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Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 91–113© Research India Publicationshttp://www.ripublication.com/gjpam.htm

Solving Systems of Fractional NonlinearEquations of Emden–Fowler Type by

using Sumudu transform method

Y. A. Amer1, A. M. S. Mahdy1,2 and E. S. M. Youssef1

1Department of Mathematics, Faculty of Science,Zagazig University, Zagazig, Egypt.

2Department of Mathematics and Statistics, Faculty of Science,Taif University, Saudi Arabia.

Abstract

The aim of this paper is finding the approximate analytical solutions for Systems ofthe fractional nonlinear equations of Emden-Fowler type by using Sumudu trans-form method. The fractional derivatives are described in the Caputo sense. Theapplications related to Sumudu transform method have been developed for differen-tial equations to the extent of access to approximate analytical solutions of systemsof nonlinear equations of Emden-Fowler type. This method is easy to work outas it gives us very accurate solutions for Solving linear and nonlinear differentialequations.

AMS subject classification:Keywords: Emden–Fowler type equations, Caputo derivative, Sumudu transform.

1. Introduction

The equations which used for experimenting with science, technology, differential equa-tions, and integrals are very complex and difficult to obtain. Accurate solutions for themand their own exact solutions are very puzzling. In this work Sumudu decompositionmethod been introduced for solving the linear and non-linear differential ([1] − [3])equations and integral equations ([4] , [5]), and it is shown that the new technique per-forms a good deal more serious.

92 Y. A. Amer, A. M. S. Mahdy and E. S. M. Youssef

Many problems in the fields of mathematical physics and astrophysics are expressedby the equation ([6] − [9]):

Dαy + γ

xy′ + f (x) g (x) = h (x) , x > 0 (1)

where Dα is the Caputo fractional derivative operator, γ > 0 is a constant.For f (x) = 1, g (x) = xm, and h(x) = 0, eq. (1), is the standard Lane–Emden

equation that has been used to model several phenomena in mathematical physics. Forthe steady-state case and γ = 2 and h (x) = 0, Eq. (1), becomes the Emden–Fowlertype equation:

Dαy + 2

xy′ + f (x) g (x) = 0, x > 0, (2)

subject toy (0) = 1, y′ (0) = 0 (3)

The Emden–Fowler equation arises in the study of fluid mechanics, relativistic mechanicsand in the study of chemically reacting systems ([10] , [11]). The singularity behaviorthat occurs at x = 0 is the main difficulty of Eq. (1), and Eq. (2).

Very lately, many powerful methods have been introduced, such as some notes onusing the variational iteration method for solving systems of equations of Emden–Fowlertype [6], the Emden–Fowler Eq. (3), and Eq. (4) was handled by using the Adomiandecomposition method and the modified decomposition method ([7], [8], [12]–[14]),the same equation was handled by using the homotopy perturbation method and thevariational iteration method (VIM) ([4], [6]).

The aim of this paper is to employ Sumudu demecomposition method ([1], [2], [3],[15]–[17]) to obtain the exact solution of systems of nonlinear equations of Emden–Fowler type.

In this work, we study systems of fractional nonlinear equations of Emden–Fowlertype subject with initial conditions given by the form ([6]–[9]):

Dαu + α

xu′ + f (u (x) , v (x)) = h1 (x) , x > 0, α > 0, (4)

Dαv + β

xv′ + g (u (x) , v (x)) = h2 (x) , x > 0, β > 0, (5)

Subject tou (0) = v (0) = 1 , u′ (0) = v′ (0) = 0 (6)

The paper is structured in six sections. In section 2, we begin with an introductionto some necessary definitions of fractional calculus theory. In section 3 we describethe homotopy perturbation sumudu transform method. In section 4, we present threeexamples to show the efficiency of using HPSTM to solve FDEs and also to compareour result with those obtained by other existing methods. Finally, relevant conclusionsare drawn in section 5.

Solving Systems of Fractional Nonlinear Equations of Emden–Fowler Type 93

2. Basic Definitions of Fractional Calculus

In this section, we present the basic definitions and properties of the fractional calculustheory, which are used further in this paper.

Definition 2.1. The Caputo fractional derivative operator Dα of order α is defined inthe following form ([18] − [21]):

Dαf (x) =

1

� (m − α)

∫ x

0

f (m) (ξ)

(x − ξ)α−m+1dξ, 0 ≤ m − 1 < α < m,

f (m) (x) , α = m ∈ N.

Definition 2.2. A real function f (t), t > 0, is said to be in the space Cα, α ∈ R, if thereexists a real number p > α such that f (t) = tpf1(t) where f1(t) ∈ C [0, ∞) , and it issaid to be in the space Cm

α if f m ∈ Cα, m ∈ N.

Definition 2.3. The Sumudu transform is defined over the set of functions ([2], [3], [22],[23]):

A ={f (t)

∣∣∣∣∃M, τ 1, τ 2 > 0, |f (t)| < Me|t |τj , if t ∈ (−1)j × [0, ∞)

},

(7)by the following formula:

f̄ (u) = S [f (t)] =∫ ∞

0f (ut)e−t dt, where u ∈ (τ 1, τ 2) (8)

whereSome special properties of the sumudu transform are as follows [16]:

1. S [1] = 1;

2. S [t] = u;

3. S

[tn−1

�(n)

]= un−1; n > 0

4. S

[tn−1

(n − 1)!]

= un−1; n = 1, 2, · · ·

Definition 2.4. The Sumudu transform of Caputo fractional derivative is defined asfollows [16]:

S[Dα

t f (t)] = uαS [f (t)] −

m−1∑k=0

u−α+kf (k)(0), m − 1 < α ≤ m.


Theorem 2.5. ([2], [24])

S[f (n) (t)

]= u−n

[F (u) −

n−1∑k=0

ukf k (0)

]for n ≥ 1 (9)

At very special case for n = 1

S[F ′ (t)

] = 1

u[F (u) − F (0)] . (10)

This theorem is very important to calculate approximate solution of the problems formore details in ([25]–[27]).

3. The Homotopy Perturbation Sumudu Transform Method

In order to elucidate the solution procedure of this method, we consider a general frac-tional nonlinear differential equation of the form ([25]–[32]):

Dα∗ x(t) + Lx(t) + Nx(t) = q(t), (11)

with m − 1 < α ≤ m, and subject to the initial condition

xj (0) = cj , j = 0, 1, . . . , m − 1, (12)

where Dα∗ x(t) is the Caputo fractional derivative, q(t) is the source term, L is the linearoperator and N is the general nonlinear operator.

Applying the Sumudu transform (denoted throughout this paper by S) on both sidesof Eq. (11), we have

S[Dα∗ x(t)

] + S [Lx(t) + Nx(t)] = S [q(t)] .

Using the property of the Sumudu transform and the initial conditions in Eq. (12), wehave

S [x(t)] =m−1∑k=0

u−α+kxk(0) + uαS [q(t)] − uαS [Lx(t) + Nx(t)] , (13)

Operating with the Sumudu inverse on both sides of Eq. (13) we get

x(t) = G(t) − S−1 [uαS [Lx(t) + Nx(t)]

]. (14)

Where G(t) represents the term arising from the source term and the prescribed initialconditions. Now, pplying the classical perturbation technique. And assuming that thesolution of Eq. (14) is in the form

x(t) =∞∑

m=0pmxm(t), (15)


where p ∈ [0, 1] is the homotopy parameter. The nonlinear term of Eq. (14) can bedecomposed as

Nx(t) =∞∑

m=0pmAm(t), (16)

for some Adomian’s polynomials Am, which can be calculated with the formula ([33],[34])

Am = 1

m!dm

dpm

[N

( ∞∑i=0

pixi(t)

)]p=0

, n = 0, 1, 2, . . . (17)

Substituting Eq. (15) and (17) in Eq. (14), we get

∞∑m=0

pmxm(t) = G(t) − pS−1[uαS

[L

( ∞∑m=0

pmxm(t)

)+

∞∑m=0

pmAm

]]. (18)

Equating the terms with identical powers of p, we can obtain a series of equations as thefollows:

p0 : x0(t) = G(t),

p1 : x1(t) = −S−1 [uαS [Lx0(t) + A0]

],

p2 : x2(t) = −S−1 [uαS [Lx1(t) + A1]

],

p3 : x3(t) = −S−1 [uαS [Lx2(t) + A2]

],

... (19)

Finally, we approximate the analytical solution x(t) by truncated series as

x(t) = limM→∞

M∑m=0

pmxm(t). (20)

4. Applications

In this section, to illustrate the method and to show the ability of the method threeexamples are presented.

Example 4.1. Cosider the systems of fractional of Emden–Fowler type as

Dαu (x) + 2

xu′ + v2 (x) − u2 (x) + 6v (x) = 6 + 6x2 (21)

Dαv (x) + 2

xv′ + u2 (x) − v2 (x) − 6v (x) = 6 − 6x2 (22)

subject tou (0) = 1 , v (0) = −1, 1 < α ≤ 2 (23)


By taking the Sumudu transform on both sides of Eq.(21) , (22), thus we get

S[Dαu (x)

] + S

[2

xu′ + v2 − u2 + 6v

]= 6 + 12u2,

S[Dαv (x)

] + S

[2

xv′ + u2 − v2 − 6v

]= 6 − 12u2.

S[Dαu (x)

] = 6 + 12u2 − S

[2

xu′ + v2 − u2 + 6v

]

S[Dαv (x)

] = 6 − 12u2 − S

[2

xv′ + u2 − v2 − 6v

].

(24)

Using the property of the Sumudu transform and the initial condition in Eq. (23), wehave

S [u] = u (0) + 6uα + 12uα+2 − uαS

[2

xu′ + v2 − u2 + 6v

],

S [v] = v (0) + 6uα − 12uα+2 − uαS

[2

xv′ + u2 − v2 − 6v

]

S [u] = 1 + 6uα + 12uα+2 − uαS

[2

xu′ + v2 − u2 + 6v

],

S [v] = −1 + 6uα − 12uα+2 − uαS

[2

xv′ + u2 − v2 − 6v

] (25)


u = 1 + 6

� (α + 1)xα + 12

� (α + 3)xα+2 − S−1

[uαS

[2

xu′ + v2 − u2 + 6v

]],(26)

v = −1 + 6

� (α + 1)xα − 12

� (α + 3)xα+2 − S−1

[uαS

[2

xv′ + u2 − v2 − 6v

]]

By assuming that

u (x) =∞∑

n=0un (x) , v (x) =

∞∑n=0

vn (x) (27)


By substituting Eq. (27) in Eq. (26) we have∞∑

n=0un (x) = 1 + 6

� (α + 1)xα + 12

� (α + 3)xα+2 (28)

−S−1[uαS

[2

x

d

dx

∞∑n=0

un (x) +∞∑

n=0An (x) −

∞∑n=0

Bn (x) + 6∞∑

n=0vn (x)

]],

∞∑n=0

vn (x) = −1 + 6

� (α + 1)xα − 12

� (α + 3)xα+2

−S−1[uαS

[2

x

d

dx

∞∑n=0

vn (x) +∞∑

n=0Bnx (x) −

∞∑n=0

An (x) − 6∞∑

n=0vn (x)

]]

Where An, Bn are Adomian polynomials that represent nonlinear term. So Adomianpolynomials are given as follows:

An (x) = v2 (x) , Bn (x) = u2 (x)

The few components of the Adomian polynomials are given as follows:

A0 = v20

A1 = 2v0v1

A2 = 2v0v2 + v21

...

B0 = u20

B1 = 2u0u1

B2 = 2u0u2 + u21

...

Then, we have

u0 = 1 + 6

� (α + 1)xα + 12

� (α + 3)xα+2

v0 = −1 + 6

� (α + 1)xα − 12

� (α + 3)xα+2

A0 = 1 − 12

� (α + 1)xα + 36

� (α + 1) � (α + 1)x2α + 24

� (α + 3)xα+2

− 144

� (α + 1) � (α + 3)x2α+2 + 144

� (α + 3) � (α + 3)x2α+4

B0 = 1 + 12

� (α + 1)xα + 36

� (α + 1) � (α + 1)x2α + 24

� (α + 3)xα+2

+ 144

� (α + 1) � (α + 3)x2α+2 + 144

� (α + 3) � (α + 3)x2α+4


Uk+1 (x) = −S−1[uαS

[2

x

d

dxuk (x) + Ak (x) − Bk (x) + 6vk (x)

]](29)

Vk+1 (x) = −S−1[uαS

[2

x

d

dxvk (x) + Bk (x) − Ak (x) − 6vk (x)

]]

U1 (x) = −S−1[uαS

[2

x

d

dxu0 (x) + A0 (x) − B0 (x) + 6v0 (x)

]]

V1 (x) = −S−1[uαS

[2

x

d

dxv0 (x) + B0 (x) − A0 (x) − 6v0 (x)

]]

u1 = 6

� (α + 1)xα − 12

(α − 1) � (2α − 1)x2α−2 − 12

(α2 + 5α + 6

)(α + 1) (α + 2) � (2α + 1)

x2α

+ 72

� (2α + 3)x2α+2 + 288 (2α + 2)!

� (α + 1) � (α + 3) � (3α + 3)x3α+2

v1 = − 6

� (α + 1)xα − 12

(α − 1) � (2α − 1)x2α−2 + 12

(α2 + 5α + 6

)(α + 1) (α + 2) � (2α + 1)

x2α

− 72

� (2α + 3)x2α+2 − 288 (2α + 2)!

� (α + 1) � (α + 3) � (3α + 3)x3α+2

...

Since

u (x) = u1 + u2 + u3 + · · ·v (x) = v1 + v2 + v3 + · · ·

then

u (x) = 1 + 12

� (α + 1)xα + 12

� (α + 3)xα+2 − 12

(α − 1) � (2α − 1)x2α−2

− 12(α2 + 5α + 6

)(α + 1) (α + 2) � (2α + 1)

x2α

+ 72

� (2α + 3)x2α+2 + 288 (2α + 2)!

� (α + 1) � (α + 3) � (3α + 3)x3α+2 + · · ·

v (x) = −1 − 12

� (α + 1)xα − 12

� (α + 3)xα+2 − 12

(α − 1) � (2α − 1)x2α−2

+ 12(α2 + 5α + 6

)(α + 1) (α + 2) � (2α + 1)

x2α

− 72

� (2α + 3)x2α+2 − 288 (2α + 2)!

� (α + 1) � (α + 3) � (3α + 3)x3α+2 + · · ·


Figure 1: The behavior of Exact and Approximate Solution of U (x) at α = 2

Example 4.2. Consider the systems of fractional Emden–Fowler type as

Dαu (x) + 1

xu′ (x) + v (x) = x3 + 5, (30)

Dαv (x) + 2

xv′ (x) + w (x) = x4 + 12x + 1,

Dαw (x) + 3

xw′ (x) + u (x) = 25x2 + 1

subject to

u (0) = 1, v (0) = 1, w (0) = 1, 1 < α ≤ 2 (31)

By taking the Sumudu transform on both sides of Eq. (30), thus we get

S[Dαu (x)

] + S

[1

xu′ (x) + v (x)

]= 6u3 + 5,

S[Dαv (x)

] + S

[2

xv′ (x) + w (x)

]= 24u4 + 12u + 1,

S[Dαw (x)

] + S

[3

xw′ (x) + u (x)

]= 50u2 + 1.


Figure 2: The behavior of Exact and Approximate Solution of V (x) at α = 2

S[Dαu (x)

] = 6u3 + 5 − S

[1

xu′ (x) + v (x)

]

S[Dαv (x)

] = 24u4 + 12u + 1 − S

[2

xv′ (x) + w (x)

]

S[Dαw (x)

] = 50u2 + 1 − S

[3

xw′ (x) + u (x)

].

(32)

Using the property of the Sumudu transform and the initial condition in Eq. (32), wehave

S [u] = u (0) + 6uα+3 + 5uα − uαS

[1

xu′ (x) + v (x)

],

S [v] = v (0) + 24uα+4 + 12uα+1 + uα − uαS

[2

xv′ (x) + w (x)

],

S [w] = w (0) + 50uα+2 + uα − uαS

[3

xw′ (x) + u (x)

].


Figure 3: The behavior of U (x) , V (x) at α = 0.9

S [u] = 1 + 6uα+3 + 5uα − uαS

[1

xu′ (x) + v (x)

],

S [v] = 1 + 24uα+4 + 12uα+1 + uα − uαS

[2

xv′ (x) + w (x)

],

S [w] = 1 + 50uα+2 + uα − uαS

[3

xw′ (x) + u (x)

].

(33)


u = 1 + 5

� (α + 1)xα + 6

� (α + 4)xα+3 − S−1

[uαS

[1

xu′ (x) + v (x)

]],

v = 1 + 1

� (α + 1)xα + 12

� (α + 2)xα+1 + 24

� (α + 5)xα+4 − S−1

[uαS

[2

xv′ (x) + w (x)

]],

w = 1 + 1

� (α + 1)xα + 50

� (α + 3)xα+2 − S−1

[uαS

[3

xw′ (x) + u (x)

]].

(34)


By assuming that

u (x) =∞∑

n=0un (x) , v (x) =

∞∑n=0

vn (x) , w (x) =∞∑

n=0wn (x) (35)

By substituting Eq. (35) in Eq. (34) we have

u = 1 + 5

� (α + 1)xα + 6

� (α + 4)xα+3

−S−1[uαS

[1

x

d

dx

∞∑n=0

un (x) +∞∑

n=0vn (x)

]],

v = 1 + 1

� (α + 1)xα + 12

� (α + 2)xα+1

+ 24

� (α + 5)xα+4 − S−1

[uαS

[2

x

d

dx

∞∑n=0

vn (x) +∞∑

n=0wn (x)

]],

w = 1 + 1

� (α + 1)xα + 50

� (α + 3)xα+2

−S−1[uαS

[3

x

d

dx

∞∑n=0

wn (x) +∞∑

n=0un (x)

]]

u0 = 1 + 5

� (α + 1)xα + 6

� (α + 4)xα+3

v0 = 1 + 1

� (α + 1)xα + 12

� (α + 2)xα+1 + 24

� (α + 5)xα+4

w0 = 1 + 1

� (α + 1)xα + 50

� (α + 3)xα+2

Uk+1 (x) = −S−1[uαS

[1

x

d

dxuk (x) + vk (x)

]], (36)

Vk+1 (x) = −S−1[uαS

[2

x

d

dxvk (x) + wk (x)

]],

Wk+1 (x) = −S−1[uαS

[3

x

d

dxwk (x) + uk (x)

]]

u1 = −S−1[u2S

[1

x

d

dxu0 (x) + v0 (x)

]],

v1 = −S−1[u2S

[2

x

d

dxv0 (x) + w0 (x)

]],

w1 = −S−1[u2S

[3

x

d

dxw0 (x) + u0 (x)

]].


u1 = − 1

� (α + 1)xα − 5

(α − 1) � (2α − 1)x2α−2 − 1

� (2α + 1)x2α

− (12α + 30)

(α + 2) � (2α + 2)x2α+1 − 24

� (2α + 5)x2α+4

v1 = − 1

� (α + 1)xα − 2

(α − 1) � (2α − 1)x2α−2 − 24

α� (2α)x2α−1

− 1

� (2α + 1)x2α − (50α + 198)

(α + 3) � (2α + 3)x2α+2

w1 = − 1

� (α + 1)xα − 3

(α − 1) � (2α − 1)x2α−2 − 5

(α2 + 33α + 62

)(α + 1) (α + 2) � (2α + 1)

x2α

− 6

� (2α + 4)x2α+3

...

Since

u (x) = u1 + u2 + u3 + · · ·v (x) = v1 + v2 + v3 + · · ·w (x) = w0 + w1 + w2 + · · ·

then

u (x) = 1 + 4

� (α + 1)xα + 6

� (α + 4)xα+3 − 5

(α − 1) � (2α − 1)x2α−2

− 1

� (2α + 1)x2α − (12α + 30)

(α + 2) � (2α + 2)x2α+1 − 24

� (2α + 5)x2α+4 + · · ·

v (x) = 1 + 12

� (α + 2)xα+1 + 24

� (α + 5)xα+4 − 2

(α − 1) � (2α − 1)x2α−2

− 24

α� (2α)x2α−1 − 1

� (2α + 1)x2α − (50α + 198)

(α + 3) � (2α + 3)x2α+2 + · · ·

w (x) = 1 + 50

� (α + 3)xα+2 − 3

(α − 1) � (2α − 1)x2α−2

− 5(α2 + 33α + 62

)(α + 1) (α + 2) � (2α + 1)

x2α − 6

� (2α + 4)x2α+3 + · · ·

Example 4.3. Consider the systems of fractional of Emden–Fowler type as

Dαu (x) + 1

xu′ + u2 (x) v (x) −

(4x2 + 5

)u (x) = 0 (37)


Figure 4: The behavior of Exact and Approximate Solution of U (x) at α = 2

Dαv (x) + 2

xv′ + u (x) v2 (x) −

(4x2 − 5

)v (x) = 0 (38)

subject tou (0) = 1 , v (0) = 1, 1 < α ≤ 2 (39)

By taking the Sumudu transform on both sides of Eq. (37), (38), thus we get

S[Dαu (x)

] + S

[1

xu′ + u2 (x) v (x) −

(4x2 + 5

)u (x)

]= 0,

S[Dαv (x)

] + S

[2

xv′ + u (x) v2 (x) −

(4x2 − 5

)v (x)

]= 0.

S[Dαu (x)

] = S

[(4x2 + 5

)u (x) − 1

xu′ − u2 (x) v (x)

],

S[Dαv (x)

] = S

[(4x2 − 5

)v (x) − 2

xv′ − u (x) v2 (x)

].

(40)

Using the property of the Sumudu transform and the initial condition in Eq. (39), we


Figure 5: The behavior of Exact and Approximate Solution of V (x) at α = 2

have

S [u] = u (0) + uαS

[(4x2 + 5

)u (x) − 1

xu′ − u2 (x) v (x)

],

S [v] = v (0) + uαS

[(4x2 − 5

)v (x) − 2

xv′ − u (x) v2 (x)

]

and

S [u] = 1 + uαS

[(4x2 + 5

)u (x) − 1

xu′ − u2 (x) v (x)

],

S [v] = 1 + uαS

[(4x2 − 5

)v (x) − 2

xv′ − u (x) v2 (x)

] (41)


u (x) = 1 + S−1[uαS

[(4x2 + 5

)u (x) − 1

xu′ − u2 (x) v (x)

]],

v (x) = 1 + S−1[uαS

[(4x2 − 5

)v (x) − 2

xv′ − u (x) v2 (x)

]].

(42)


Figure 6: The behavior of Exact and Approximate Solution of W (x) at α = 2

By assuming that

u (x) =∞∑

n=0un (x) , v (x) =

∞∑n=0

vn (x) (43)

By substituting Eq. (43) in Eq. (42) we have

∞∑n=0

un (x) = 1 + S−1[uαS

[(4x2 + 5

) ∞∑n=0

un (x) − 1

x

d

dx

∞∑n=0

un (x) −∞∑

n=0An (x)

]],

∞∑n=0

vn (x) = 1 + S−1[uαS

[(4x2 − 5

) ∞∑n=0

vn (x) − 2

x

d

dx

∞∑n=0

vn (x) −∞∑

n=0Bn (x)

]].

(44)Where An, Bn are Adomian polynomials that represent nonlinear term. So Adomianpolynomials are given as follows:

An (x) = u2 (x) v (x) , Bn (x) = u (x) v2 (x) (45)


Figure 7: The behavior of U (x) , V (x) , W (x) at α = 0.9

The few components of the Adomian polynomials are given as follows:

A0 (x) = u20 (x) v0 (x)

A1 (x) = u20 (x) v1 (x) + 2u0 (x) v0 (x) u1 (x)

A2 (x) = u20 (x) v2 (x) + 2u0 (x) v1 (x) u1 (x) + 2u0 (x) v0 (x) u2 (x) + u2

1 (x) v0 (x)

...

and

B0 (x) = u0 (x) v20 (x)

B1 (x) = u1 (x) v20 (x) + 2u0 (x) v0 (x) v1 (x)

B2 (x) = u2 (x) v20 (x) + 2v0 (x) u1 (x) v1 (x) + 2v0 (x) u0 (x) v2 (x) + u0 (x) v2

1 (x)

...

Then we have

u0 (x) = 1, v0 (x) = 1, A0 (x) = 1, B0 (x) = 1


Figure 8: The behavior of Exactand Approximate Solution of U (x) at α = 2

Uk+1 (x) = S−1[uαS

[(4x2 + 5

)uk (x) − 1

x

d

dxuk (x) − Ak (x)

]](46)

Vk+1 (x) = S−1[uαS

[(4x2 − 5

)vk (x) − 2

x

d

dxvk (x) − Bk (x)

]]

Then

u1 = S−1[uαS

[(4x2 + 5

)u0 − 1

x

d

dxuk (x) − A0

]]

v1 = S−1[uαS

[(4x2 − 5

)v0 − 2

x

d

dxv0 − B0

]]

u1 = 8

� (α + 3)xα+2 + 4

� (α + 1)xα

v1 = 8

� (α + 3)xα+2 − 6

� (α + 1)xα


Figure 9: The behavior of Exactand Approximate Solution of V (x) at α = 2

And

A1 = 2

� (α + 1)xα + 24

� (α + 3)xα+2

B1 = − 8

� (α + 1)xα + 24

� (α + 3)xα+2

u2 = − 4

(α − 1) � (2α − 1)x2α−2 +

(18α2 + 46α + 20

)(α + 1) (α + 2) � (2α + 1)

x2α

+16(α2 + 3α + 3

)� (2α + 3)

x2α+2 + 32 (α + 3) (α + 4)

� (2α + 5)x2α+4

v2 = 12

(α − 1) � (2α − 1)x2α−2 +

(38α2 + 98α + 44

)(α + 1) (α + 2) � (2α + 1)

x2α

−8(3α2 + 9α + 14

)� (2α + 3)

x2α+2 + 32 (α + 3) (α + 4)

� (2α + 5)x2α+4

...


Figure 10: The behavior of U (x) , V (x) at α = 0.9

Since

u (x) = u0 + u1 + u2 + · · ·v (x) = v1 + v2 + v3 + · · ·

then

u (x) = 1 + 4

� (α + 1)xα + 8

� (α + 3)xα+2 − 4

(α − 1) � (2α − 1)x2α−2

+(18α2 + 46α + 20

)(α + 1) (α + 2) � (2α + 1)

x2α

+16(α2 + 3α + 3

)� (2α + 3)

x2α+2 + 32 (α + 3) (α + 4)

� (2α + 5)x2α+4 + · · ·


v (x) = 1 − 6

� (α + 1)xα + 8

� (α + 3)xα+2 + 12

(α − 1) � (2α − 1)x2α−2

+(38α2 + 98α + 44

)(α + 1) (α + 2) � (2α + 1)

x2α

−8(3α2 + 9α + 14

)� (2α + 3)

x2α+2 + 32 (α + 3) (α + 4)

� (2α + 5)x2α+4 + · · ·

5. Conclusions

The main aim of this paper is to know that the sumud transform method is one of themost important and simplest methods used in solving linear and nonlinear differentialequations. This method has been successfully applied to systems of fractional nonlinearequations of Emden–Fowler type in this method we do not need to do the difficultcomputation for finding the Adomian polynomials, Generally speaking, the proposedmethod is promising and applicable to a broad class of linear and nonlinear problems inthe theory of fractional calculus.

References

[1] J. Singh, D. Kumar and Sushila, Homotopy Perturbation Sumudu TransformMethod for Nonlinear Equations. Adv. Theor. Appl. Mech., Vol. 4. No. 4, pp.165–175, (2011).

[2] N. ecdet Bildik and S. inan Deniz, The Use of Sumudu Decomposition Method forSolving Predator-Prey Systems, An International Journal of Mathematical SciencesLetters 3, 285–289 (2016).

[3] S. Z. Rida, A. S. Abedl-Rady, A. A. M. Arafa and H. R. Abedl-Rahim, A Do-mian Decomposition Sumudu Transform Method for Solving Fractional NonlinearEquations, Math. Sci. Lett. 5, No. 1, 39–48, (2016).

[4] S. Irandoust-pakchin and S. Abdi-Mazraeh, “Exact solutions for some of the frac-tional integro-differential equations with the nonlocal boundary conditions by usingthe modifcation of He’s variational iteration method,” International Journal of Ad-vanced Mathematical Sciences, vol. 1, no. 3, pp. 139–144, (2013).

[5] R. C. Mittal and R. Nigam, “Solution of fractional integro-differential equations byAdomain decomposition method,” International Journal of Applied Mathematicsand Mechanics, vol. 4, no. 2, pp. 87–94, (2008).

[6] A. M. Wazwaz, The variational iteration method for solving systems of equationsof Emden–Fowler type, International Journal of Computer Mathematics, 88(16),pp. 3406–3415, (2011).

[7] A. M.Wazwaz, Adomian decomposition method for a reliable treatment of theEmden–Fowler equation, Appl. Math. Comput., 161, pp. 543–560, (2005).


[8] A. M. Wazwaz, Analytical solution for the time-dependent Emden–Fowler typeof equations by Adomian decomposition method, Appl. Math. Comput., 166, pp.638–651, (2005).

[9] M.usa R. Gad-Allah and Tarig. M. Elzaki, A New Homotopy Perturbation Methodfor Solving Systems of Nonlinear Equations of Emden-Fowler Type, Volume 11,Issue 2, ISSN: 2395-0218, February (2017).

[10] H. Eltayeb, Coupled singular and nonsingular thermoelastic system and DoubleLaplace Decomposition Method, BISKA, intentional open access journal, pp. 212–222, 4(3), (2016).

[11] H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differentialequations usingAdomian decomposition, J. Comput.Appl. Math., 196 (2) 644–651,(2006).

[12] He, J. H. Acoupling method of homotopy technique and perturbation techniquefor nonlinear problems, International Journal of Non-Linear Mechanics, 35(1), pp.37–43 (2000).

[13] A. M. Wazwaz, Anew method for solving differential equations of the Lane–Emdentype, Appl. Math. Comput., 118(2/3), pp. 287–310, (2001).

[14] A. M. Wazwaz, Anew method for solving singular initial value problems in thesecond order ordinary differential equations, Appl. Math. Comput., 128, pp. 47–57, (2002).

[15] S. Rathore, D. Kumar, J. Singh and S. Gupta, Homotopy Analysis Sumudu Trans-form Method for Nonlinear Equations, Int. J. Industrial Mathematics, Vol. 4, No.4, (2012).

[16] F. B. M. Belgacem and A. A. Karaballi, Sumudu transform fundamental propertiesin vestigations and applications, International J. Appl. Math. Stoch. Anal, Vol.,DOI:10.1155/JAMSA/2006/91083, 2005, pp 1–23, (2006).

[17] V. G. Gupta and B. Sharma, Application of Sumudu Transform in Reaction-Diffusion systems and Nonlinear Waves, Applied Mathematical Sciences, Vol. 4,pp. 435–446, (2010).

[18] El-Sayed, AMA, Salman, SM, On a discretization process of fractional order Ric-cati’s differential equation, J. Fract. Calc. Appl. 4, 251–259, (2013).

[19] Agarwal, RP, El-Sayed, AMA, Salman, SM, Fractional-order Chua’s system:discretization, bifurcation and chaos. Adv. Differ. Equ. 2013, Article ID 320.doi:10.1186/1687-1847-2013-320, (2013).

[20] Elsadany, AA, Matouk, AE, Dynamical behaviors of fractional-order Lotka-Volterapredator-prey model and its discretization, Appl. Math. Comput., 49, 269–283,(2015).

[21] Moustafa El-Shahed, Juan J. Nieto, A.M. Ahmed and I.M.E. Abdelstar, Fractional-order model for biocontrol of the lesser date moth in palm trees and its discretization,Advances in Difference Equations 2017:295, (2017).


[22] S. Momani and Z. Odibat, Numerical approach to differential equations of fractionalorder, J. Comput. Appl. Math. 2006, DOI: 10.1016/j.cam.07.015, (2006).

[23] S. Deniz and N. Bildik, Comparison of Adomian Decomposition Method and Tay-lor Matrix Method in Solving Different Kinds of Partial Differential Equations,International Journal of Modeling and Optimization 4.4, 292–298, (2014).

[24] H. Bulut, H.M. Baskonus and F. B. M. Belgacem, The Analytical solutions of somefractional ordinary differential equations by sumudu transform method, Abstractapplied analysis, volume 2013, Article id 203875, 6 pages, (2013).

[25] Rathore S, Kumar D, Singh J, Gupta S. Homotopy analysis sumudu transformmethod for nonlinear equations, Int. J. Industrial Mathematics, 4(4), (2012).

[26] Singh J, Kumar D, Sushila, Homotopy perturbation sumudu transform method fornonlinear equations, Adv. Theor. Appl. Mech., 4(4):165–175, (2011).

[27] Ganji D., The application of He’s homotopy perturbation method to nonlinearequations arising in heat transfer, Physics Letters A., 355:337–341, (2006).

[28] Bulut H, Baskonus H.M., Belgacem F.B.M., The analytical solutions of some frac-tional ordi-nary differential equations by sumudu transform method. Abstract Ap-plied Analysis, 2013: Article id 203875:6, (2013).

[29] Hashim I, Chowdhurly M, Mawa S., On multistage homotopy perturbation methodapplied to nonlinear biochemical reaction model, Chaos, Solitons and Fractals,36:823–827, (2008).

[30] He J., Homotopy perturbation technique, Comput. Methods, Appl. Mech Engng.,178(3-4):257–262, (1999).

[31] Liao S., Comparison between the homotopy analysis method and homotopy pertur-bation method. Applied Mathematics and Computation, 169:1186–1194, (2005).

[32] Y. A. Amer, A. M. S. Mahdy and E. S. M. Youssef, Solving Systems of FractionalDifferential Equations Using Sumudu Transform Method, Asian Research Journalof Mathematics, 7(2): 1–15, Article no.ARJOM.32665, (2017).

[33] A. Ghorbani, Beyond, Adomian polynomials: He polynomials, Chaos, Solitonsand fractals, Vol. 39, No. 3, pp. 1486–1492, (2009).

[34] Jafari H, Daftardar-Gejji V., Solving a system of nonlinear fractional differentialequations usingAdomian decomposition, J. Comput.Appl. Math., 196(2):644–651,(2006).

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