On fractional kinetic equations and their Sumudu transform multiparameter Struve functions based solutions

0.15 MB·

by  K. S. Nisar

ON FRACTIONAL KINETIC EQUATIONS AND THEIR SUMUDU
TRANSFORM MULTIPARAMETER STRUVE FUNCTIONS BASED
SOLUTIONS
K.S. NISAR*, F.B.M. BELGACEM, M. S. ABOUZAID
Abstract. This research paper treats fractional kinetic equations using the Sumudu
transformoperator. The exactsolutions obtainedare presentedin terms ofStruve func-
7 tions offour parameters. Byway ofobtainingsolutions some novelanduseful andnovel
1
kinetictheoremsarepresentedinlightoftheSumuduproperties. Resultsobtainedinthis
0
2 studymaybepragmaticallyusedinmanybranchestheoreticalandexperimentalscience
n applications, not the least of which are mathematical physics, and various engineering
a fields.
J

7
1

1. Introduction
]
A
C The Struve function H (z) and modified Struve function L (z) are are given by the
v v
. following infinite series, respectively,
h
t
a z υ+1 ∞ ( 1)k z 2k
m H (z) = − , (1.1)
υ 2 Γ k + 3 Γ k +υ + 3 2
[ (cid:16) (cid:17) Xk=0 2 2 (cid:16) (cid:17)

1 and (cid:0) (cid:1) (cid:0) (cid:1)
v z υ+1 ∞ 1 z 2k
9 L (z) = . (1.2)
8 υ 2 Γ k + 3 Γ k +υ + 3 2
5 (cid:16) (cid:17) Xk=0 2 2 (cid:16) (cid:17)
4 The Struve function of order υ (see [40])(cid:0)is the(cid:1)sol(cid:0)ution of th(cid:1)e non-homogeneous Bessel
0
differential equation:
.
1
0 4(x/2)υ+1
7 x2y′′(x)+xy′(x)+ x2 υ2 y(x) = , υ C. (1.3)
− √πΓ(υ +1/2) ∈
1
: (cid:0) (cid:1)
v where Γ(z) is the gamma function and, L (z) is related to H (z), by the relation
υ υ
i
X
L (z) = ie−nπi/2H (iz). (1.4)
r n − n
a
While, thehomogeneoussolutionsof (1.3)aretheBesselfunctions, theparticularsolutions
may be given to correspond to Struve functions. For complex parameter, υ, modified
2010 Mathematics Subject Classification. 26A33, 44A10, 44A20, 33E12.
Keywords andphrases. Fractionalkineticequations,Sumudutransforms,generalizedStruvefunction,
Fractional calculus .
*corresponding author.
1
2 K.S. NISAR*,F.B.M. BELGACEM, M. S. ABOUZAID
Struve functions, L , turn out to be solutions of a modified version of equation (1.3),
υ
where the LHS zeroth coefficient is replaced by: (x2 + v2). Applications of Struve
−
functions can be found in various branches of applied science (see [1,21,29,35,36]).
Generalized versions of Struve function are done by extending its domain or expanding
the type and number of parameters. In particular, a four- parameter generalized struve
function, studied by Singh [37], is defined by:
∞ ( 1)k
Hλ,α(x) := − x 2k+p+1,p C (1.5)
p,µ Γ(αk +µ)Γ λk +p+ 3 2 ∈
k=0 2
X (cid:0) (cid:1)
where λ > 0,α > 0 and µ an arbitrary para(cid:0)meter. (cid:1)
For subsequent need towards our objective below we also recall the generalized Mittag-
Leffler function [24] defined by,
∞ xn
E (x) = . (1.6)
α,β
Γ(αn+β)
n=0
X
The main motive of this paper is to study the solution of generalized form of the fractional
kinetic equation involving generalized Struve function of four parameters with the help of
Sumudu transform.
The Sumudu transform introduced by Watugala (see [41,42]). For more details about
Sumudu transform, see ( [2–10,13,22]). The Sumudu transform over the set functions
A = f (t) M,τ ,τ > 0, f (t) < Me|t|/τj , if t ( 1)j [0, )
1 2
∃ | | ∈ − × ∞
n o
(cid:12)
is defined by
(cid:12)
∞
G(u) = S[f (t);u] = f (ut)e−tdt, u ( τ ,τ ). (1.7)
1 2
∈ −
Z0
The significance of fractional differential equations in the field of applied science in-
creased more attention not only in mathematics but also in physics, dynamical systems,
control systems and systems engineering, to create the mathematical model of numer-
ous physical phenomena ( [14–16,18,19,25–28,30–34,43]. To carry our investigation,we
would like to recall the following results due to Haubold and Mathai [20]. The fractional
differential equation between rate of change of reaction was established by Haubold and
Mathai [20]. The destruction rate and the production rate are given as follows
dN
= d(N )+p(N ) (1.8)
t t
dt −
where N = N (t) the rate of reaction ,d = d(N) the rate of destruction, p = p(N) the
rate of production and N denote the function defined by N (t∗) = N (t t∗),t∗ > 0
t t
−
ON FRACTIONAL KINETIC EQUATIONS AND THEIR SUMUDU TRANSFORM... 3
The special case of (1.8), for spatial fluctuations or in homogeneities in N (t) the
quantity are neglected, that is the equation
dN
= c N (t) (1.9)
i i
dt −
with the initial condition that N (t = 0) = N is the number of density of species i at
i 0
time t = 0 and c > 0. If we reject the index i and integrate the standard kinetic equation
i
(1.9), we have
N (t) N = c D−1N (t) (1.10)
− 0 − 0 t
where D−1 is the special case of the Riemann-Liouville integral operator D−υ defined
0 t 0 t
as
t
1
D−υf (t) = (t s)υ−1f (s)ds,t > 0, (υ) > 0 (1.11)
0 t Γ(υ) − ℜ
Z
0
The fractional generalization of the standard kinetic equation (1.10) given by Haubold
and Mathai [20] as:
N (t) N = cυD−1N (t) (1.12)
− 0 − 0 t
and obtained the solution of (1.12) as follows
∞ ( 1)k
N (t) = N − (ct)υk (1.13)
0
Γ(υk+1)
k=0
X
Further, Saxena and Kalla [34] considered the following fractional kinetic equation:
N (t) N f (t) = cυ D−1N (t) ( (υ) > 0) (1.14)
− 0 − 0 t ℜ
where N(t) denotes the number density o(cid:0)f a given(cid:1) species at time t, N = N (0) is the
0
number density ofthatspecies attimet = 0, cisaconstant andf L(0, ). Byapplying
∈ ∞
the Laplace transform to (1.14),
F (p)
L[N (t)](p) = N
0
1+cυp−υ
∞
c
= N ( cυ)np−nυ F (p) n N , < 1 , (1.15)
0 0
− ∈ p
!
n=0 (cid:18) (cid:12) (cid:12) (cid:19)
X (cid:12) (cid:12)
where the Laplace transform ( [38]) is defined by (cid:12) (cid:12)
(cid:12) (cid:12)
∞
F (p) = L[f(t)] = e−ptf (t)dt (p) > 0 (1.16)
ℜ
Z
0
The Sumudu transform of (1.11) is (see, ( [3]), p.106,eqn (2.1)) :
S D−vf (t) = uvG(u) (1.17)
0 t
(cid:2) (cid:3)
4 K.S. NISAR*,F.B.M. BELGACEM, M. S. ABOUZAID
where G(u) is defined in (1.7). It is clear that the Sumudu transform of f (t) = tα is
given by
∞
S[f (t)] = (tu)αe−tdt = uαΓ(α+1), (α) > 1 (1.18)
ℜ −
Z0
2. Solution of generalized fractional Kinetic equations
We devote this section to the derivative of the exact solution of time fractional kinetic
equation via the well-known Sumudu transform. The results are as follows:
Theorem 1. If d > 0,υ > 0,α,c,l,t C , (l) > 1 and (u) > 0 with u < d−1, then
∈ ℜ − ℜ | |
the solution of the equation
N (t) N Hλ,α(t) := dυ D−υN (t), (2.1)
− 0 l,µ − 0 t
is given by the following formula
N ∞ ( 1)kΓ(2k +l+2) t 2k+l
N (t) = 0 − E ( dυtυ) (2.2)
2 Γ(αk +µ)Γ λk +l+ 3 2 υ,2k+l+1 −
k−0 2 (cid:18) (cid:19)
X
where E (.) is the generalized M(cid:0)ittag-Leffler(cid:1)function given in (1.6).
v,2k+l+1
Proof. Taking the Sumudu transform to the both sides of (2.1), we obtain the following
relation,
S N (t) N S Hλ,α(t);p = dυS D−υN (t)
{ }− 0 l,µ − 0 t
Now, using the definition given in (n1.5), we geto (cid:8) (cid:9)
∗ ∞ ( 1)k t 2k+l+1 ∗
N (u) N S − = dυuvN (u)
− 0 Γ(αk +µ)Γ λk +l + 3 2 −
( !)
k−0 2 (cid:18) (cid:19)
X
∗ (cid:0) (cid:1)
where N (u) = S N (t),u and S tλ−1 = uλ−1Γ(λ) gives,
{ }
∗ ∞ ( 1(cid:8))k (cid:9) ∗
N (u) N − u2k+l+1Γ(2k +l +2) = dυuvN (u),
− 0 Γ(αk +µ)Γ λk +l + 3 −
k−0 2
X
(cid:0) (cid:1)
∗ ∞ ( 1)k u 2k+l+1Γ(2k +l+2)
N (u)(1+dυuv) = N − 2
0 Γ(αk +µ)Γ λk +l + 3
k−0 (cid:0) (cid:1) 2
X
∗ N ∞ ( 1)k (cid:0)u 2k+l+1Γ((cid:1)2k +l+2)
N (u) = 0 − 2
(1+dυuv) Γ(αk +µ)Γ λk +l + 3
k−0 (cid:0) (cid:1) 2
X
∞ ( 1)k u 2k+l+1Γ(2k +(cid:0)l+2) ∞ (cid:1)
= N − 2 ( 1)r(du)vr
0 Γ(αk +µ)Γ λk +l + 3 −
k−0 (cid:0) (cid:1) 2 r−0
X X
(cid:0) (cid:1)
ON FRACTIONAL KINETIC EQUATIONS AND THEIR SUMUDU TRANSFORM... 5
Now, taking the inverse transform of the above expression and using the relation
tλ−1
S−1 uv = , (u) > 0, (v) > 0,
{ } Γ(v) ℜ ℜ
we get
∞ ( 1)k(2)−(2k+l+1)Γ(2k +l +2) ∞ ( 1)rdvrt2k+vr+l
N (t) = N − −
0 Γ(αk +µ)Γ λk +l + 3 Γ(2k +l+1+vr)
k−0 2 r−0
X X
∞ ( 1)kΓ(2k(cid:0)+l +2)t2k(cid:1)+l ∞ ( 1)rdvrtvr
= N − −
0 Γ(αk +µ)Γ λk +l + 3 (2)2k+l+1 Γ(2k +l +1+vr)
k−0 2 r−0
X X
In view of (1.6), we obtain the desi(cid:0)red result. (cid:1) (cid:3)
Corollary 2.1. If we put α = λ = 1 and µ = 3/2 in (2.1) then we obtain the solution of
fractional kinetic equation involving Struve function H (z) as :
v
For d > 0,υ > 0,c,l,t C, (l) > 1 and (u) > 0, then the solution of the equation
∈ ℜ − ℜ
N (t) N H1,1 (t) := dυ D−υN (t),
− 0 l,3/2 − 0 t
is given by the following formula
N ∞ ( 1)kΓ(2k +l+2) t 2k+l
N (t) = 0 − E ( dυtυ)
2 Γ k +l+ 3 Γ k + 3 2 v,2k+l+1 −
k−0 2 2 (cid:18) (cid:19)
X
(cid:0) (cid:1) (cid:0) (cid:1)
Theorem 2. If d > 0,υ > 0,α,c,b,l,t C, (l) > 1 and (u) > 0 then for the
∈ ℜ − ℜ
solution of the equation
N (t) N Hλ,α(dυtυ) = dυ D−υN (t) (2.3)
− 0 p,µ − 0 t
there holds the formula
dv l+1
N (t) = N tlv+v−1
0
2
(cid:18) (cid:19)
∞ ( 1)kΓ(2kv +vl+ν +1) dυtv 2k
− E ( dυtυ), (2.4)
× Γ(αk+µ)Γ λk +l+ 3 2 v,(2k+l+1)υ −
k=0 2 (cid:18) (cid:19)
X
(cid:0) (cid:1)
where E (.) is the generalized Mittag-Leffler function (1.6).
v,2kυ+lv+v+1
Proof. The proof can be proved in parallel with the proof of Theorem 1, so the details of
(cid:3)
proofs are omitted.
6 K.S. NISAR*,F.B.M. BELGACEM, M. S. ABOUZAID
Corollary 2.2. If we set, α = λ = 1 and µ = 3/2 in (2.3) then we obtain the solution of
fractional kinetic equation involving Struve function H (z) as : For d > 0,υ > 0,c,l,t
v
∈
C, (l) > 1 and (u) > 0, then the solution of the equation ,
ℜ − ℜ
N (t) N H1,1 (dvtv) := dυ D−υN (t),
− 0 l,3/2 − 0 t
is given by
dv l+1
N (t) = N tlv+v−1
0
2
(cid:18) (cid:19)
∞ ( 1)kΓ(2kv +vl+ν +1) dυtv 2k
− E ( dυtυ)
× Γ k + 3 Γ k +l + 3 2 v,(2k+l+1)υ −
k=0 2 2 (cid:18) (cid:19)
X
Theorem 3. If d > 0,υ >(cid:0) 0,a,(cid:1)c,b(cid:0),l,t C ,(cid:1) (l) > 1, (u) > 0 with u < d−1 and
∈ ℜ − ℜ | |
a = d then for the solution of the equation
6
N (t) N Hλ,α(dυtυ) = aυ D−υN (t), (2.5)
− 0 l,µ − 0 t
there hold the formula
dv l+1 ∞ ( 1)kΓ(2kυ +υl+υ +1)
N (t) = N tlv+v−1 −
0 2 Γ(αk +µ)Γ λk +l+ 3
(cid:18) (cid:19) k=0 2
X
dυtv 2k (cid:0) (cid:1)
E ( aυtυ), (2.6)
v,(2k+l+1)υ
× 2 −
(cid:18) (cid:19)
where E ( aυtυ) is the Mittag-Leffler function in (1.6).
v,(2k+l+1)υ
−
Proof. The proof of theorem 3 is derived similarly as that of theorems 1 and 2. (cid:3)
Corollary 2.3. By setting α = λ = 1 and µ = 3/2 in (2.5) we obtain the solution of
fractional kinetic equations as : For d > 0,υ > 0,c,l,t C,a = d , (l) > 1 (u) >
∈ 6 ℜ − ℜ
0 and then the solution of the equation ,
N (t) N H1,1 (dvtv) := aυ D−υN (t),
− 0 l,3/2 − 0 t
is given by
dv l+1 ∞ ( 1)kΓ(2kυ +υl+υ +1)
N (t) = N tlv+v−1 −
0 2 Γ k + 3 Γ k +l+ 3
(cid:18) (cid:19) k=0 2 2
X
dυtv 2k (cid:0) (cid:1) (cid:0) (cid:1)
E ( aυtυ)
v,(2k+l+1)υ
× 2 −
(cid:18) (cid:19)
ON FRACTIONAL KINETIC EQUATIONS AND THEIR SUMUDU TRANSFORM... 7
2.1. Special Cases.
1 . Consider the generalized Struve function given by Bhowmic [11]
∞ ( 1)k t 2k+l+1
Hλ(x) = − 2 (2.7)
l Γ λk +l+ 3 Γ k + 3
k−0 (cid:0)2(cid:1) 2
X
Now, we have the following corollaries du(cid:0)e to theorem(cid:1) s(cid:0)1, 2 an(cid:1)d 3 respectively.
Corollary 2.4. If d > 0,υ > 0,λ,l,t C, (l) > 1 and (u) > 0 then the solution of
∈ ℜ − ℜ
the equation
N (t) N Hλ(t) := dυ D−υN (t),
− 0 l − 0 t
is given by the following formula
N ∞ ( 1)kΓ(2k +l+2) t 2k+l
N (t) = 0 − E ( dυtυ)
2 Γ λk +l + 3 Γ k + 3 2 υ,2k+l+1 −
k−0 2 2 (cid:18) (cid:19)
X
Corollary 2.5. If d > 0,υ(cid:0)> 0,λ,l,t(cid:1) C(cid:0) , (cid:1)(l) > 1 and (u) > 0 then for the
∈ ℜ − ℜ
solution of the equation
N (t) N Hλ(dυtυ) = dυ D−υN (t)
− 0 l − 0 t
there holds the formula
dv l+1 ∞ ( 1)kΓ(2kv +vl+ν +1) dυtv 2k
N (t) = N tlv+v−1 − E ( dυtυ)
0 2 Γ λk +l + 3 Γ k + 3 2 v,(2k+l+1)υ −
(cid:18) (cid:19) k=0 2 2 (cid:18) (cid:19)
X
Corollary 2.6. If d > 0,υ > 0,λ,(cid:0)b,l,t C ,(cid:1)a =(cid:0)d , ((cid:1)l) > 1 and (u) > 0 then for
∈ 6 ℜ − ℜ
the solution of the equation
N (t) N Hλ(dυtυ) = aυ D−υN (t)
− 0 l − 0 t
there hold the formula
dv l+1 ∞ ( 1)kΓ(2kυ +υl+υ +1)
N (t) = N tlv+v−1 −
0 2 λk +l+ 3 Γ k + 3
(cid:18) (cid:19) k=0 2 2
X
dυtv 2k (cid:0) (cid:1) (cid:0) (cid:1)
E ( aυtυ)
v,(2k+l+1)υ
× 2 −
(cid:18) (cid:19)
2 . Consider the generalized Struve function given by Kant [23]
∞ ( 1)k x 2k+l+1
Hλ,α(x) = − 2 (2.8)
l Γ λk +l + 3 Γ αk + 3
k−0 (cid:0)2 (cid:1) 2
X
Now, we have the following corollaries(cid:0)due to theo(cid:1)rem(cid:0) 1, 2 an(cid:1)d 3 respectively.
8 K.S. NISAR*,F.B.M. BELGACEM, M. S. ABOUZAID
Corollary 2.7. If d > 0,υ > 0,λ,l,t C , (l) > 1 and (u) > 0, then the solution
∈ ℜ − ℜ
of the equation
N (t) N Hλ,α(t) := dυ D−υN (t),
− 0 l − 0 t
is given by the following formula
N ∞ ( 1)kΓ(2k +l+2) t 2k+l
N (t) = 0 − E ( dυtυ)
2 λk +l + 3 Γ αk + 3 2 υ,2k+l+1 −
k−0 2 2 (cid:18) (cid:19)
X
Corollary 2.8. If d > 0,υ(cid:0)> 0,λ,l,t (cid:1)C(cid:0), (l) >(cid:1) 1 and (u) > 0 then for the solution
∈ ℜ − ℜ
of the equation
N (t) N Hλ,α(dυtυ) = dυ D−υN (t)
− 0 l − 0 t
there holds the formula
dv l+1 ∞ ( 1)kΓ(2kv +vl+ν +1) dυtv 2k
N (t) = N tlv+v−1 − E ( dυtυ)
0 2 Γ λk +l+ 3 Γ αk + 3 2 v,(2k+l+1)υ −
(cid:18) (cid:19) k=0 2 2 (cid:18) (cid:19)
X
Corollary 2.9. If d > 0,υ > 0,λ(cid:0),b,l,t C(cid:1),a =(cid:0) d , ((cid:1)l) > 1 and (u) > 0 then for
∈ 6 ℜ − ℜ
the solution of the equation
N (t) N Hλ,α(dυtυ) = aυ D−υN (t)
− 0 l − 0 t
there hold the formula
dv l+1 ∞ ( 1)kΓ(2kυ +υl+υ +1)
N (t) = N tlv+v−1 −
0 2 λk +l + 3 Γ αk + 3
(cid:18) (cid:19) k=0 2 2
X
dυtv 2k (cid:0) (cid:1) (cid:0) (cid:1)
E ( aυtυ)
v,(2k+l+1)υ
× 2 −
(cid:18) (cid:19)
3 . Consider the generalized Struve function given by Singh [37]
∞ ( 1)k x 2k+l+1
Hλ (x) = − 2 (2.9)
l,µ
Γ λk + l +(cid:0) 3(cid:1) Γ k + 3
Xk−0 µ 2 2
(cid:16) (cid:17)
Now, we have the following corollaries due to theorem(cid:0) 1, t(cid:1)heorem 2 and theorem 3
respectively.
Corollary 2.10. If d > 0,υ > 0,λ,l,t C , (l) > 1 and (u) > 0 then the solution
∈ ℜ − ℜ
of the equation
N (t) N Hλ (t) := dυ D−υN (t),
− 0 l,µ − 0 t
is given by the following formula
N ∞ ( 1)kΓ(2k +l+2) t 2k+l
N (t) = 0 − E ( dυtυ)
υ,2k+l+1
2 Xk−0 Γ λk + µl + 23 Γ k + 32 (cid:18)2(cid:19) −
(cid:16) (cid:17)
(cid:0) (cid:1)
ON FRACTIONAL KINETIC EQUATIONS AND THEIR SUMUDU TRANSFORM... 9
Corollary 2.11. If d > 0,υ > 0,λ,l,t C , (l) > 1 and (u) > 0 then for the
∈ ℜ − ℜ
solution of the equation
N (t) N Hλ (dυtυ) = dυ D−υN (t)
− 0 l,µ − 0 t
there holds the formula
dv l+1 ∞ ( 1)kΓ(2kv+vl+ν +1) dυtv 2k
N (t) = N tlv+v−1 − E ( dυtυ).
0 v,(2k+l+1)υ
2 Γ λk + l + 3 Γ k + 3 2 −
(cid:18) (cid:19) Xk=0 µ 2 2 (cid:18) (cid:19)
(cid:16) (cid:17)
Corollary 2.12. If d > 0,υ > 0,λ,b,l,t C ,(cid:0)a = d(cid:1), (l) > 1,µ is an arbitrary
∈ 6 ℜ −
parameter and (u) > 0 then for the solution of the equation
ℜ
N (t) N Hλ (dυtυ) = aυ D−υN (t)
− 0 l,µ − 0 t
there hold the formula
dv l+1 ∞ ( 1)kΓ(2kυ +υl+υ +1)
N (t) = N tlv+v−1 −
0
2 λk + l + 3 Γ k + 3
(cid:18) (cid:19) Xk=0 µ 2 2
dυtv 2k (cid:16) (cid:17) (cid:0) (cid:1)
E ( aυtυ)
v,(2k+l+1)υ
× 2 −
(cid:18) (cid:19)
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