics and Computation, 2014, 227, 274–290.10.1016/j.amc.2013.11.033Search in Google Scholar

[15] Kumar D., Singh J., Sushila, Application of Homotopy Analysis Transform Method to Fractional Biological Population Model, Romanian Reports in Physics, 2013, 65, 63–75.Search in Google Scholar

[16] Gurtin M. E., Maccamy R. C., On the diffusion of biological population, Mathematical Biosciences, 1977, 33, 35–49.10.1016/0025-5564(77)90062-1Search in Google Scholar

[17] Gurney W. S. C., Nisbet R. M., The regulation of inhomogeneous populations, Journal of Theoretical Biology, 1975, 52, 441–457.10.1016/0022-5193(75)90011-9Search in Google Scholar

[18] Lu Y. G., Holder estimates of solutions of biological population equations, Applied Mathematics Letters., 2000, 13, 123–126.10.1016/S0893-9659(00)00066-5Search in Google Scholar

[19] Mainardi F., On the initial value problem for the fractional diffusion-wave equation, S. Rionero and T. Ruggeeri, editors, Waves and stability in continuous media, World Scientific, Singapore, 1994, 246–251.Search in Google Scholar

[20] Mainardi F., Fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 1996, 9, 23–28.10.1016/0893-9659(96)00089-4Search in Google Scholar

[21] Luchko Yu., Gorenflo R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math Vietnamica, 1999, 24, 207–233.Search in Google Scholar

[22] Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999.Search in Google Scholar

[23] Samko G., Kilbas A. A., Marichev O. I., Fractional integrals and derivatives: theory and applications, Gordon and Breach, Yverdon, 1993.Search in Google Scholar

[24] Mainardi F., Panini G., The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput., 2003, 141, 51–62.10.1016/S0096-3003(02)00320-XSearch in Google Scholar

[25] Mainardi F., Gorenflo R., On Mittag Leffler type functions in fractional evolution processes, J. Comput. Appl. Math., 2000, 118, 283–299.10.1016/S0377-0427(00)00294-6Search in Google Scholar

Received: 2015-10-8

Accepted: 2016-5-4

Published Online: 2016-6-24

Published in Print: 2016-1-1

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.