Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

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Commun. Theor. Phys. 60 (2013) 269–277 Vol. 60, No. 3, September 15, 2013

Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

Pradip Roul∗

Visvesvaraya National Institute of Technology, Department of Mathematics, Nagpur 440010, India

(Received December 28, 2012; revised manuscript received April 1, 2013)

Abstract The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation

with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy–perturbation

method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of

Caputo. Comparisons are made with those derived by Adomian’s decomposition method, revealing that the homotopy

perturbation method is more accurate and convenient than the Adomian’s decomposition method. Furthermore, the

results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed

method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential

equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be

further applied to solve other similar fractional partial differential equations.

PACS numbers: 02.30.Jr, 87.10.Ed, 05.45.YvKey words: reaction-diffusion equation, fractional calculus, Homotopy-perturbation method, biological pop-

ulation model, Mittag–Leffler function

1 Introduction

In recent years extensive research has been devoted

to the study of differential equations with fractional or-

ders due to their importance for applications in various

branches of applied sciences and engineering.[1−14] Many

important phenomena in viscoelasticity, signal processing,

electromamagnetics, crowded systems, and fluid mechan-

ics are well described by fractional differential equation.

Moreover, the fractional differential model has certain ad-

vantage in comparison with classical integer order differen-

tial equation model, for instance the anomalous diffusion

behavior cannot be modelled accurately using the integer

order differential equation. It can be used to accurately

describe the memory and hereditary properties for vari-

ous engineering materials and biological processes. It also

provides exact description of nonlinear phenomena.

In this study, we shall focus on nonlinear time-

fractional degenerate parabolic equation arising in the

spatial diffusion of biological population model. This

model was originally developed by Gurten et al.[15] The

diffusion of a biological species in a domain D is described

by the following non-linear degenerate parabolic partial

differential equation of integer order in the form

∂u(x, y, t)

∂t= ∆ϕ(u(x, y, t)) + g(u(x, y, t)) , (1)

where the field u(x, y, t) gives the population density at

(x, y) at time t, v(x, y, t) is the diffusion velocity, g(x, y, t)

is the population supply due to births and deaths, ϕ(u) is

a nonlinear function of u, and ∆ is the Laplacian operator.

The function ϕ(u) satisfies the following conditions

(i) ϕ′(u) = 0, for u = 0.

(ii) ϕ′(u) > 0, for u > 0.

Equation (1) is a second-order parabolic equation

when u > 0, but degenerates to first order when u = 0.

In this study, we consider the nonlinear fractional-

order biological population model in the following form:

∂αu(x, y, t)

∂tα=

∂2u2(x, y, t)

∂x2+

∂2u2(x, y, t)

∂y2

+ g(u(x, y, t)) , 0 < α ≤ 1 , (2)

subject to the initial condition u(x, y, 0) = f(x, y), t = 0,

where α is the order of the time derivative.

The time-fractional biological population model (2) is

obtained from the classical reaction-diffusion equation by

replacing the first-order time derivative with fractional

derivative of order 0 < α ≤ 1. This generalized equation

is shown to model a diffusion process in which diffusion of

biological species is anomalous subdiffusion.

We consider a more general form of the population

supply term g(u) in Eq. (2) as g(u) = huc(1− rud), where

h, c, d, and r are real numbers.

It is worth pointing out that the above problem (2)

with α = 1 and g(u(x, y, t)) = −kup(x, y, t), where

k ≥ 0, 0 < p < 1, arises in the study of flow

through porous media.[16−17] Moreover, this problem with

first order time derivative leads to Malthusian law[15]

for g(u(x, y, t)) = c1u(x, y, t) and Verhulst law[15] for

g(u(x, y, t)) = c2u(x, y, t) − c3u2(x, y, t). Here c1, c2, c3

are positive constants.

The main objective of this study is to employ

homotopy-perturbation method for finding the exact so-

lution and approximate analytical solutions of nonlinear

∗Corresponding author, E-mail: drpkroul@yahoo.com

c© 2013 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

270 Communications in Theoretical Physics Vol. 60

fractional-order biological population models (2) by de-

scribing the fractional derivatives in the Captou sense. To

the author’s knowledge there is no study concerning the

implementation of homotopy perturbation method for the

degenerate partial differential equation.

We would like to point out that there exists no method

for finding the exact solution of fractional order differential

equations, only approximate solutions can be obtained by

employing perturbation method, analytic approximation

method or numerical method. The traditional perturba-

tion methods were widely used over the last three decades

for solving linear and nonlinear problems. However, this

method depends on a small parameter in the system which

is difficult to optimally be chosen in order to obtain ap-

propriate results. An effective and convenient method

for solving such equations is needed. The most com-

mon analytical techniques are the Adomian’s decomposi-

tion method,[2,18−19] the Variational iteration method,[5]

the homotopy perturbation method,[20] and the fractional

complex transform method,[21−22] which are appropriate

for finding approximate analytical solutions of linear and

nonlinear equations of fractional order.

One of the most powerful among these analytical

methods is the Homotopy perturbation method, which

was originally proposed by He[23] for solving differ-

ential equations of integer order. Later the method

has been employed successfully by several researchers

for finding approximate analytical and numerical solu-

tions of functional equations either of integer or frac-

tional order. HPM is a combination of the classi-

cal perturbation method and the homotopy concept as

used in topology. The major advantage of the method

lies in its ability to solve the nonlinear problem accu-

rately. This approach has been effectively applied to var-

ious equations such as diffusion and wave equations,[24]

nonlinear coupled KDV equations,[25] reaction-diffusion

equations,[26] Flierl–Petviashivili equation,[27] Jaulent-

miodeck equation,[28] Fokker–Planck equation,[29] Boussi-

nesq equation,[30] Burgers equation[31] etc. This technique

is employed in Ref. [32] to solve fractional partial differ-

ential equation in fluid mechanics. Also this procedure

is used in Refs. [33–37] for solving the integro-differential

equation. Authors of Ref. [38] employed the homotopy

perturbation method for solving the inverse problem of

diffusion equation. In contrast to the traditional pertur-

bation methods, the HPM solves the problems without

requiring perturbation or small parameter in the system.

The main feature of this method is the condition of ho-

motopy by introducing an embedment parameterp, which

takes the value from 0 to 1. If p = 0, the homotopy equa-

tion generally reduces to a sufficiently simplified form,

which yields a rather simple solution. While p = 1, it

turns out to be the original problem, and gives the re-

quired solution. In the HPM, the approximate solution is

calculated in the form of power series which usually con-

verges rapidly to the exact solution

This article is organized as follows. In Sec. 2, we

begin by introducing the basic principles of homotopy-

perturbation method and apply this method to nonlinear

differential equations of fractional order (2). Section 3 is

devoted to some necessary definitions and properties of the

fractional differentiations and integrations. In Sec. 4, we

present the approximate solutions of our model problem.

Finally in Sec. 5, we summarize and discuss the results.

2 Analysis of Fractional Homotopy Perturba-

tion Method

To illustrate the basis ideas of the fractional homotopy-

perturbation method, we consider the following a nonlin-

ear differential equation of time-fractional derivative:

Dαu(t)+Lu(t)+Nu(t)+g(r) = 0 , 0 < α ≤ 1 , r ∈ Ω , (3)

subject to the boundary condition of the form

B(

u,∂u

∂n

)

= 0 , r ∈ Γ , (4)

respectively, where L represents the linear operator, N de-

notes a non-linear operator, and g(r) is a known analytical

function, u(t) is an unknown function, Dα = ∂α/∂tα de-

notes the Caputo fractional derivative of order α, Γ is the

boundary of the domain Ω and B is a boundary operator.

In view of Homotopy perturbation method,[23] we con-

struct the following homotopy v(t, p) for Eq. (3): v(t, p) :

Ω × [0, 1] → R, which satisfies

H(v(t), p) = (1 − p)Dαv(t) + p[Dαv(t)

+ Lu0(t) + Nv(t) + g(r)] = 0 , (5)

or equivalently,

H(v(t), p) = Dαv(t)+ p[Lu0(t) + Nv(t) + g(r)] = 0 , (6)

where p ∈ [0, 1] is an embedding parameter, u0(t) is an ini-

tial approximation of Eq. (3), which satisfies the boundary

conditions.

If p = 0, then Eq. (6) becomes

H(v(t), 0) = Dαv(t) = 0 , (7)

and when p = 1, Eq. (6) turns out to be the original sys-

tem given in Eq. (3), i.e.

H(v(t), 1) = Dαv(t) + Lu0(t) + Nv(t) + g(r) = 0 . (8)

Applying the perturbation technique,[39−40] we have the

following power series presentation for v in terms of the

homotopy parameter p:

v = v0 + pv1 + p2v2 + p3v3 + · · · (9)

Setting p = 1 in Eq. (9), we can obtain the HPM series

solution of Eq. (3) as follows

u = limp→1

v = limp→1

(v0 + pv1 + p2v2 + p3v3 + · · ·)

= v0 + v1 + v2 + v3 + · · · (10)

No. 3 Communications in Theoretical Physics 271

Inserting Eq. (10) into Eq. (6) and then equating the iden-

tical powers of p, we obtain the following series of linear

equations:

p0 : Dαv0(t) = 0 , (11)

p1 : Dαv1(t) = −Lv0(t) − N1(v0(t)) − g(r) , (12)

p2 : Dαv2(t) = −Lv1(t) − N2(v0(t), v1(t)) , (13)

p3 : Dαv3(t) = −Lv2(t) − N3(v0(t), v1(t), v2(t)) , (14)

···

and so on.

Here the non-linear functions N1, N2, N3, . . . satisfy

the following equation:

N(v0(t) + pv1(t) + · · ·) = N1(v0(t)) + pN2(v0(t), v1(t))

+ p2N3(v0(t), v1(t), v2(t)) + · · · (15)

To solve these linear equations (11)–(14), we apply the

operator Jα, which is the inverse operator of Dα, on both

sides of the equations, yields the following n-term trun-

cated series solution for Eq. (3):

un(t) = v0(t) + v1(t) + v2(t) + v3(t) + · · · + vn−1(t) ,

where

v0(t) = u0(t) , (16)

v1(t) = −Jα[Lv0(t) + N1v0(t) + g(t)] , (17)

v2(t) = −Jα[Lv1(t) + N2(v0(t), v1(t))] , (18)

v3(t) = −Jα[Lv2(t) + N2(v0(t), v1(t), v2(t))] ,

···

vn−1(t) = −Jα[Lvn−2(t)

+ Nn−2(v0(t), v1(t), v2(t), . . . , vn−2(t))] . (20)

3 Preliminaries

In this section, we give some basis definitions and prop-

erties of the fractional calculus theory which are required

for solving our model problem (2). There are several

definitions of fractional derivatives and integrations ex-

ist in Refs. [8, 41–43], such as Riemann–Liouville def-

inition, Grunwald–Letnikov’s definition, Captou’s, defi-

nition, Weyl’s approach, and Reize’s definition. Among

these definitions, the most commonly used definitions are

the Riemann–Liouville and Caputo. However, the ad-

vantage of the Captou’s derivative over the definition

of Riemann–Liouville derivatives is that the fractional

derivative of a constant is zero. In this study, the frac-

tional derivatives are described in the sense of Caputo.

Definition 1 A real function f(x), x > 0, is said to be

in space cσ, σ ∈ R, if there exists a real number h > σ,

such that f(x) = xhf1(x), where f1(x) ∈ C(0,∞), and it

is said to be in the space cnσ if and only if fn ∈ cσ, n ∈ N .

Definition 2 Let f : R → R, x → f(x), and f(x) ∈

cσ, σ ≥ −1. Then the Riemann–Liouville fractional inte-

gration of f(x), of order α, may be defined as

Jαf(x) =1

Γ(α)

∫ x

0

(x − τ)α−1f(τ)dτ ,

α > 0, x > 0 , (21)

J0f(x) = f(x) . (22)

Definition 3 The fractional derivative of f(x) of order

α in the Caputo sense is defined as

Dαf(x) =1

Γ(m − α)

∫ x

0

(x − τ )m−α−1 ∂m

∂τmf(τ)dτ ,

for m − 1 < α < m , m ∈ N , x > 0 , (23)

Dαf(x) =∂m

∂τmf(x) , for α = m . (24)

Definition 4 For f(x) ∈ cσ, σ ≥ −1, α > 0, β > 0, and

γ > −1, we have in the following some of the properties

of the operator Jα:

(i) JαJβf(x) = JβJαf(x) ,

(ii) JαJβf(x) = Jα+βf(x) ,

(iii) Jα(xγ) =Γ(γ + 1)

Γ(α + γ + 1)xα+γ . (25)

Lemma 1 If m − 1 < α ≤ m, m ∈ N and f(x) ∈ cσ,

σ ≥ −1, then the following two properties of the Caputo

fractional derivative hold:

DαJα(f(x)) = f(x) , (26)

JαDα(f(x)) = f(x) −m−1∑

i=0

f (i)(0+)xi

i!, x > 0 . (27)

4 Numerical Experiments

In this section, we present two numerical examples to

demonstrate the behaviour of the solution of a fractional

partial differential equations arising in biological popula-

tion models. Moreover, we compare our solution with the

existing solution[11] for the special case of α = 1 to cor-

roborate the efficiency and the accuracy of the method.

Example 1

Let us first consider the following biological population

model:

∂αu(x, y, t)

∂tα=

∂2u2(x, y, t)

∂x2+

∂2u2(x, y, t)

∂y2− u(x, y, t)

(

1 +8

9u(x, y, t)

)

, 0 < α ≤ 1 , (28)

subject to the initial condition

u(x, y, 0) = exp(1

3(x + y)

)

. (29)

272 Communications in Theoretical Physics Vol. 60

By means of homotopy-perturbation method, we can construct the homotopy for Eq. (28) which satisfies:

∂uα(x, y, t)

∂tα+ p

(

−∂2u2(x, y, t)

∂x2−

∂2u2(x, y, t)

∂y2+ u(x, y, t)

(

1 +8

9u(x, y, t)

))

. (30)

Substituting Eq. (10) into Eq. (30) and equating the terms with same powers of p, we have the following set of linear

partial differential equations:

p0 :∂αu0(x, y, t)

∂tα= 0 , u0(x, y, t) = exp

(1

3(x + y)

)

. (31)

p1 :∂αu1(x, y, t)

∂tα−

∂2u20(x, y, t)

∂x2−

∂2u20(x, y, t)

∂y2+ u0(x, y, t) +

8

9u2

0(x, y, t) = 0 , u1(x, y, 0) = 0 , (32)

p2 :∂αu2(x, y, t)

∂tα− 2

∂2

∂x2(u0(x, y, t)u1(x, y, t)) − 2

∂2

∂y2(u0(x, y, t)u1(x, y, t))

+ u1(x, y, t) +16

9u0(x, y, t)u1(x, y, t) = 0 , u2(x, y, 0) = 0 . (33)

···

and so on. Next, applying the operator Jα, the inverse

operator of Dα, on both sides of the above equations and

using the initial conditions (29) we obtain the following a

few terms of the approximation solution of Eq. (28)

u0(x, y, t) = exp(1

3(x + y)

)

, (34)

u1(x, y, t) = − exp(1

3(x + y)

) tα

Γ(α + 1), (35)

u2(x, y, t) = exp(1

3(x + y)

) t2α

Γ(2α + 1). (36)

···

In a similar manner other higher-order components of

HPM series namely u3, u4, u5, etc can be derived recur-

rently. Now, the approximate solution of Eq. (28) can be

written in a series form as follows

u(x, y, t) = u0(x, y, t) + u1(x, y, t)

+ u2(x, y, t) + u3(x, y, t) + · · ·

= limn→∞

[

exp(1

3(x + y)

)

− exp(1

3(x + y)

) tα

Γ(α + 1)+ · · ·

+ exp(1

3(x + y)

) (−1)ntnα

Γ(nα + 1)

]

= exp(1

3(x + y)

)

∞∑

k=0

(−tα)k

Γ(kα + 1)

= exp(1

3(x + y)

)

Eα(−tα) , (37)

which is the exact solution of Eq. (28), where Eα(tα) is

the Mittag–Leffler function,[8] it can be defined as follows

Eα(w) =

∞∑

k=0

wk

Γ(kα + 1). (38)

Setting α = 1 in Eq. (37), we therefore obtain as

u(x, y, t) = exp(1

3(x + y)

)

∞∑

n=0

(−1)ntn

n!

= exp(1

3(x + y)

)

exp(−t)

= exp(1

3(x + y) − t

)

, (39)

which is the exact solution obtained in Ref. [11] using the

Adomian decomposition method for the standard form bi-

ological population model, and in addition, Verhulst law

is verified for this case.

It is worth pointing out that the approximate solu-

tions in Eq. (37) obtained in the form of power series

with easily computable components adopting homotopy-

perturbation technique without considering any lineariza-

tion or discretization of the variables. One can see from

the above solution process that the approximate solution

converges very fast to its exact solution.

Example 2

We consider the following biological population model:

∂αu(x, y, t)

∂tα=

∂2u2(x, y, t)

∂x2+

∂2u2(x, y, t)

∂y2+ hu−1(x, y, t)(1 − ru(x, y, t)) , 0 < α ≤ 1 , x, y ∈ R , t > 0 , (40)

subject to the initial condition

u(x, y, 0) =

√

hr

4x2 +

hr

4y2 +

y

16+

5

16. (41)

By means of homotopy-perturbation technique, the homotopy for Eq. (40) is given by

∂uα(x, y, t)

∂tα+ p

(

−∂2u2(x, y, t)

∂x2−

∂2u2(x, y, t)

∂y2− hu−1(x, y, t)(1 − ru(x, y, t))

)

= 0 , (42)

No. 3 Communications in Theoretical Physics 273

Substituting Eq. (10) into Eq. (42) and equating the terms with same powers of p, we have the following set of linear

partial differential equations

p0 :∂αu0(x, t)

∂tα= 0 , u(x, y, 0) =

√

hr

4x2 +

hr

4y2 +

y

16+

5

16. (43)

p1 :∂αu1(x, y, t)

∂tα−

∂2u20(x, y, t)

∂x2−

∂2u20(x, y, t)

∂y2− h(u−1

0 (x, y, t) − r) = 0 , u1(x, y, 0) = 0 , (44)

p2 :∂αu2(x, y, t)

∂tα− 2

∂2

∂x2(u0(x, y, t)u1(x, y, t)) − 2

∂2

∂x2(u0(x, y, t)u1(x, y, t))

− h(u−20 (x, y, t)u1(x, y, t) − r) = 0 , u2(x, y, 0) = 0 . (45)

···

By applying the operator Jα, the inverse operator of Dα,

on both sides of the above equations and using the initial

conditions (41) we obtain the following a few terms of the

approximation solution of Eq. (40):

u0(x, y, t) =

√

hr

4x2 +

hr

4y2 +

y

16+

5

16, (46)

u1(x, y, t) = h(hr

4x2 +

hr

4y2 +

y

16+

5

16

)

−1/2

×tα

Γ(α + 1), (47)

u2(x, y, t) = −2h2(hr

4x2 +

hr

4y2 +

y

16+

5

16

)

−3/2

×t2α

Γ(2α + 1), (48)

···

and in the same manner the rest of the terms of the ap-

proximation solution of Eq. (40) can be obtained. Substi-

tuting u0, u1, u2 into Eq. (10), we obtain the third-order-

term of the approximate solution of Eq. (40) in series form

as follows

u(x, y, t) = u0(x, y, t) + u1(x, y, t) + u2(x, y, t)

=

√

hr

4x2 +

hr

4y2 +

y

16+

5

16

+ h(hr

4x2 +

hr

4y2 +

y

16+

5

16

)

−1/2 tα

Γ(α + 1)

− 2h2(hr

4x2 +

hr

4y2 +

y

16+

5

16

)

−3/2

×t2α

Γ(2α + 1). (49)

The exact solution of Eq. (40) for α = 1 is given by

u(x, y, t) =

√

hr

4x2 +

hr

4y2 +

y

16+ 2ht +

5

16. (50)

The approximate solution of Eq. (40) for α = 1

obtained using the third-order-term of the homotopy-

perturbation procedure is depicted in Fig. 1(a), while the

exact solution in Fig. 1(b). It is clear from Table 1, and

Figs. 1 that our approximate solution is nearly identical

with the exact solution and those of the solutions obtained

by Adomian’s decomposition method.[11] We note that the

approximate solutions obtained in Ref. [11] using the three

terms of the decomposition series. We compare the results

of the absolute error between these two techniques as illus-

trated in Tables 2 and 3 for t = 10 and t = 20, respectively,

and comparison shows that our method is more accurate

than the ADM.

Fig. 1 The surface generated for the solution of Eq. (60) for t = 10 : (a) Approximate solution and (b) Exactsolution.

274 Communications in Theoretical Physics Vol. 60

Table 1(a) Comparison between the exact solution and the approximation solutions obtained by HPMand ADM for t = 10.

(x, y) Exact ADM HPM

(−450,−450) 224.938 649 043 096 5 224.938 651 616 480 3 224.938 649 042 619 80

(−400,−400) 199.938 792 717 504 8 199.938 795 974 648 9 199.938 792 716 825 99

(−300,−300) 149.939 223 798 622 3 149.939 229 590 169 9 149.939 223 797 012 77

(−250,−250) 124.939 568 725 577 6 124.939 577 066 597 2 124.939 568 722 795 74

(0, 0) 0.721 687 836 487 03 −0.087 783 840 520 90 0.683 242 993 124 935 75

(50, 50) 25.072 810 638 883 97 25.073 017 531 231 13 25.072 810 294 449 695

(100, 100) 50.067 662 551 125 08 50.067 714 479 152 43 50.067 662 507 890 844

(200, 200) 100.065 082 987 690 2 100.065 095 990 674 8 100.065 082 982 275 24

(350, 350) 175.063 976 401 009 9 175.063 980 649 505 7 175.063 976 399 998 64

(500, 500) 250.063 533 593 631 6 250.063 535 675 883 0 250.063 533 593 284 66

Table 1(b) Comparison between the exact solution and the approximation solutions obtained by HPMand ADM for t = 20.

(x, y) Exact ADM HPM

(−450,−450) 224.939 112 131 853 7 224.939 122 425 388 8 224.939 112 129 946 95

(−400,−400) 199.939 313 709 602 1 199.939 326 738 178 7 199.939 313 706 886 88

(−300,−300) 149.939 918 522 942 6 149.939 941 689 132 9 149.939 918 516 504 58

(−250,−250) 124.940 402 459 199 2 124.940 435 823 277 3 124.940 402 448 071 58

(0, 0) 0.853 912 563 829 97 −2.400 864 341 458 41 0.683 242 993 124 935 75

(50, 50) 25.076 964 861 535 11 25.077 792 430 695 54 25.076 963 483 569 813

(100, 100) 50.069 743 025 770 24 50.069 950 737 872 44 50.069 742 852 826 110

(200, 200) 100.066 123 971 435 4 100.066 175 983 373 3 100.066 123 949 775 15

(350, 350) 175.064 571 420 566 6 175.064 588 414 550 1 175.064 571 416 521 74

(500, 500) 250.063 950 154 088 9 250.063 958 483 094 3 250.063 950 152 701 10

Table 2 Comparison of the absolute error between results obtained by HPM and ADM for t = 10.

(x, y) ADM HPM

(−450,−450) 2.573 383 767 120 942 × 10−6 4.766 889 105 667 360 1 × 10−10

(−400,−400) 3.257 144 184 114 394 × 10−6 6.787 672 646 169 085 1 × 10−10

(−300,−300) 5.791 547 592 024 089 × 10−6 1.609 492 983 334 348 5 × 10−9

(−250,−250) 8.341 019 542 657 662 × 10−6 2.781 888 497 338 513 8 × 10−9

(0, 0) 8.094 716 770 079 3 × 10−1 3.844 484 336 209 652 6 × 10−2

(50, 50) 2.068 923 471 546 624 × 10−4 3.444 342 766 556 474 1 × 10−7

(100, 100) 5.192 802 735 058 470 × 10−5 4.323 424 462 882 030 6 × 10−8

(200, 200) 1.300 298 454 726 411 × 10−5 5.414 989 345 808 862 7 × 10−9

(350, 350) 4.248 495 877 095 704 × 10−6 1.011 215 999 824 344 2 × 10−9

(500, 500) 2.082 251 364 180 334 × 10−6 3.469 722 287 263 721 2 × 10−10

Table 3 Comparison of the absolute error between results obtained by HPM and ADM for t = 20.

(x, y) ADM HPM

(−450,−450) 1.029 353 506 533 579 × 10−5 1.906 784 063 976 374 5 × 10−9

(−400,−400) 2.316 619 033 145 293 × 10−5 2.715 182 745 305 355 6 × 10−9

(−300,−300) 2.316 619 033 145 293 × 10−5 6.438 028 776 756 25 50 × 10−9

(−250,−250) 3.336 407 806 809 765 × 10−5 1.112 761 083 277 291 6 × 10−8

(0, 0) 3.254 776 905 288 38 1.706 695 707 050 308 6 × 10−1

(50, 50) 8.275 691 604 2753 43 × 10−4 1.377 965 297 4221790 × 10−6

(100, 100) 2.077 121 022 009 413 × 10−4 1.729 441 194 697 756 1 × 10−7

(200, 200) 5.201 193 793 784 941 × 10−5 2.166 018 475 691 089 4 × 10−8

(350, 350) 1.699 398 349 659 212 × 10−5 4.044 863 999 297 376 7 × 10−9

(500, 500) 8.329 005 354 886 454 × 10−6 1.387 803 649 777 197 3 × 10−9

No. 3 Communications in Theoretical Physics 275

Fig. 2 Absolute error functions (a) e1 (b) e2 for t = 10.

Fig. 3 Absolute error functions (a) e1 (b) e2 for t = 20.

Fig. 4 Population density u(x, y, t) as a function of timet for Example 2 for different α.

Moreover, to determine the accuracy of the homo-

topy perturbation solution against the exact solution, let

en(x, y, t) = u(x, y, t)−un−1(x, y, t) denote the error func-

tion. Figures 2 and 3, respectively, display the results of

the absolute error functions for t = 10, t = 20 for n = 1, 2.

It can be seen from the figures that the accuracy of the

method is apparently increased in calculating more terms

in the approximate solutions.

Fig. 5 Population density u(x, y, t) as a function of po-sition (x, y) for Example 2 for different α.

To demonstrate the influence of varying the order of

the fractional derivative on the behaviour of solution, we

take four different values of α. We plot the population

density solution at (x, y) = (0.1, 0.1) as a function of time

t for α = 1, α = 0.9, α = 0.6, and α = 0.1 in Fig. 4. It

276 Communications in Theoretical Physics Vol. 60

is evident that the solution continuously depends on the

time fractional derivative. Figure 5 shows the diffusion of

biological population as a function (x, y) for various α. In

this case the order of the fractional derivative also influ-

ences on the behaviour of the solution.

5 Conclusions

To conclude, the Homotopy-perturbation method has

been implemented in this study for computing the ap-

proximate analytical solutions and numerical solutions of

fractional order partial differential equations in biological

sciences. The fractional derivatives are described using the

definition of Caputo. In special case of α = 1, the gen-

eral solution reduces to the diffusion solution. Two typical

examples have been discussed in order to illustrate the ef-

ficiency and the accuracy of the Homotopy perturbation

method. It is evident that the present method provides an

approximate solution in a rapidly convergent to the exact

solutions. On the other hand, a good approximation to

the exact solution is achieved in the last example by con-

sidering the first three terms of HPM series. Furthermore,

it is also shown that the approximate solution is affected

by the order of the time-fractional derivative.

Moreover, comparison was made with the existing ap-

proximate solution obtained by Adomian’s decomposition

method, and it shows that two methods are in good agree-

ment with each other. However, our method is more ac-

curate than the ADM, and even for small iteration the

approximation has high accuracy. In addition, the present

method requires less computational work for providing

quantitatively reliable results if compared with the ADM,

as well as traditional perturbation methods. It may be

concluded that the homotopy-perturbation method is an

effective and very powerful method for obtaining analyt-

ical solutions of a wide class of problems involving frac-

tional derivatives.

Acknowledgments

The author is very grateful to anonymous referees for

their valuable suggestions and comments which improved

the paper.

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