Solving fuzzy fractional differential equations using fuzzy Sumudu transform

Research Article

Solving fuzzy fractional differential equations using fuzzy Sumudu transform

Norazrizal Aswad Abdul Rahman, Muhammad Zaini Ahmad^∗

Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia.

Abstract

In this paper, we apply fuzzy Sumudu transform (FST) for solving fuzzy fractional differential equations (FFDEs) involving Caputo fuzzy fractional derivative. It followed by suggesting a new result on the property of FST for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of FFDEs and finally we demonstrate a numerical example.

Keywords: Caputo fuzzy fractional derivative, fuzzy Sumudu transform, fuzzy fractional differential equation.

MSC: 26A33, 44A05, 34A08, 34A07

1. Introduction

Fractional calculus is the generalization of ordinary calculus. This includes the functions’ derivative of arbitrary order. The topic has been explored and studied by various researchers in many fields such as engineering, mathematics and so forth [29, 13, 31, 3, 19]. One of the major contributions in this field was the work studied by [33], which discussed the topic intensively. Later, it was studied by [27], where the authors proposed some applications. When dealing with fractional differential equations, the terms such as Riemann-Liouville, Gr¨unwald-Letnikov and Caputo fractional derivative are considered by many authors [21, 18, 26]. Of the three definitions of derivative stated, Riemann-Liouville and Caputo fractional derivatives appeared to be more popular.

As times moving on, the fractional differential equations seems to have some drawbacks. One of them is the initial value assigned to the model. In general, the determination of initial values is very difficult.

It always involves uncertainty quantities. This is true when dealing with real physical phenomena. To

∗Corresponding author

Email addresses: [email protected](Norazrizal Aswad Abdul Rahman),[email protected](Muhammad Zaini Ahmad)

Received 201X-X-X

handle uncertainty quantities, many researchers proposed several new concepts. The one that stands out among the concepts is fuzzy set theory [43]. This theory is able to deal with differential equations possessing uncertainties at initial values. The first contribution on handling fractional differential equations with uncertainties was studied by [2]. This has influenced many researchers to further explore the subject [9, 8, 6, 5, 37, 25].

Integral transforms have long been used in solving ordinary differential equations, as well as fractional differential equations. The integral transforms were preceded by Fourier transform. Later, several new integral transforms have been proposed, namely, Laplace, Mellin, and Hankel transforms [35, 28, 38]. One of the recent integral transforms introduced in the literature is the Sumudu transform [39, 40]. The virtue of this transform is that it holds a scale preserving property which resulting in the original function to be similar with the transformed function. It can also be seen in the literature, there exist several discussions on solving few types of fractional differential equations, as we stated previously, using Sumudu transform [23, 14, 15]. Recently, fuzzy Laplace transform [7] has been used to solve FFDEs involving Riemann-Liouville fractional derivative [34]. However, this type of fractional derivative has a drawback. It requires a quantity of fractional H-derivative of an unknown solution at the fuzzy initial point, which is not practical in real life situation. In this paper, we propose a new solution of FFDEs involving Caputo fuzzy fractional derivative using FST. The FST is first proposed by [4] followed by [1].

The arrangement of this paper is as the following. In Section 2, we revise some fundamental theories on fuzzy numbers and fuzzy functions. Plus, some definitions and theorems on Caputo fuzzy fractional derivative will also be provided. It is followed by the definition of the FST in Section 3. In this section, we also propose a new property of FST for Caputo fuzzy fractional derivative. Next, in Section 4, we provide a procedure on solving FFDEs possessing Caputo fuzzy fractional derivative using the FST in detail. A numerical example is demonstrated in Section 5 and finally in Section 6, the conclusions is drawn.

2. Basic concepts and theories

Here, we revisit several definitions and theorems for a better understanding of this paper.

2.1. Fuzzy numbers and fuzzy functions

Throughout this paper, Rdenotes the set of real numbers. Fuzzy number is defined as follows.

Definition 2.1. [42] A fuzzy number is a mapping eu:R→[0,1] with the following criteria:

1. eu is normal, i.e. there existsx₀∈Rsuch thatu(xe ₀) = 1, 2. eu is convex, i.e. for all andλ∈[0,1], x, y∈R,

u(λxe + (1−λ)y)≥min{eu(x),u(y)},e holds,

3. eu is upper semi continuous, i.e. for anyx₀ ∈R, u(xe ₀)≥ lim

x→x^±₀ u(x)e

4. suppeu={x∈R|u(x)e >0} is the support of eu, and its closure cl(supp u) is compact, .e

Definition 2.2. [22] Let eu be a fuzzy number defined in F(R). The α-level set of u, for anye α ∈ [0,1], denoted byeuα, is a crisp set that contains all elements inR, such that the membership value ofueis greater or equal toα, that is:

ueα={x∈R|u(x)e ≥α}.

Whenever we represent the fuzzy number with α-level set, we can see that it is closed and bounded. It is denoted by [u_α, uα], where they represent the lower and upper bound α-level set of the fuzzy number, respectively.

As the fuzzy number is resolved by the interval eu_α, researchers [17, 30] defined another representation, parametrically, of fuzzy numbers as in the following definition.

Definition 2.3. A fuzzy number ue in parametric form is a pair [u_α, uα] of functions u_α and uα, for any α∈[0,1] which satisfy the following requirements:

1. u_α is a bounded non-decreasing left continuous function in (0,1], 2. uα is a bounded non-increasing left continuous function in (0,1], 3. u_α ≤uα.

Some researchers classified the fuzzy numbers into several types of fuzzy membership function. To the deepest of our study, triangular fuzzy membership function or also often referred to as triangular fuzzy number is the most widely used membership function.

Definition 2.4. [24] A triangular fuzzy number eu can be defined by a triplet (a₁, a₂, a₃), the membership function is defined as follows:

eu(x) =



















0, ifx < a1, x−a1

a2−a1

, ifa1 ≤x < a2, a3−x

a₃−a₂, ifa₂ ≤x≤a₃, 0, ifx > a3,

The α-level of the fuzzy number ueisueα= [a1+ (a2−a3)α, a3−(a3−a2)α], for any α∈[0,1].

The definition of the operations on fuzzy numbers can be referred in the paper by [36].

Theorem 2.5. [41] Let fe: R → F(R) and it is represented by [f_α(x), f_α(x)]. For any fixed α ∈ [0,1], assume f_α(x) and f_α(x) are Riemann-integrable on [a, b] for every b ≥ a, and assume there are two pos- itive M_α and M_α such that Rb

a|f

α(x)|dx ≤ M_α and Rb

a|f_α(x)|dx ≤ M_α for every b ≥ a. Then, f(x)e is improper fuzzy Riemann-integrable on [a,∞) and the improper fuzzy Riemann-integrable is a fuzzy number.

Furthermore, we have

Z ∞ a

fe(x)dx= Z ∞

a

f_α(x)dx, Z ∞

a

f_α(x)dx

. H-difference of fuzzy numbers is defined as follows.

Definition 2.6. Ifu,e ev∈ F(R) and if there exists a fuzzy subsetξ ∈ F(R) such thatξ+ue=ev, thenxi is unique. In this case,ξ is called the Hukuhara difference, or simply H-difference, ofu and v and is denoted by ev−^H u.e

In the next definition, the strongly generalized differentiability concept is provided.

Definition 2.7. [11, 12] Let fe: (a, b) → F(R) and x0 ∈ (a, b). We say that feis strongly generalized differentiable atx₀, if there exists an element fe⁰(x₀)∈ F(R), such that

1. for all h >0 sufficiently small, there exist fe(x₀+h)−^H fe(x₀),fe(x₀)−^Hfe(x₀−h) and the limits (in the metricD)

h→0lim

fe(x0+h)−^H f(xe 0)

h = lim

h→0

fe(x0)−^H fe(x0−h)

h =fe⁰(x0), or:

2. for all h >0 sufficiently small, there exist fe(x₀)−^H fe(x₀+h),fe(x₀−h)−^H fe(x₀) and the limits (in the metricD)

h→0lim

fe(x₀)−^H fe(x₀+h)

−h = lim

h→0

fe(x₀−h)−^Hfe(x₀)

−h =fe⁰(x₀).

In this paper, we denote the space of all continuous fuzzy functions on [a, b]⊆ R and the space of all Lebesgue integrable fuzzy functions on the bounded interval [a, b] by C^F[a, b] and L^F[a, b], respectively.

Definition 2.8. [34] Letfe∈C^F[a, b]∩L^F[a, b] be a fuzzy function. The fuzzy Riemann-Liouville integral of the fuzzy functionfeis defined as follows

I^βfe

(x) = 1 Γ(β)

Z x 0

fe(t)

(x−t)^1−βdt, x, β ∈R+.

Theorem 2.9. [10] Letfe∈C^F[a, b]∩L^F[a, b]be a fuzzy function. The fuzzy Riemann-Liouville integral of the fuzzy functionfeis as as follows:

h I^βfe

(x)

i

α = h

I^βf_α(x), I^βf_α(x) i

, 0≤α≤1, where

I^βf_α

(x) = 1 Γ(β)

Z x 0

f_α(t)

(x−t)^1−βdt, x, β∈R+,

I^βf_α

(x) = 1 Γ(β)

Z x 0

f_α(t)

(x−t)^1−βdt, x, β∈R+. 2.2. Caputo Fuzzy Fractional Derivative

In this subsection, we provide some definitions and theorems on Caputo fuzzy fractional derivative. [32]

extended the Caputo fractional derivative of crisp case into fuzzy setting. Here, we provide some of the concept proposed.

Lemma 2.10. [32] Letf(x)be a crisp continuous function and(dβe)-times differentiable in the independent variable x over the interval of differentiation (integration) [0, x]. Then the relation

CD^βf(x) =^RL D^β



f(x)−

dβe

X

k=0

x^k k!f₀^(k)



, β∈(n−1, n], n∈N, is hold,

where f₀^(k)= d^kf(x) dx^k

x=0

and^CD^β denotes Caputo derivative operator. Whiledβeand bβc are the value β rounded up and down to the closest integer number, respectively. ^RLD^β is the common Riemann-Liouville fractional derivative operator which is defined as follows

RLD^βf(x) = 1 Γ(dβe −β)

d^dβe dx^dβe

Z x 0

f(t)

(x−t)^1−dβe+βdt

Definition 2.11. [32] Let fe(x) ∈ C^F[0, b]∩L^F[0, b], G(x) =e _{Γ(dβe−β)}¹ Rx 0

fe(t)−Pdβe k=0

t^k k!^fe

(k) 0

(x−t)^1−dβe+β dt, and H(xe ₀) = lim_h→0+

G(xe 0+h) G(xe 0)

h = lim_h→0+

G(xe 0) G(xe 0−h)

h andL(xe ₀) = lim_h→0+

G(xe 0) G(xe 0+h)

−h = lim_h→0+

G(xe 0−h) G(xe 0)

−h . fe(x) is Caputo fuzzy fractional differentiable function of order 0 < β ≤ 1, if there exists an element

CD^βfe(x₀)∈C^F such that for all 0≤α≤1 and forh >0 sufficiently near zero, either:

1. ^CD^βfe(x0) = lim_h→0⁺ G(xe 0+h) G(xe 0)

h = lim_h→0⁺ G(xe 0) G(xe 0−h)

h , or

2. ^CD^βfe(x0) = lim_h→0⁺ G(xe 0) G(xe 0+h)

−h = lim_h→0⁺ G(xe 0−h) G(xe 0)

−h ,

for 0< β≤1.

If the fuzzy functionfe(x) is differentiable as in Definition 2.11 (1), it is called Caputo fuzzy differentiable in the first form. Iffe(x) is differentiable as in Definition 2.11 (2), it is called Caputo fuzzy differentiable in the second form.

Theorem 2.12. [32] Let fe(x) ∈ C^F[0, b]∩L^F[0, b] be a fuzzy function and [fe(x)]α = [f_α(x), f_α(x)], for α∈[0,1] and x0 ∈(0, b). Then

1. If fe(x) is Caputo fuzzy fractional differentiable in the first form, then for every 0< β≤1, [^CD^βfe(x₀)]_α= [^CD^βf_α(x₀),^CD^βf_α(x₀)].

2. If fe(x) is Caputo fuzzy fractional differentiable in the second form, then for every0< β≤1, [^CD^βfe(x₀)]_α= [^CD^βf_α(x₀),^CD^βf_α(x₀)],

where

CD^βf_α(x₀) =

"

1 Γ(dβe −β)

Z _x

0

D^dβef

α(t) (x−t)^1−dβe+β

#

x=x0

.

CD^βf_α(x₀) =

"

1 Γ(dβe −β)

Z x 0

D^dβef_α(t) (x−t)^1−dβe+β

#

x=x0

,

D^kf(t) = d^kf(t) dt^k .

Next, we give the definition for the classical Sumudu transform when dealing with Caputo’s fractional derivative of crisp type.

Definition 2.13. [16, 20] The classical Sumudu transform of the Caputo’s fractional derivative of the functionf is given by

s[^CD^βf(x)](u) =u^−βG(u)−

n−1

X

k=0

f^k(0)

u^β−k, β ∈(n−1, n].

Since in this paper, we only consider 0< β <1, Definition 2.13 can be simplified as follows.

s[^CD^βf(x)](u) =G(u)−f⁰(0)

u^β , β∈(0,1].

Note that when β = 1, the definition is similar to the definition of Sumudu transform for first order derivative.

3. Fuzzy Sumudu transform for Caputo fuzzy fractional derivative

In this part, we recall the definition of FST and later we propose a new result on the property of FST for Caputo fuzzy fractional derivative.

Definition 3.1. [4, 1] Letfe:R→ F(R) be a continuous fuzzy function. Supposefe(ux)e^−x is improper fuzzy Riemann-integrable on [0,∞), thenR∞

0 fe(ux)e^−xdxis called fuzzy Sumudu transform and is denoted by

G(u) =S[fe(x)](u) = Z ∞

0

fe(ux)e^−xdx, u∈[−τ₁, τ₂],

where the variableuis used to factor the variablexin the argument of the fuzzy function andτ1, τ2 >0.

The FST can also be written into the following parametric form.

S[fe(x)](u) = [s[f_α(x)](u),s[f_α(x)](u)].

In the following theorem, we introduce a new property of FST for Caputo fuzzy fractional derivative. This is done by directly extending the definition for classical Sumudu transform of Caputo fractional derivative into fuzzy setting.

Theorem 3.2. Letfe(x)∈C^F[0, b]∩L^F[0, b]be a continuous fuzzy function, and^CD^βfeis the Caputo fuzzy fractional derivative offeon [0,∞). Then, for 0< β≤1, we have

S[^CD^βf(x)](u) =e G(u)−^H fe(x0)

u^β ,

where feis Caputo fuzzy fractional differentiable in the first form, or S[^CD^βf(x)](u) =e −fe(x₀)−^H(−G(u))

u^β ,

where feis Caputo fuzzy fractional differentiable in the second form.

Proof. First, we assume feis Caputo fuzzy fractional differentiable in the first form (Theorem 2.12 (1)).

Therefore:

G(u)−^H fe(0)

u^β =

s[f(x)](u)−f(0)

u^β ,s[f(x)](u)−f(0) u^β

. From the classical Sumudu transform for Caputo fractional derivative, we know that

s[^CD^βf(x)](u) = s[f(x)](u)−f(0)

u^β ,

and

s[^CD^βf(x)](u) = s[f(x)](u)−f(0)

u^β .

Then,

G(u)−^H fe(0)

u^β =h

s[^CD^βf(x)](u), s[^CD^βf(x)](u)i . Since feis Caputo fuzzy fractional differentiable in the first form

G(u)−^H f(xe ₀)

u^β =S[^CD^βfe(x)](u).

Now, we assume thatfeis Caputo fuzzy fractional differentiable in the second form (Theorem 2.12 (2)).

Therefore,

−fe(x0)−^H (−G(u))

u^β =

−f(x0)−(−s[f(x)](u))

u^β ,−f(x0)−(−s[f(x)](u)) u^β

. This is analogous to

−fe(x0)−^H(−G(u))

u^β =

s[f(x)](u)−f(0)

u^β ,s[f(x)](u)−f(0) u^β

. From the classical Sumudu transform for Caputo fractional derivative, finally we have

−fe(x0)−^H(−G(u))

u^β =

h

s[^CD^βf(x)](u), s[^CD^βf(x)](u) i

.

Since feis Caputo fuzzy fractional differentiable in the second form, then we finally have

−fe(x₀)−^H(−G(u))

u^β =S[^CD^βfe(x)](u).

The proof is complete.

4. Procedures for Solving FFDEs using FST

Consider the following FFDE

(CD^βy(x) =e f[x,ey(x)], ye_α(x₀) = [y

α(0), y_α(0)]. (4.1)

where f ∈C^F(a, b)∩L^F(a, b) and x₀ ∈(a, b).

By using FST on both side of Eq. (4.1), we have Sh

CD^βy(x)e i

(u) =S[f(x,y(x))] (u).e

Case 1: If we considerf to be Caputo fuzzy fractional differentiable in the first form, then from Theorem 2.12 (1), we get [^CD^βy(xe 0)]α= [^CD^βy_α(x0),^CD^βy_α(x0)]. Now, we obtain the following system

(CD^βy_α(x) =f[x,y(x)] =e f_α[x,y(x)],e y_α(x₀) =y_α(0),

CD^βy_α(x) =f[x,y(x)] =e f_α[x,y(x)],e y_α(x0) =y_α(0), whereβ ∈(0,1].

From Theorem 3.2, we have

S[^CD^βey(x)](u) = G(u)−^H ey(t0)

u^β .

Therefore,















s[f_α(x,y(x))](u) =e s[y_α(x)](u)−y_α(0)

u^β ,

s[f_α(x,y(x))](u) =e s[y_α(x)](u)−y_α(0)

u^β ,

(4.2)

where,

fα(x,ey(x)) = min{fe(x, u)|u∈[y

α(x), y_α(x)]}, and,

f_α(x,y(x)) = max{e f(x, u)|ue ∈[y_α(x), y_α(x)]}.

To solve Eq. (4.2), first we assume that

s[y_α(x)](u) =L¹_α(u), s[y_α(x)](u) =U_α¹(u).

L¹_α(u) and U_α¹(u) are the solutions of Eq. (4.2) under this case. We obtain y_α(x) and y_α(x) using the inverse FST as the following.

y_α(x) =s⁻¹[L¹_α(u)], y_α(x) =s⁻¹[U_α¹(u)].

Case 2: If we considerf to be Caputo fuzzy fractional differentiable in the second form, then from Theorem 2.12 (2), we get [^CD^βfe(x0)]α= [^CD^βy_α(x0),^CD^βy_α(x0)]. Now, we obtain the following system

(CD^βy_α(x) =f[x,y(x)] =e f_α[x,y(x)],e y_α(x₀) =y_α(0),

CD^βy_α(x) =f[x,y(x)] =e f_α[x,y(x)],e y_α(x0) =y_α(0), where β∈(0,1].

From Theorem 3.2,

S[^CD^βy(x)](u) =e −y(te ₀)−^H (−G(u))

u^β .

Therefore,















s[f_α(x,y(x))](u) =e s[y_α(x)](u)−y_α(0)

u^β ,

s[f_α(x,y(x))](u) =e s[y_α(x)](u)−y_α(0)

u^β ,

(4.3)

where,

f_α(x,ey(x)) = min{fe(x, u)|u∈[y_α(x), y_α(x)]},

and,

f_α(x,y(x)) = max{e f(x, u)|ue ∈[y_α(x), y_α(x)]}.

To solve Eq. (4.3), first we assume that

s[y_α(x)](u) =L²_α(u), s[y_α(x)](u) =U_α²(u).

L²_α(u) and U_α²(u) are the solutions of Eq. (4.3) or this case. We have y_α(x) and y_α(x) by the inverse of FST as the following.

yα(x) =s⁻¹[L²_α(u)], y_α(x) =s⁻¹[U_α²(u)].

5. A numerical example

In this part, the method proposed will be demonstrated on a FFDE. This is to show that the method is practicable.

Example 5.1. The following FFDE is considered.

(CD^βy(x) =e y(x),e

y(xe ₀) = [y_α(0), y_α(0)]. (5.1)

Case 1: By taking fuzzy Sumudu transform on both side of (5.1), we have Sh

CD^βey(x) i

(u) =S[y(x)](u).e

From Theorem 2.12 (1) for Caputo fuzzy fractional differentiability in the first form, we have [^CD^βy(xe ₀)]_α= [^CD^βy

α(x₀),^CD^βy_α(x₀)], and from Theorem 3.2,

S[^CD^βey(x)](u) = G(u)−^H ey(x₀)

u^β ,

we have















s[yα(x)](u) = s[y_α(x)](u)−y_α(0)

u^β ,

s[y_α(x)](u) = s[y_α(x)](u)−y_α(0)

u^β .

Then, we obtain

((1−u^β)s[y_α(x)](u) =y_α(0), (1−u^β)s[y_α(x)](u) =y_α(0).

By applying inverse Sumudu transform, we obtain











y_α(x) =y_α(0)s⁻¹ 1

1−u^β

, y_α(x) =y_α(0)s⁻¹

1 1−u^β

. By relation,

s[x^γ−1Eβ,γ(∓x^β)](u) = u^γ−1 1±u^β, finally, we have:

(y_α(x) =y_α(0)E_β,1[x^β], y_α(x) =y_α(0)E_β,1[x^β].

E_β,1[x^β] is the Mittag-Leffler function defined by E_β,1[x^β] =

∞

X

k=0

(x^β)^k Γ(βk+ 1). Case 2: Using fuzzy Sumudu transform on Eq. (5.1), we have

Sh

CD^βey(x)i

(u) =S[y(x)](u).e

From Theorem 2.12 (2) for Caputo fuzzy fractional differentiability in the second form:

[^CD^βy(xe 0)]α= [^CD^βy_α(x0),^CD^βy_α(x0)], and from Theorem 3.2

S[^CD^βf(x)](u) =e −fe(x0)−^H(−G(u))

u^β ,

we have















s[yα(x)](u) = −y_α(0)−[−s[y_α(x)](u)]

u^β .

s[y_α(x)](u) = −y

α(0)−[−s[y

α(x)](u)]

u^β ,

equivalent to















s[y_α(x)](u) = s[y_α(x)](u)−y_α(0)

u^β .

s[y_α(x)](u) = s[y_α(x)](u)−y_α(0)

u^β ,

Then, we obtain

((1 +u^2β)s[y

α(x)](u) =y

α(0)−u^βy_α(0), (1 +u^2β)s[y_α(x)](u) =y_α(0)−u^βy_α(0).

By applying inverse Sumudu transform, we have











y_α(x) =y_α(0)s⁻¹ 1

1 +u^2β

−y_α(0)s⁻¹ u^β

1 +u2^β

, y_α(x) =y_α(0)s⁻¹

1 1 +u^2β

−y_α(0)s⁻¹ u^β

1 +u2^β

. By relation,

s[x^γ−1E_β,γ(∓x^β)](u) = u^γ−1 1− ±u^β, finally, we have

(y_α(x) =y_α(0)E_2β,1[−x^2β]−y_α(0)x^βE_2β,β+1[−x^2β], y_α(x) =y_α(0)E_2β,1[−x^2β]−y

α(0)x^βE_2β,β+1[−x^2β], where,

E_2β,1[−x^2β] =

∞

X

k=0

(−x^2β)^k Γ(2βk+ 1), and

E_2β,β+1[−x^2β] =

∞

X

k=0

(−x^2β)^k Γ(2βk+β+ 1). are the Mittag-Leffler functions.

Assume thaty(xe 0) = [1 +α,3−α]. The numerical solutions of (5.1) for Cases 1 and 2 atx= 2 are listed in Tables 1 and 2, respectively. For graphical results, please see in Figs. 1 and 2. Numerical solutions for both cases are obtained by expending the Mittag-Leffler functions up to 11 terms. It can be concluded that the solutions of Eq. (5.1) is in agreement with the solutions of fuzzy differential equation asβ approaches to 1.

6. Conclusions

In this paper, we have proposed a new analytical method for dealing with fuzzy fractional differential equations involving Caputo fuzzy fractional derivatives. A new property of fuzzy Sumudu transform for Caputo fuzzy fractional derivative has been introduced. The new property has been used to construct a procedure for solving FFDEs. A numerical example has been solved to show that FST is functional.

Acknowledgement

This research received funding by Ministry of Science, Technology and Innovation, Malaysia under the Fundamental Research Grant Scheme, project code 9003-00417.

Table 1: Numerical solutions of Eq. (5.1) for Case 1 using different values ofβ.

β = 0.4 β = 0.6 β= 0.8 β= 1.0

α y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) 0 16.5737 49.7212 12.0222 36.0667 9.1583 27.4749 7.3890 22.1670 0.1 18.2311 48.0638 13.2244 34.8644 10.0741 26.5590 8.1279 21.4281 0.2 19.8885 46.4064 14.4267 33.6622 10.9899 25.6432 8.8668 20.6892 0.3 21.5459 44.7491 15.6289 32.4600 11.9058 24.7274 9.6057 19.9503 0.4 23.2032 43.0917 16.8311 31.2578 12.8216 23.8116 10.3446 19.2114 0.5 24.8606 41.4343 18.0333 30.0556 13.7374 22.8957 11.0835 18.4725 0.6 26.5180 39.7770 19.2356 28.8533 14.6533 21.9799 11.8224 17.7336 0.7 28.1753 38.1196 20.4378 27.6511 15.5691 21.0641 12.5613 16.9947 0.8 29.8327 36.4622 21.6400 26.4489 16.4849 20.1482 13.3002 16.2558 0.9 31.4901 34.8048 22.8422 25.2467 17.4008 19.2324 14.0391 15.5169 1.0 33.1475 33.1475 24.0444 24.0444 18.3166 18.3166 14.7780 14.7780

Figure 1: Numerical solutions of (5.1) for (a)β= 0.4 (b)β= 0.6, (c)β= 0.8 and (d)β= 1 (Case 1).

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Table 2: Numerical solutions of Eq. (5.1) for Case 2 using different values ofβ.

β = 0.4 β = 0.6 β= 0.8 β = 1.0

α y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) y^β(x) 0 -1.2888 0.1705 -1.5531 -0.4324 -2.1398 -1.2789 -3.1440 -2.1577 0.1 -1.2158 0.0976 -1.4971 -0.4884 -2.0967 -1.3219 -3.0947 -2.2071 0.2 -1.1428 0.0246 -1.4410 -0.5444 -2.0537 -1.3650 -3.0454 -2.2564 0.3 -1.0699 -0.0484 -1.3850 -0.6005 -2.0106 -1.4080 -2.9961 -2.3057 0.4 -0.9969 -0.1213 -1.3289 -0.6565 -1.9676 -1.4511 -2.9468 -2.3550 0.5 -0.9239 -0.1943 -1.2729 -0.7125 -1.9245 -1.4941 -2.8975 -2.4043 0.6 -0.8510 -0.2673 -1.2169 -0.7686 -1.8815 -1.5372 -2.8481 -2.4536 0.7 -0.7780 -0.3402 -1.1608 -0.8246 -1.8385 -1.5802 -2.7988 -2.5029 0.8 -0.7051 -0.4132 -1.1048 -0.8807 -1.7954 -1.6232 -2.7495 -2.5523 0.9 -0.6321 -0.4862 -1.0488 -1.9367 -1.7524 -1.6663 -2.7002 -2.6016 1.0 -0.5591 -0.5591 -0.9927 -0.9927 -1.7093 -1.7093 -2.6509 -2.6509

Figure 2: Numerical solutions of (5.1) for (a)β= 0.4 (b)β= 0.6, (c)β= 0.8 and (d)β= 1 (Case 2).

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