Application of Laplace Transform on Solution of Fractional Differential Equation

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ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Application of Laplace Transform on Solution of Fractional Differential Equation

A. K. Thakur

¹

, Ravi Kumar

²

and Gopi Sahu

²

1

Associate Professor and

²

Research Scholar Department of Mathematics

DR.C.V.Raman University Kota Bilaspur, India [email protected]

(Received on: May 14, 2018) ABSTRACT

The Laplace transform method is applied to fractional differential equation with a right-sided derivative and variable potential. After solving the intermediate difference equation we arrive at the general solution of the problem in the form of a Meijer-G-function series.

AMSC: 26A33, 42A38 ,44A10

Keywords: Integral transforms, Meijer-G-function, Fractional differential equation.

1. INTRODUCTION

Fractional calculus, involving derivatives and integrals of non-integer order, is now an integral part of mathematical modelling methods. Non-integer order calculus, both in one and Multidimensional space, it has becomes an important tool in many areas of physics, mechanics, chemistry, engineering, finances and bioengineering.

In applications of fractional calculus, a new class of integral-differential equation has been developed. They include integrals and derivatives of non-integer order and in general variable coefficients. The methods of solving include fixed point theorem, integral transform method as well as operational methods based on properties of new classes of special function.

Still, some fractional differential equations, including Euler-Lagrange equations of fractional mechanics, remain an open problem.

In this paper we test the Laplace transform method for fractional differential equation

with the right-sides derivative and variable potential - . The main motivation to study this

class of equation is the fact that the solution are known in the literature only for its analogue

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with left-sides derivative. Kilbas and Saigo solved such an equation using the Banach theorem on a fixed point. This method seems more suitable for equations with right-sided operators as in this case, fixed point theorems yield only the existence and uniqueness of solutions.

Attempts to calculate the exact form of solutions by means of straightforward integration did not produce an explicit and closed form of solution.

Fractional integrals and derivatives

We shall recall the definitions of right-sided fractional operators. The following formulas describe fractional integrals for order α fulfilling condition Re(α) > 0:

{f(t)} =

Г( )

∫

₍ ^{( )}₎

0˂t˂b (1) {f(t)} =

Г( )

∫

₍ ^{( )}₎

(2) The first of the above integrals is defined in a finite time interval and it is called a Riemann-

Liouville right-sided integral. The second formula describes a right-sided integral defined on the real halfaxis, namely it is a Liouville integral.

Using the above fractional integrals we define the following fractional derivatives

{f(T)} =( − ) {f(t)} 0˂t˂b (3) {f(T)} = ( − ) {f(t)} (4) Where Re(α) > 0 and [Re(α)] = n-1.

Operator is correctly determined in a finite time interval and it is known as a right-sided Riemann-Liouville derivative. Derivative acts on functions defined on the real halfaxis and it is called a right-sided Liouville derivative.

The following formulas present some results of fractional integration and differentiation for power functions:

{( − ) } =

^Г( ⁾

Г( )

( − ) , Re( ) > 0 (5) { } =

^Г( ⁾

Г( )

, Re(α+ ) ˂ 1 (6) { ( − ) } =

^Г( ⁾

Г( )

( − ) , Re( ) > 0 (7) { } =

^Г( ⁾

Г( )

, Re(α+ )- [Re(α)] ˂ 1 (8) In what follows we also apply the composition rule for fractional derivatives and integrals of the same order. When Re(α) > 0 and, f ∈ (0, ) the following formula

{f(t)} = f(t) (9) is valid almost everywhere in interval [0,b]. In the case when Re(α) >1 the above

composition rule holds at any point t ∈ [0,b].

2. LAPLACE TRANSFORM AND ITS PROPERTIES

Let us recall the definition of the Laplace transform

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L{F(t)} = f(p)= ∫ ( ) (10) Laplace transform has the convolution property is defined by the formula

F(t)G(t) = ∫ ( ) ( − ) (11) Where the relation FG is called resultant or faltun of F & G. The convolution theorem states that

F*G = ∫ ( ) ( − ) = {f(p)g(p)} (12) The following two properties shall be applied to solving fractional differential equation L{ ( )} = f (p-a)

3. FRACTIONAL DIFFERENTIAL EQUATION WITH RIGHT-SIDED RIEMANN- LIOUVILLE DERIVATIVE AND VARIABLE POTENTIAL

Let us consider the following fractional differential equation in a finite time interval ( -λ )f(t) = 0, t ∈ [0,b] (13) We shall extend this equation to the real positive halfaxis. To this aim we rewrite solution f in the following form

( ) = (t)∆H(t) t∈

where we denote _H(t): = H(t) – H(t–b) with H - being the Heaviside’s function. We notice that function vanishes outside interval [0,b] and coincides with solution f inside time interval [0,b]. Hence we replace equation (13) with an equation in the form of

( -λ ) (t)=0 (14) The above equation can be transformed to its equivalent integral form using composition rule (9). Thus, we shall obtain function , solving integral equation of order α:

(1-λ ) (t)= (15) where function is an arbitrary stationary function of the right-sided Riemann-Liouville

derivative, given by formula

( ) = ∑ ( − ) ∆H(t) (16) Let us note that the following two fractional integrals coincide on positive halfaxis for solution (t) = (t)

and owing to this property we can rewrite equation (17) as an equation with a Liouville fractional integral

(1-λ ) (t)= (17) The next step is the application of the Laplace transform. The result is a difference equation for the Laplace transform of solution L { }

(1-g(p) )L{ (p)} = L{ (p)} Re(p) > 0 (18)

Where is a translation operator acting on fuction as follows ℎ( ) = h(p+α+ ) and

We assume Re(α+ ) > 0. Function g is given as the quotient of Euler Gamma functions

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G(p) =

^{Г( )}

Г( )

The above difference equation is solved by the following series absolutely convergent in complex halfplane Re(p) > 0 when Re(α+ ) > 0:

L{ ( )} = ∑ ( ( ) ) L{ ( )} (19) In order to derive an exact and explicit form of solution we should now invert the above Laplace transform. Let us note that for each m the last term is simply the Laplace transform of the stationary function given in formula (16). The main difficulty is therefore the inverse Laplace transform for the corresponding products of the gfunctions. However, they can be

identified as kernels G of the Laplace transforms defining the Meijer G-functions;

∏ ( + ( + )) =

_,^,

[

_⃗^⃗

| ] (20) Where the vectors are defined as follows

⃗ = (α + )⃗

=α ⃗ ⃗ = (α + ) ⃗

(21) ⃗

= [0, 1,...m-1] ϵ ⃗

= [1, 1,...1] ϵ

Let us note that formula (18), describing the Laplace transform of solution, holds in complex halfplane Re(p) > 0. Hence, inverting this transform we should use vertical contour

_.

According to Theorem 1.1 and above formulas from monograph [16] we calculate parameters ∆ = 0

^∗

= 0 = -mα

And obtained the following codition

∆ +Re( ) < -1 implies Re(α) >

Applying formula (12) for convolution and formula (20) we obtain the solution given as the series of Meijer G-functions in the convolution with the stationary function multiplied by a power function

(t) = ∑

_,^,

[

^⃗_⃗

| ] *

⁽ ⁾

(t) (22) In order to calculate the Laplace convolutions we apply the integration formula for Fox H- functions rewritten for a subset of Meijer G-functions (compare Theorem 2.7 from the monograph by Kilbas and Saigo [16]). When ∆ = 0 and condition

max [ ( ) -1]+ ( )+ α –k + 1 < 0 (23) is fulfilled, then the following general formula for the integration of Meijer G-functions is valid

I u G

_,^,

[ |u]

_⃗^⃗

(x) = x G

_,^,

[ |x

[ , ⃗]

[ ⃗, ]

]

As assumption (23) is fulfilled for each element of series (22), we apply the above formula for integration and we calculate the explicit form of all Laplace convolutions in series (22) G

_,^,

[

_⃗^⃗

| ] *

⁽ ⁾

( − ) ∆ ( )

=

⁽ ⁾

Г( − + 1)I

⁽

G

_,^,

[

_⃗^⃗

| ] ( )

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=

⁽ ⁾

Г( − + 1) G

_,^,

[

⃗ ,

⃗ _,

where the vectors defining the Meijer G-functions look as follows

⃗

_,

= [ ⃗ ,m( + ) + − + 1] ∈ (24) B⃗

_,

= [m( + ), ⃗ ]∈ (25)

The above integration formula yields the Laplace convolution corresponding to the (b-t)_-k component of the stationary function. Using this formula we arrive at the component series of the solution connected to the respective component of the stationary function:

(t)= Г( − + 1) ∑ ( ) G

_,^,

[

⃗ ,

⃗ _,

The general solution of equations (17) and (13) is an arbitrary linear combination of the above series

( ) =∑

(t) t ∈ [0,b]

The presented considerations can be summarized in the following Proposition.

Proposition 3.1

Let Re( ) ∈ (n-1, n) and Re( + ) > 0, Re( ) > ½. Then equation ( - λ )f(t) = 0

has in finite time interval [0,b] a general solution in the form of a linear combination of solutions

( ) =∑

(t) (26) where are arbitrary real coefficients and components f

are the following series of Meijer G-functions

(t)=( − ) + Г( − + 1) ∑ ( ) G

_,^,

[

⃗ ,

⃗ _,

(27) with defining vectors given by formulas (24, 25).

Remark 3.2

Let us note that the Riemann-Liouville derivatives obey in a finite time interval the following formula [12]:

Q D

= D

Q (28) where operator Q is a reflection operator in interval [0,b] acting as follows on functions determined in this interval Qh(t): = h(b-t) and D

is the left-sided Riemann-Liouville derivative [8, 12].

Let us observe that this remark yields the following Proposition valid.

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Proposition 3.3

Let Re( )∈(n -1, n) and Re ( + ) > 0, Re( ) >1/2. Then equation ( - λ( − ) )f(t) = 0

has in finite time interval [0,b] a general solution in the form of a linear combination of solutions

( ) =∑

(t) (29)

where are arbitrary real coefficients and components

are the following series of Meijer G-functions

(t)= + Г( − + 1) ∑ ( ) G

_,^,

[ 1 −

⃗ ,

⃗ _,

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with defining vectors given by formulas (24, 25).

3.1. Example: solution of equation (13) for + =

Let us consider an example for = 0. Equation (13) becomes an eigenvalue equation with complex eigenvalue ;

The Meijer G-functions appearing in formulas (26, 27) for the general solution can be transformed using their reduction property (see Property 2.2 in monograph [14]).

We conclude that in this case they can be identified with the power functions:

G

_,^,

[

⃗ ,

⃗ _,

= G

_,^,

[

[ ]

=

^Г( ⁾

Г( )

(1 − )

Hence component solutions

for k = 1,…,n are of the form

(t) = ( − )

_,

(λ( − ) )

where functions E are the generalized Mittag-Lefller functions (see their definition and properties in monograph [8]).

3.2. Example: solution of equation (15) for α + = α/2

As a next example we solve an equation for = α/2. For this value of parameter , equation (13) has the following form:

( - λ

^/

) (t) = 0

Similarly to the previous example we can also apply the reduction formula and obtain for m >

0 the following simple formulas for Meijer G-functions

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G

_,^,

[

⃗ ,

⃗ _,

= G

_,^,

[

[ , / ]

[ / ( ), / / ( ) ]

Component solution

for k =1,…,n looks as follows

(t)= ( − ) + Г( − + 1) ∑ ( ) G

_,^,

[

[ , / ]

[ / ( ), / / ( ) ]

REFERENCES

1. Agrawal O.P., Tenreiro-Machado J.A., Sabatier J., (Eds.) Fractional Derivatives and Their Application: Nonlinear Dynamics, vol. 38, Springer-Verlag, Berlin (2004).

Application of Laplace Transform on Solution of Fractional Differential Equation