Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 303472, 15 pages
doi:10.1155/2011/303472
Research Article
On Efficient Method for System of
Fractional Differential Equations
Najeeb Alam Khan,1 Muhammad Jamil,2, 3 Asmat Ara,1
and Nasir-Uddin Khan1
1
Department of Mathematics, University of Karachi, Karachi 75270, Pakistan
Abdul Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
3
Department of Mathematics, NEDUET, Karachi 75270, Pakistan
2
Correspondence should be addressed to Najeeb Alam Khan, njbalam@yahoo.com
Received 14 December 2010; Accepted 5 February 2011
Academic Editor: J. J. Trujillo
Copyright q 2011 Najeeb Alam Khan et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The present study introduces a new version of homotopy perturbation method for the solution
of system of fractional-order differential equations. In this approach, the solution is considered as
a Taylor series expansion that converges rapidly to the nonlinear problem. The systems include
fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix
Riccati-type differential equation. The new approximate analytical procedure depends only on two
components. Comparing the methodology with some known techniques shows that the present
method is relatively easy, less computational, and highly accurate.
1. Introduction
Fractional differential equations have received considerable interest in recent years and
have been extensively investigated and applied for many real problems which are modeled
in different areas. One possible explanation of such unpopularity could be that there are
multiple nonequivalent definitions of fractional derivatives 1. Another difficulty is that
fractional derivatives have no evident geometrical interpretation because of their nonlocal
character. However, during the last 12 years fractional calculus starts to attract much more
attention of scientists. It was found that various, especially interdisciplinary, applications 2–
6 can be elegantly modeled with the help of the fractional derivatives.
The homotopy perturbation method is a powerful devise for solving nonlinear
problems. This method was introduced by He 7–9 in the year 1998. In this method,
the solution is considered as the summation of an infinite series that converges rapidly.
2
Advances in Difference Equations
This technique is used for solving nonlinear chemical engineering equations 10, timefractional Swift-Hohenberg S-H equation 11, viscous fluid flow equation 12, FourthOrder Integro-Differential equations 13, nonlinear dispersive Km, n, 1 equations 14,
Long Porous Slider equation 15, and Navier-Stokes equations 16. It can be said that He’s
homotopy perturbation method is a universal one, which is able to solve various kinds of
nonlinear equations. The new homotopy perturbation method NHPM was applied to linear
and nonlinear ODEs 17.
In this paper, we construct the solution of system of fractional-order differential
equations by extending the idea of 17, 18. This method leads to computable and efficient
solutions to linear and nonlinear operator equations. The corresponding solutions of the
integer-order equations are found to follow as special cases of those of fractional-order
equations.
We consider the system of fractional-order equations of the form
Dαi yi t Fi t, y1 , y2 , y3 , . . . , yn fi t,
yi t0 ci ,
0 < αi ≤ 1, i 1, 2, . . . , n.
1.1
2. Basic Definitions
We give some basic definitions, notations, and properties of the fractional calculus theory
used in this work.
Definition 2.1. The Riemann-Liouville fractional integral operator J μ of order μ on the usual
Lebesgue space L1 a, b is given by
1
J fx Γ μ
x
μ
x − tμ−1 ftdt,
μ > 0,
0
2.1
J fx fx.
0
It has the following properties:
i J μ exists for any x ∈ a, b,
ii J μ J β J μβ ,
iii J μ J β J β J μ ,
iv J α J β fx J β J α fx,
v J μ x − aγ Γγ 1/Γα γ 1x − aμγ ,
where f ∈ L1 a, b, μ, β ≥ 0 and γ > −1.
Definition 2.2. The Caputo definition of fractal derivative operator is given by
1
Dμ fx J m−μ Dn fx Γ m−μ
t
0
x − τm−μ−1 f m τdτ,
2.2
Advances in Difference Equations
3
where m − 1 < μ ≤ m, m ∈ N, x > 0. It has the following two basic properties for m − 1 <
μ ≤ m and f ∈ L1 a, b:
Dμ J μ fx fx,
J μ Dμ fx fx −
m−1
f k 0 k0
x − ak
,
k!
x > 0.
2.3
3. Analysis of New Homotopy Perturbation Method
Let us consider the system of nonlinear differential equations
Ai yi fi t,
t ∈ Ω,
3.1
where Ai are the operators, fi are known functions and yi are sought functions. Assume that
operators Ai can be written as
Ai yi Li yi Ni yi ,
3.2
where Li are the linear operators and Ni are the nonlinear operators. Hence, 3.1 can be
rewritten as follows:
Li yi Ni yi fi t,
t ∈ Ω.
3.3
We define the operators Hi as
Hi Yi ; p ≡ 1 − p Li Yi − Li yi,0 p Ai Y − fi ,
where p ∈ 0, 1 is an embedding or homotopy parameter, Yi t; p : Ω × 0, 1 →
are the initial approximation of solution of the problem in 3.3 can be written as
Hi Yi ; p ≡ Li Yi − Li yi,0 pLi yi,0 p Ni Yi − fi .
3.4
Ê and yi,0
3.5
Clearly, the operator equations Hi v, 0 0 and Hi v, 1 0 are equivalent to the equations
Li Yi − Lyi,0 0 and Ai Y − fi t 0, respectively. Thus, a monotonous change of
parameter p from zero to one corresponds to a continuous change of the trivial problem
Li Yi − Li yi,0 0 to the original problem. Operator Hi Yi , p is called a homotopy map.
Next, we assume that the solution of equation Hi Yi , p can be written as a power series in
embedding parameter p, as follows:
Yi Yi,0 pYi,1 ,
i 1, 2, 3, . . . , n.
3.6
Now, let us write 3.5 in the following form:
Li Yi yi,0 t p fi − Ni Yi − yi,0 t .
3.7
4
Advances in Difference Equations
By applying the inverse operator, L−1
i to both sides of 3.7, we have
−1
−1
−1
.
y
f
−
L
N
y
Yi L−1
p
L
−
L
t
Y
t
i,0
i
i
i,0
i
i
i
i
3.8
Suppose that the initial approximation of 3.3 has the form
yi,0 t ∞
i 1, 2, 3, . . . , n,
ai,n Pn t,
3.9
n0
where ai,n , n 0, 1, 2, . . . are unknown coefficients and Pn t, n 0, 1, 2, . . . are specific
functions on the problem. By substituting 3.6 and 3.9 into 3.8, we get
Yi,0 pYi,1 L−1
i
∞
ai,n Pn t
p
L−1
i fi
−
L−1
i Ni Yi,0
pYi,1 −
L−1
i
∞
n0
ai,n Pn t
n0
.
3.10
Equating the coefficients of like powers of p, we get the following set of equations:
−1
coefficient of p : Y0 L
0
∞
ai,n Pn t ,
n0
−1
coefficient of p1 : Y1 L−1
fi L−1
i
i Yi,1 − Li Ni Yi,0 .
3.11
Now, we solve these equations in such a way that Yi,1 t 0. Therefore, the approximate
solution may be obtained as
−1
yi t Yi,0 t L
∞
ai,n Pn t .
3.12
n0
4. Applications
Application 1
Consider the following linear fractional-order 2-by-2 stiff system:
Dtα ut k−1 − εut k1 − εvt,
Dtα vt k1 − εut k−1 − εvt
4.1
with the initial conditions
u0 1,
v0 3,
4.2
Advances in Difference Equations
5
where k and ε are constants. To obtain the solution of 4.1 by NHPM, we construct the
following homotopy:
1 − p Dtα Ut − u0 t p Dtα Ut − k−1 − εUt − k1 − εV t 0,
1 − p Dtα V t − v0 t p Dtα V t − k1 − εUt − k−1 − εV t 0.
4.3
Applying the inverse operator, Jtα of Dtα both sides of the above equation, we obtain
Ut U0 Jtα u0 t − pJtα u0 t − k−1 − εUt − k1 − εV t,
V t V 0 Jtα v0 t − pJtα v0 t − k1 − εUt − k−1 − εV t.
4.4
The solution of 4.1 to has the following form:
Ut U0 t pU1 t,
V t V0 t pV1 t.
4.5
Substituting 4.5 in 4.4 and equating the coefficients of like powers of p, we get the
following set of equations:
U0 t U0 Jtα u0 t,
V0 t V 0 Jtα v0 t,
U1 t Jtα −u0 t k−1 − εU0 t k1 − εV0 t,
4.6
V1 t Jtα −v0 t k1 − εU0 t k−1 − εV0 t.
20
k
Assuming u0 t 20
n0 an Pn , v0 t n0 bn Pn , Pk t , U0 u0, and V 0 v0 and
solving the above equation for U1 t and V1 t lead to the result
U1 t a1 tα1
2a2 tα2
6a3 tα3
24a4 t2α
2k − 4εk − a0 tα
−
−
−
−
··· ,
Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
4.7
b1 tα1
2b2 tα2
6b3 tα3
24b4 t2α
−2k − 4εk − b0 tα
−
−
−
−
··· .
V1 t Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
Vanishing U1 t and V1 t lets the coefficients ai , bi , i 0, 1, 2, . . . by taking α 1 the following
values:
a0 2k1 − 2ε,
−8k4 1 − 2ε4
,
a3 3
8k7 1 − 2ε7
,
a6 45
−8k10 1 − 2ε10
,
a9 2835
a1 −4k2 1 − 2ε2 ,
4k5 1 − 2ε5
a4 ,
3
−16k8 1 − 2ε8
a7 ,
315
8k11 1 − 2ε11
a10 ,
14175
a2 4k3 1 − 2ε3 ,
−8k6 1 − 2ε6
a5 ,
15
4k9 1 − 2ε9
a8 ,
315
−16k12 1 − 2ε12
a11 ,
155925
6
8k13 1 − 2ε13
,
a12 467775
−16k16 1 − 2ε16
a15 ,
155925
8k19 1 − 2ε19
,
a18 97692469875
b0 −2k1 2ε,
8k4 1 2ε4
,
b3 3
−8k7 1 2ε7
b6 ,
45
8k10 1 2ε10
,
b9 2835
−8k13 1 2ε13
b12 ,
467775
16k16 1 2ε16
,
b15 155925
−8k19 1 2ε19
,
b18 97692469875
−16k14 1 − 2ε14
a13 ,
6081075
4k17 1 − 2ε17
a16 ,
638512875
−16k20 1 − 2ε20
a19 ,
1856156927625
b1 4k2 1 2ε2 ,
Advances in Difference Equations
16k15 1 − 2ε15
a14 ,
42567525
−8k18 1 − 2ε18
a17 ,
10854718875
8k21 1 − 2ε21
a20 ,
9280784638125
b2 −4k3 1 2ε3 ,
−4k5 1 2ε5
8k6 1 2ε6
b4 ,
b5 ,
3
15
16k8 1 2ε8
−4k9 1 2ε9
b7 ,
b8 ,
315
315
−8k11 1 2ε11
16k12 1 2ε12
b10 , b11 ,
14175
155925
16k14 1 2ε14
−16k15 1 2ε15
b13 ,
b14 ,
6081075
42567525
−4k17 1 2ε17
8k18 1 2ε18
b16 ,
b17 ,
638512875
10854718875
16k20 1 2ε20
−8k21 1 2ε21
b19 ,
b20 .
1856156927625
9280784638125
4.8
Therefore, we obtain the solutions of 4.1 as
2k1 − 2εtα 4k2 1 − 2ε2 tα1 8k3 1 − 2ε3 tα2 16k4 1 − 2ε4 tα3
ut 1 −
−
··· ,
Γα 1
Γα 2
Γα 3
Γα 4
2k1 2εtα 4k2 1 2ε2 tα1 8k3 1 2ε3 tα2 16k4 1 2ε4 tα3
vt 3 −
−
−
··· .
Γα 1
Γα 2
Γα 3
Γα 4
4.9
Our aim is to study the mathematical behavior of the solution ut and vt for different
values of α. This goal can be achieved by forming Pade’ approximants, which have the
advantage of manipulating the polynomial approximation into a rational function to gain
more information about ut and vt. It is well known that Pade’ approximants will converge
on the entire real axis, if ut and vt are free of singularities on the real axis. It is of interest to
note that Pade’ approximants give results with no greater error bounds than approximation
by polynomials. To consider the behavior of solution for different values of α, we will take
advantage of the explicit formula 4.9 available for 0 < α ≤ 1 and consider the following two
special cases.
Advances in Difference Equations
7
Case 1. Setting α 1, k 50, ε 0.01 in 4.9, we obtain the approximate solution in a series
form as
u10,11 t 1 148.73t 1203.65t2 51963.1t3 · · ·
,
1 50.7628t 1227.89t2 18726.5t3 · · ·
4.10
3 69.439t 3823.59t2 40311.9t3 · · ·
v10,11 t .
1 57.1463t 1550.5t2 26447.2t3 · · ·
Case 2. In this case, we will examine the linear fractional stiff equation 4.1. Setting α 1/2, k 50, ε 0.01 in 4.9 gives
ut 1 196t1/2 39992t3/2 7999984t5/2 228571424t7/2
−
−
··· ,
√
√
√
√
π
3 π
15 π
15 π
4.11
204t1/2 13336t3/2 2666672t5/2 533333344t7/2
vt 3 − √
−
− ··· .
√
√
√
π
π
5 π
35 π
For simplicity, let t1/2 z, then
196z 39992z3 7999984z5 228571424z7
uz 1 √ −
−
··· ,
√
√
√
π
3 π
15 π
15 π
4.12
204z 13336z3 2666672z5 533333344z7
−
− ··· .
vz 3 − √ √
√
√
π
π
5 π
35 π
Calculating the 10/11 Pade’ approximants and recalling that z t1/2 , we get
u10,11 9.58 × 10−8 0.0000126t1/2 0.0002357t − 0.0001051t3/2 − · · ·
9.581 × 10−8 2.0904 × 10−6 t1/2 4.56399 × 10−6 t 0.000096t2 · · ·
v10,11 2.66 × 10123 − 1.216 × 10125 t1/2 8.69947 × 10125 t · · ·
8.88 × 10122 − 6.45605 × 10123 t1/2 4.22967 × 10124 t · · ·
,
4.13
.
Application 2
Consider the following nonlinear fractional-order 2-by-2 stiff system:
Dtα ut −1002ut 1000v2 t,
Dtα vt ut − vt − v2 t
4.14
with the initial conditions
u0 1,
v0 1.
4.15
8
Advances in Difference Equations
To obtain the solution of 4.14 by NHPM, we construct the following homotopy:
1 − p Dtα Ut − u0 t p Dtα Ut 1002Ut − 1000V 2t 0,
1 − p Dtα V t − v0 t p Dtα V t − Ut V t V 2 t 0.
4.16
Applying the inverse operator, Jtα of Dtα both sides of the above equation, we obtain
Ut U0 Jtα u0 t − pJtα u0 t 1002Ut − 1000V 2 t ,
V t V 0 Jtα v0 t − pJtα v0 t − Ut V t V 2 t .
4.17
The solution of 4.14 to have the following form:
Ut U0 t pU1 t,
V t V0 t pV1 t.
4.18
Substituting 4.18 in 4.17 and equating the coefficients of like powers of p, we get the
following set of equations:
U0 t U0 Jtα u0 t,
V0 t V 0 Jtα v0 t,
U1 t Jtα −u0 t − 1002U0t 1000V02t ,
4.19
V1 t Jtα −v0 t U0 t − V0 t − V02 t .
20
k
Assuming u0 t 20
n0 an Pn , v0 t n0 bn Pn , Pk t , U0 u0, and V 0 v0 and
solving the above equation for U1 t and V1 t lead to the result
U1 t a1 tα1
2a2 tα2
6a3 tα3
24a4 tα4
−a0 2tα
−
−
−
−
− ··· ,
Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
b1 tα1
2b2 tα2
6b3 tα3
24b4 tα4
−b0 1tα
−
−
−
−
−··· .
V1 t Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
4.20
Vanishing U1 t and V1 t lets the coefficients ai , bi , i 0, 1, 2, . . . to take the following values:
a0 −2, a1 4, a2 −4, a3 8
−4
−8
, a4 , . . . , a20 ,
3
3
9280784638125
−1
1
−1
−1
, b3 , b4 , . . . , b20 .
b0 −1, b1 1, b2 2
6
24
2432902008176640000
4.21
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Therefore, we obtain the solution of 4.14 as
ut 1 −
2tα
4tα1
8tα2
16tα3
32tα4
−
−
··· ,
Γα 1 Γα 2 Γα 3 Γα 4 Γα 5
tα1
tα2
tα3
tα4
tα
−
−
··· .
vt 1 −
Γα 1 Γα 2 Γα 3 Γα 4 Γα 5
4.22
The exact solution of 4.14 for α 1 is ut e−2t , vt e−t .
Application 3
Consider the following nonlinear Genesio system with fractional derivative:
Dtα ut vt,
Dtα vt wt,
4.23
Dtα wt −cut − bvt − awt u2 t
with the initial conditions
u0 0.2,
v0 −0.3,
w0 0.1,
4.24
where a, b, and c are constants. To obtain the solution of 4.23 by NHPM, we construct the
following homotopy:
1 − p Dtα Ut − u0 t p Dtα Ut − V t 0,
1 − p Dtα V t − v0 t p Dtα V t − Wt 0,
1 − p Dtα Wt − w0 t p Dtα Wt cUt bV t aWt − U2 t 0.
4.25
Applying the inverse operator, Jtα of Dtα both sides of the above equation, we obtain
Ut U0 Jtα u0 t − pJtα u0 t − V t,
V t V 0 Jtα v0 t − pJtα v0 t − Wt,
Wt Wt Jtα w0 t − pJtα w0 t cUt bV t aWt − U2 t .
4.26
The solution of 4.23 to have the following form:
Ut U0 t pU1 t,
V t V0 t pV1 t,
Wt W0 t pW1 t.
4.27
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Advances in Difference Equations
Substituting 4.27 in 4.26 and equating the coefficients of like powers of p, we get the
following set of equations:
U0 t U0 Jtα u0 t,
V0 t V 0 Jtα v0 t,
W0 t W0 Jtα w0 t,
U1 t Jtα −u0 t V0 t,
4.28
V1 t Jtα −v0 t W0 t,
W1 t Jtα −w0 t − cU0 t − bV0 t − aV0 t W02 t .
20
20
k
Assuming u0 t 20
n0 an Pn , v0 t n0 bn Pn , w0 t n0 cn Pn , Pk t , U0 u0,
V 0 v0, W0 w0, a 1.2, b 2.92, and c 6, and solving the above equation for
U1 t, V1 t and W1 t lead to the result
U1 t −a0 3/10tα
a1 tα1
2a2 tα2
6a3 tα3
24a4 tα4
−
−
−
−
− ··· ,
Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
V1 t W1 t b1 tα1
2b2 tα2
6b3 tα3
24b4 tα4
b0 − 1/10tα
−
−
−
−
− ··· ,
Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
4.29
c1 tα1
2c2 tα2
6c3 tα3
24c4 tα4
−c0 101/250tα
−
−
−
−
− ··· .
Γα 1
Γα 2 Γα 3 Γα 4 Γα 5
Vanishing U1 t, V1 t, and W1 t lets the coefficients ai , bi , ci , i 0, 1, 2, . . . to take the
following values:
a0 b0 −3
,
10
1
,
10
a1 −101
,
250
a2 −101
,
500
a3 2341
,
7500
a4 −377
,...,
6250
a20 −64170831419403533391899
,
1160098079765625000000000000000000
b1 −101
,
250
c1 b2 2341
,
2500
b3 −754
,
3125
b4 −5153
,...,
75000
33855543777297749556491
,
89238313828125000000000000000000
b20
c0 1
,
10
23411
,
1250
c20 c2 −2262
,
3125
c3 −5153
,
75000
c4 5838803330656480870609733
.
19334967996093750000000000000000000
−508141
,...,
3750000
4.30
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11
Therefore, we obtain the solutions of 4.23 as
ut 3tα
tα1
101tα2
2341tα3
4524tα4
1
−
−
−
··· ,
5 10Γα 1 10Γα 2 250Γα 3 1250Γα 4 3125Γα 5
vt −3
tα
101tα1
2341tα2
4524tα3
5153tα4
−
−
−
··· ,
10 10Γα 1 250Γα 2 1250Γα 3 3125Γα 4 3125Γα 5
wt 1
101tα
2341tα1
4524tα2
5153tα3
508141tα4
−
−
−
−
· · · .
10 250Γα 1 1250Γα 2 3125Γα 3 3125Γα 4 156250Γα 5
4.31
Application 4
Finally, we consider the following nonlinear matrix Riccati differential equation with
fractional derivative:
Dtα Y t −Y 2 t Q,
Y 0 0,
4.32
1 0 1 1 where Q 1/2 11 −1
−1 1 . To find the solution of this equation by NHPM, we will
1
0 100
treat the matrix equation as a system of fractional-order differential equations
Dtα ut −u2 t − vtzt 101
,
2
Dtα vt −utvt − vtwt −
Dtα zt
99
,
2
99
−utvt − vtwt − ,
2
Dtα wt −u2 t − vtzt 4.33
101
,
2
with the initial conditions
u0 0,
v0 0,
z0 0,
w0 0.
4.34
Therefore, we obtain the solution of 4.33 as
ut wt vt zt 1
3tα
tα1
101tα2
2341tα3
4524tα4
−
−
−
· · · ,
5 10Γα 1 10Γα 2 250Γα 3 1250Γα 4 3125Γα 5
−3
tα
101tα1
2341tα2
4524tα3
5153tα4
−
−
−
· · · .
10 10Γα 1 250Γα 2 1250Γα 3 3125Γα 4 3125Γα 5
4.35
Advances in Difference Equations
3
3
2.5
2.5
u, v
u, v
12
2
1.5
2
1.5
1
1
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
t
0.3
0.4
0.5
0.3
0.4
0.5
t
a
b
3
6
4
u, v
u, v
2.5
2
2
0
1.5
−2
1
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
t
c
t
d
Figure 1: Solutions of linear stiff system for k 50, ε 0.01, α 1, a Exact, b Numerical, c NHPMPade 10/11, d NHPM-Pade 10/11, k 50, ε 0.01, α 0.5 color figure can be viewed in the online
issue.
5. Concluding Remarks
The NHPM for solving system of fractional-order differential equations are based on two
component procedure and polynomial initial condition. The NHPM applied on fractionalorder Stiff equation, fractional Genesio equation, and the matrix Riccati-type differential
equation. The Applications in problems 1–4 are plotted in Figures 1, 2, 3, and 4, which
show the accuracy of NHPM. The computations associated with the applications discussed
above, were performed by MATHEMATICA. The NHPM is very simple in application
and is less computational more accurate in comparison with other mentioned methods.
By using this method, the solution can be obtained in bigger interval. Unlike the ADM
19, the NHPM is free from the need to use Adomian polynomials. In this method,
we do not need the Lagrange multiplier, correction functional, stationary conditions,
and calculating integrals, which eliminate the complications that exist in the VIM 20.
In contrast to the HPM and HAM, in this method, it is not required to solve the
functional equations in each iteration. The efficiency of HAM is very much depended on
choosing auxiliary parameter . All the applications are taken from 20 with fractional
derivatives.
13
1
1
0.8
0.8
u, v
u, v
Advances in Difference Equations
0.6
0.6
0.4
0.4
0.2
0.2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
t
0.8
1
t
a
b
1
1
0.9
0.8
0.7
u, v
u, v
0.8
0.6
0.5
0.6
α 0.9
α 1/3
0.4
0.4
α 0.9
0.2
0.3
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
α 1/3
0.8
0.6
1
t
t
c
d
Figure 2: Solutions of nonlinear stiff system for α 1, a Exact, b Numerical, c NHPM, and c NHPM,
α 0.9, 1/3 color figure can be viewed in the online issue.
0.4
w
0.3
0.3
0.2
0.2
0.1
u, v
u, v
0.4
u
0
w
0.1
u
0
−0.1
−0.1
−0.2
v
−0.2
v
−0.3
−0.3
0
0.5
1
1.5
2
0
2.5
0.5
1
1.5
a
0.4
α
1
2
w
0.2
0.1
0.1
0
u
−0.1
−0.2
α 0.75
w
0.3
0.2
0
2.5
b
0.4
u, v
u, v
0.3
2
t
t
u
−0.1
−0.2
v
v
−0.3
−0.3
0
0.5
1
1.5
2
0
0.5
1
c
1.5
2
t
t
d
Figure 3: Solutions of nonlinear Genesio system for a Numerical, b NHPM α 1, c NHPM, α 0.5,
d NHPM, α 0.75 color figure can be viewed in the online issue.
14
Advances in Difference Equations
4
4
2
2
u, v
u, v
6
0
0
−2
−2
−4
−6
−4
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
t
2
2.5
3
t
a
b
10
u, v
5
0
−5
−10
0
0.5
1
1.5
2
2.5
3
t
c
Figure 4: Solutions of matrix Riccati equations u w, v z for α 1 a Numerical, b NHPM-Pade
9/11, c NHPM-Pade 9/11, α 0.5 color figure can be viewed in the online issue.
Acknowledgment
M. Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences,
GC University, Lahore, Pakistan, the Department of Mathematics & Basic Sciences, NED
University of Engineering & Technology, Karachi-75270, Pakistan, and also the Higher
Education Commission of Pakistan for generously supporting and facilitating this research
work.
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