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Analytic Solution of Linear Fractional Differential Equation with Jumarie Derivative in Term of Mittag-Leffler Function

Uttam Ghosh*, Srijan Sengupta, Susmita Sarkar, Shantanu Das

Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India

Abstract: There is no unified method to solve the fractional differential equation. The type of derivative here used in this paper is of Jumarie formulation, for the several differential equations studied. Here we develop an algorithm to solve the linear fractional differential equation composed via Jumarie fractional derivative in terms of Mittag-Leffler function; and show its conjugation with ordinary calculus. In these fractional differential equations the one parameter Mittag-Leffler function plays the role similar as exponential function used in ordinary differential equations.

Keywords: jumarrie fractional derivative, riemann-liouvelli fractional derivative, mittag-leffler function, fractional differential equations

Received March 05, 2015; Revised April 11, 2015; Accepted April 27, 2015

Citation: Uttam Ghosh, Srijan Sengupta, Susmita Sarkar, Shantanu Das, Analytic Solution of Linear Fractional Differential Equation with Jumarie Derivative in Term of Mittag-Leffler Function. American Journal of Mathematical Analysis, 3 (2). 32-38. January 2015. doi:10.12691/ajma-3-2-2

1. Introduction

The analytical solutions of the fractional differential equation are emerging branch of applied science also in basic science. Different methods are developing to solve the fractional differential equations. Since the definition of fractional derivative is modifying to relate it with the classical derivative. Mathematicians are trying to develop the formulas of fractional calculus but geometry of fractional derivative has no concrete shape [1]. Depending on different type of derivatives different methods of solution are developing [2-7]. Riemann-Liouville definition the fractional derivative of a constant is non-zero which creates a difficulty to relate between the classical calculus. To overcome this difficulty Jumarie [2,3,4,5] modified the definition of fractional derivative of Riemann-Liouvell type and with this new formulation, we obtain the derivative of a constant as zero. Thus using this definition linking between the fractional and classical calculus becomes easier. There is no unique method to solve the linear fractional differential equations. Using the Jumarie modified definition of fractional derivative we obtain the derivative of Mittag-Leffler function as Mittal-Leffler function, as in case of classical whole-order derivative the derivative of is itself exponential function. Thus via use of Jumarie modified Riemann-Liouvelli derivative, there exists conjugation with classical calculus, which eases in many cases to solve fractional differential equation composed with Jumarie fractional derivative. Here we want to develop an algorithm to solving the linear fractional differential equation using the Mittag-Leffler function. We have obtained applied this method to homogeneous fractional differential equations and got corresponding fundamental solution.

Organization of the paper is as follows. In section 2.0 some definition of fractional derivative is reproduced with essential examples. In section 3.0 and 4.0 some properties of Mittag-Leffler function is described. Finally in section 5.0 the methods for solving the linear fractional differential equation composed by Jumarie fractional derivative is developed using the Mittag-Leffler function.

2. Some Definitions of Fractional

There are many definition of fractional derivative. Grunwald-Letnikov fractional derivative [6], Liouville fractional derivative [8], Riemann-Liouville fractional derivative [10], Caputo fractional derivative [8,9,10,11], Kolwanker-Gangal local fractional derivative [12,13,14,15,16], Jumarie modified fractional derivative [2]. Here we use the Riemann-Liouville fractional derivative and its modified form by Jumarie [2].

2.1. Riemann-Liouville Definition of Fractional Derivative

Let the function is one time integrable then the integro-differential expression

is known as the Riemann-Liouville (R-L)definition of fractional derivative [6] withas integer with condition

In Riemann-Liouville definition the function, is integrated fold and then differentiate times. We can re-write the above as follows

The left R-L fractional derivative is defined by

And the right R-L derivative is

Where in aboveis integer such that that is just greater than fractional number.

Using the left R-L derivative we get the fractional derivative of the function as non-zero, as demonstrated below.

Similarly the right R-L derivative of is

This shows that the fractional derivative of a constant (K) is non-zero but in classical calculus derivative of a constant is zero which is contradiction between the classical derivative and the fractional derivative of a constant. To overcome this difference Jumarie [2] modified the left R-L fractional derivative.

We get the R-L left derivative of a power function as

2.2 Jumarie Modified Definition of the Fractional Derivative Is

Using this definition we get

The above formula in line-1, becomes fractional order integration if we replace by which is

Using the above formula we get for, the fractional integral for order

Using the substitution we have for; , and for ,; , ; , we get the following

We used Beta-function defined as

Applying the above obtained result the fractional integral of order, with is

Taking one whole derivative of the above we get

We have therefore calculated fractional derivative by R-L left formula forsuch that thus our nearest integer is one that is and we write that below

Thus for, the fractional RL derivative of is

For a constant function, putting in above expression, we get

Let us now see what Jumarrie derivative is from above R-L derivative obtained for. The composition of the Jumarie derivative, with start point of integration as and is

The above expression show that this Jumarrie derivative is composed of two RL derivatives those are minus RL derivative of a constant. Going by similar steps as done for, we get first the fractional integral in terms of incomplete Gamma function as

The fractional derivative of is by taking one whole derivative of above expression we get the following

The fractional derivative of is

Therefore

For , we have

We will be using Jumarie derivative for power function with start point of differentiation as, in subsequent sections. When start point of differentiation is non-zero we will be shifting the origin to that non-zero point and use the above formula.

3. Mittag-Leffler Function

In 1903 Mittag-Leffler [17,18,19,20] was introduce a function defined by is named as the one parameter Mittag-Leffler function. For we get

The integral representation of the Mittag-Leffler function is Here the path of the integral C is a loop which starts and ends at and encloses the circles of disk

The two parameter Mittag-Leffler function [21] was defined by

and the corresponding integral representation of the two parameter Mittag-Leffler function is where the contour C is already defined.

3.1. Some Properties Mittag-Leffler Function and Its Application

We rewrite the Mittag-Leffler [17,18,19] function in the following form by an infinite series

is the one parameter Mittag-Leffler function.

Using Jumarie derivative of order, with with start point as for, that is, for; and also using Jumarie derivative of constant as zero, we get the following very useful identity. In all the subsequent sections we will say is the Jumarie derivative with zero as start point

Thus

[picture] | (1)

This shows that is a solution is a solution of the fractional differential equation

[picture] | (2)

Where A is arbitrary constant.

Therefore

with has solution

Using this property of the Mittag-Leffler one can easily prove the following theorem.

Again integrating the Mittag-Leffler function in the interval [0, x] we get

Theorem 1: The Mittag-Leffler function satisfies the relation

Proof: Let then

[picture] | (3)

Using the solution of the equation (2) we get the solution of the equation (3) in the following form

From the definition of y we get. Therefore we have .

Thus we get

[picture] | (4)

We get useful property of one parameter Mittag-Leffler function.

Using the above property of Mittag-Leffler function we get

[picture] | (5)

[picture] | (6)

4. Complex Mittag-Leffler Function and Its Properties

Jumarie [22] defined the complex Mittag-Leffler in the following form

On the other hand Jumarie [22] defined period () of the function in the following form, taking and therefore

The series presentation of is

Taking term by term Jumarie derivative we get

5. Solution of Linear Second Order Fractional Differential Equation

Let us consider the function

with A and B is constants. Differentiating times with respect to t, for , with Jumarie derivative we get

Differentiating above by Jumarrie derivative and re-arranging, we get

This shows that the fractional differential equation

has solution in the form

On the other hand consider the differential equation

it can be express in the following form

[picture] | (7)

Let, then equation (7) reduce to the form

Solution of the above equation is same as the solution of the equation (2) which is

On integrating both side we get that is applying on both sides of above, we get

Therefore is a solution of the differential equation. Thus we can state the following theorem

Theorem 2: The fractional differential equation

has solution of the form

where A and B are constants.

Proof of the theorem is follows from the previous arguments.

Similarly one can generalized the solution of the differential equation in the following form

If

with all are distinct be a fractional differential equation with then solution of the differential equation will be

where Ai are arbitrary constants and is one parameter Mittag-Leffler function.

Let us consider the function

Where A and B are constants.

Then

Thus solution of the differential equation

is

A and B are constants.

On the other hand consider the differential equation

[picture] | (8)

Let then equation (8) reduce to the form

[picture] | (9)

Solution of this differential equation is

A and B are constants.

Thus the following theorem can be stated

Theorem 3: The fractional differential equation

has solution of the form

where A and B are constants.

The proof of the theorem is already explained in the previous arguments.

Theorem 4: Solution of the fractional differential equation

is of the form

Proof: The given differential equation can be written in the following form

[picture] | (10)

Using theorem 3 we get the solution of the fractional differential (10) can be written in the following form

Thus we get useful results.

6. Conclusions

There are several methods to solve fractional differential equations, and the solution depends on the type of fractional derivative used. Here we develop an analytical method to find the solutions of linear fractional differential equation, composed by Jumarie fractional derivative in terms of one parameter Mittag-Leffler function. Some well known properties of Mittag-Leffler have been used to find solution of the fractional differential equations. The solutions obtained are similar as the solutions obtained usual calculus obtained in terms the exponential function. This conjugation with ordinary calculus when Jumarie type fractional derivative is used to compose the fractional differential equations is useful in several physical problems.

Acknowledgement

Acknowledgments are to Board of Research in Nuclear Science (BRNS), Department of Atomic Energy Government of India for financial assistance received through BRNS research project no. 37(3)/14/46/2014-BRNS with BSC BRNS, title “Characterization of unreachable (Holderian) functions via Local Fractional Derivative and Deviation Function.

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