Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals
Uttam Ghosh1,, Susmita Sarkar2, Shantanu Das3, 4
1Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India
2Department of Applied Mathematics, University of Calcutta, Kolkata, India
3Reactor Control System Design Section Bhabha Atomic Research Centre, Mumbai, India
4Department of Physics, Jadavpur University Kolkata, West Bengal, India
Abstract
In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type fractional derivative, and describe this method developed by us, to find out particular integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. The short cut rules, that are developed here in this paper to replace the operator Da or operator D2a as were used in classical calculus, gives ease in evaluating particular integrals. Therefore this method proposed by us is useful and advantageous as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.
Keywords: Mittag-Leffler functions, non-homogeneous fractional differential equations, modified riemann-liouville definition
Received August 14, 2015; Revised August 31, 2015; Accepted September 10, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.Cite this article:
- Uttam Ghosh, Susmita Sarkar, Shantanu Das. Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals. American Journal of Mathematical Analysis. Vol. 3, No. 3, 2015, pp 54-64. http://pubs.sciepub.com/ajma/3/3/1
- Ghosh, Uttam, Susmita Sarkar, and Shantanu Das. "Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals." American Journal of Mathematical Analysis 3.3 (2015): 54-64.
- Ghosh, U. , Sarkar, S. , & Das, S. (2015). Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals. American Journal of Mathematical Analysis, 3(3), 54-64.
- Ghosh, Uttam, Susmita Sarkar, and Shantanu Das. "Solutions of Linear Fractional non-Homogeneous Differential Equations with Jumarie Fractional Derivative and Evaluation of Particular Integrals." American Journal of Mathematical Analysis 3, no. 3 (2015): 54-64.
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1. Introduction
The fractional differential equations and its solutions arises in different branches of applied science, engineering, applied mathematics and biology [1-9][1]. The solutions of fractional difference equations are obtained by different methods which includes Exponential-Function Method [10], Homotopy Perturbation Method [11], Variation Iteration Method [12], Differential transform Method [13] and Fractional Sub-equation Method [14], Analytical Solutions in terms of Mittag-Leffler function [15]. In developing those methods the usually used fractional derivative is Riemann-Liouvellie (R-L) [6], Caputo derivative [6], Jumarie’s left handed modification of R-L fractional derivative [16, 17]. In [15] we have developed an algorithm to solve the homogeneous fractional order differential equations in terms of Mittag-Leffler function and fractional sine and cosine functions. However, there are no standard methods to find solutions of non-homogeneous fractional differential equations. In this paper we describe a method to solve the fractional order non-homogeneous differential equations. Organizations of the paper are as follows; in section 2.0 we describe the different definitions of fractional derivatives and properties of Mittag-Leffler function. In section-3.0 we describe the solutions of order fractional differential equations. In section 4.0 the solutions of
order fractional differential equations is described, with several types of forcing functions. In section 5.0 this methods has been applied to solve both un-damped and damped fractional order forced oscillator equations. In this paper the fractional derivative operator
will be of Jumarie type fractional derivative.
2. Definition of Fractional Derivatives
The useful definitions of the fractional derivatives are the Grunwald-Letinikov (G-L) definition and Riemann-Liouville(R-L) definition [6] and Modified R-L-definitions [16, 17].
• Grunwald-Letinikov definition
Let be any function then the
-th order derivative
of
is defined by
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Where is any arbitrary number real or complex; and the generalized binomial coefficients are described as follows [1, 16, 17]
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The above formula becomes fractional order integration if we replace by
which is Riemann fractional integration formula
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In above we have noted several notations used for fractional integration.
• Riemann-Liouville fractional derivative definition
Let the function is one time integrable then the integro-differential expression as following defines Riemann-Liouvelli fractional derivative [1, 6]
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Here the is a positive integer number just greater than real number
The above expression is known as the Riemann-Liouville definition of fractional derivative [6] with
In the above definition fractional derivative of a constant is non-zero.
• Modified Riemann-Liouville definition
To overcome the shortcoming fractional derivative of a constant, as non-zero, another modification of the definition of left R-L type fractional derivative of the functionwas proposed by Jumarie [16] in the form described below
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Here we state that for
and
. However in this paper we will be using this left-Jumarie fractional derivative that is
, for
and with condition
for all
. We will simplify the symbol and drop
and differentiating variable
and simply write
.Using the above definition Jumarie [16] proved the following
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We have recently modified the right R-L definition of fractional derivative of the function
in the following form [17],
![]() |
Using both the modified definition we investigate the characteristics of the non-differentiable points of some continuous functions in [17]. The above defined all the derivatives are non-local type, and obtained solution to homogeneous FDE, with Jumarie derivative [15]. Subsequently we will be using as fractional derivative operator of Jumarie type, with start point
, and stating the function
for all
in following sections.
The Mittag-Leffler function was introduced by Gösta Mittag-Leffler [18] in 1903. The one-parameter Mittag-Leffler function is denoted by and defined by following series
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Again from the Jumarie definition of fractional derivative we have we apply this property to get
order Jumarie Derivative of the Mittag-Leffler function
as follows
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Therefore the fractional differential equation has solution in the form
where A is an arbitrary constant.
The general format of the fractional linear differential equation is
![]() | (2.1) |
Where , of Jumarie type. The above differential equation is said to be linear non-homogeneous fractional differential equation when
, otherwise it is homogeneous. Solution of the linear fractional differential equations (composed via Jumarie derivative) can be easily obtained in terms of Mittag-Leffler function and fractional sine and cosine functions [15].
The function is forcing function. We have written this as function of
purposely for ease. For example we will use in this paper
,
,
etc. are taken as forcing functions. There will be other functions in the derivations like
,
all functions described with scaled variable that is
. Nevertheless the forcing functions can be written as simple
though.
In that paper [15] we found the following (theorems) which we will be using in this paper
(i) The fractional differential equation has solution of the form
where A and B are constants,
(ii) The fractional differential equation has solution of the form
where A and B are constants and
(iii) Solution of the fractional differential equation is of the form
where A and B are constants.
From now we indicate Jumarie fractional derivative with start point of differentiation as as
instead
.
Theorem 1: If and
are two solutions of the fractional differential equation
then
is also a solution, where c1 and c2 are arbitrary constants.
Proof: Since has solutions
and
![]() |
Hence is also a solution of the given fractional differential equation.
Hence the theorem is proved.
Similarly, we can prove if are solutions of the fractional differential equation
then
is also a solution of it.
Theorem 2: If
![]() |
then solution of the homogeneous equation is
where Ak’s are arbitrary constants and all
are distinct.
Proof: Since Jumarie type fractional derivative of Mittag-Leffler function with a as a constant is
Thus solution of the differential equation
is
where A is a constant [15].
Let be a non-trivial trial solution of the differential equation
then
or we write the following after subtracting
from both the sides as demonstrated below
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We apply the above result sequentially as demonstrated below
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Since we get
![]() | (2.2) |
Implying that
Hence the general solution is
![]() |
Hence the theorem is proved.
The above theorem implies principal of superposition holds for the linear fractional differential equations (composed via Jumarie fractional derivative) also.
Note: In the above theorem if two or more roots of the equation (2.2) are equal or roots are complex then the solution [15] form is given below.
Forand
then solution of the is
![]() |
For and
then the solution is
![]() |
where Ak’s are arbitrary constants.
For and other are
then the solution is
![]() |
Thus solutions of linear homogeneous fractional differential equation with Jumarie fractional derivative is express in terms of Mittag-Leffler functions and fractional type sine and cosine series.
Now the question arises what will be solution of linear non-homogeneous fractional differential equations. The solution corresponding to the homogeneous equation will be called as the complementary function, it contains the arbitrary constants and this solution will be denoted by. The other part, that is a solution which is free from integral constant, and depending on the forcing function will be called as Particular Integral (PI) and will be denoted by
. Thus the general solution will be
. We will develop simple method to evaluate Particular Integral.
3.
order Non-Homogeneous Fractional Differential Equations
Consider the linear order non-homogeneous fractional differential equation with
for
for
of the following form,
![]() | (3.1) |
The solution of the corresponding homogeneous part is [15]
![]() |
Multiply both side of equation (3.1) by as demonstrated below
![]() |
In the above steps we have used . Now operating
on both the sides of the obtained last expression in above derivation i.e.
![]() |
Also we add a constant since Jumarie type derivative of a constant is zero and from here we get the following
![]() |
![]() | (3.2) |
or
![]() |
the first part corresponds to solution of corresponding homogeneous equation, that is and the other part
corresponds to the effect of non-homogeneous part and free from integral constant, but depending on the nature of forcing function, this part is named as Particular Integral (PI) as in case of classical differential equations. Now we take several forms of forcing function.

Here consider the linear first order non-homogeneous fractional differential equation of order with
for
![]() |
then the particular integral (PI) described in the previous section is
![]() |
Putting in above we get the following
![]() |
For P.I. is
![]() |
• Short procedure for calculating Particular Integral for
This procedure is similar and in conjugation with classical integer order calculus. In classical order calculus . Hence the forced function reduce to
Therefore the particular integral will be
![]() |
Here we observe that the derivative operator is replaced by c in the first case, i.e. for
. In the second case the derivative operator
is replaced by
. We can replace the fractional Jumarie derivative operator
by
for the first case
and by
for second case
The short procedure as follows for Particular Integral that is,
![]() |
![]() |
Hence the general solution of equation (3.1) is
![]() |

Again when then the differential equation (3.1) becomes
![]() | (3.3) |
The solution of the homogeneous part [15] that is is
.
Let the solution of the corresponding non-homogeneous equation where
is an unknown function of
. Then using the definition by Jumarie [16] that is
![]() |
We get the following
![]() |
putting this in (3.3) we get
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Therefore we get
![]() |
We now apply fractional integration by parts by Jumarie formula [16] as depicted below
![]() |
Here we mention that the symbol implies Jumarie fractional integration as defined in section-2. We will use also [15] derived expression that is
![]() |
in the following derivation.
![]() |
Hence the general solution is
![]() |
the first part in above expression is solution of homogeneous equation and the second part of the above that is
is particular integral.
• Short procedure for Calculating Particular Integral for
This procedure is similar and in conjugation with classical integer order calculus. Here for and
, and the corresponding particular integral is
![]() |
In the same way we can have a short procedure as follows for Particular Integral that is,
![]() |
In the above derivation
![]() |
is used. Thus all the Jumarie derivatives for
, where
is Natural number. Therefore we have discussed the solutions of non-homogeneous
order differential equations for different forcing functions
.



can be factorized as
,and we use this in following derivation.
As in section 3.1 here we replace by
and by
for the operations
and
respectively, as is demonstrated below.
![]() |
![]() |
Therefore
![]() |
Similarly we get by following above procedure
![]() |
Thus to find the particular integral replace
.
This procedure is similar and in conjugation with classical integer order calculus. In classical order calculus hence the forced function reduce to
Therefore the particular integral will be
![]() |
![]() |
Therefore
![]() |
4.
order Non-Homogeneous Fractional differential equations
General formulation of non-homogeneous fractional differential equation of
![]() |
where p and q are constant here. Consider the order non-homogeneous fractional differential equation
![]() |
where then solution of the non-homogeneous part that is
given by
[15].

For we have the given equation is
![]() | (4.1) |
Here let the particular integral be where P is constant.
Then
![]() |
and putting in the given equation (4.1) we get the following
![]() |
Therefore
![]() |
and consequently the Particular integral is
![]() |
Hence the general solution is
![]() | (4.2) |
For then the solution (4.2) does not exists. In this case the fractional differential equation is
![]() | (4.3) |
If we consider in this case also then putting in (4.3) we get
which is free from P i.e. P is non-determinable. This form of PI is not suitable here; consider the modified form as following
![]() |
Then
![]() |
and putting in (4.3) we get the following
![]() |
In this case the general solution of the fractional differential equation is of following form
![]() |
When then
![]() | (4.4) |
and take the particular integral in the form
![]() |
then
![]() |
![]() |
Putting this in (4.4) and after simplification we get
![]() |
In this case the general solution will be of following form
![]() |
Thus we can summarize the result as a theorem in the following form
Theorem: The differential equation the has particular integral
![]() |
(i) When for
then solution of the fractional differential equation will be
![]() |
(ii) When for
then solution of the fractional differential equation will be
![]() |
(iii) When for
then solution of the fractional differential equation will be
![]() |

Using the direct method as describe in section 3.1 we can easily calculate the Particular integrals for different functional format of.
• For we have
![]() |
In this case the general solution is
![]() |
• For
![]() |
In this case the second part will be adjusted in the complementary function and hence the general solution is
![]() |
• For
![]() |
In this case the general solution will be
![]() |
• In generalized case for any polynomial type function the particular integral is
![]() |
For must contain a factor of the form
i.e.
![]() |
then
![]() |
• Again if Here using Leibnitz rule of fractional derivative on (Jumarie type fractional derivative) we get the following steps
![]() |
![]() |
![]() |
Thus in generalized case we have
![]() |
5. Solution the Fractional Differential-Application of Method Derived
Example 1: We take the following fractional differential equation for
![]() |
Solution: Solution of the corresponding homogeneous equation is [15]
![]() |
The particular integral calculation is done in following steps
![]() |
Hence the general solution is
![]() |
Example 2: Consider the fractional order forced differential equation for
![]() |
Solution: Here solution of the corresponding homogeneous equation is [15]
![]() |
The particular integral is
![]() |
Hence the general solution is
![]() |
For particular integral we replaced so
Example 3: Take the fractional order damped-forced differential equationfor
![]() |
Solution: Here solution of the corresponding homogeneous equation [15]
![]() |
is
![]() |
The particular integral is
![]() |
here replace
multiply the number pr and denominator by
![]() |
![]() |
Hence the general solution is
![]() |
6. Conclusions
In this paper we have developed a method to solve the linear fractional non-homogeneous fractional differential equations, composed by Jumarie type derivative. The solutions are obtained here in terms of Mittag-Leffler function and fractional sine fractional cosine functions. Here we have proved via usage of Jumarie fractional derivative operator that for obtaining the particular integrals for several forcing functions scaled in function of variable eases the method, and we obtain conjugation with classical method to solve classical non-homogeneous differential equations. The short cut rules, that are developed here in this paper to replace the operator
or operator
as were used in classical calculus, gives ease and advantage in evaluating particular integrals. These techniques obtained herein this paper is remarkable to study fractional dynamic systems, and eases to get solution in terms of Mittag-Leffler, and fractional-trigonometric functions as in conjugation with exponential and normal trigonometric function for normal integer order calculus. Therefore this developed method is useful and advantageous as it is having conjugation with the classical methods of solving non-homogeneous fractional linear differential equations composed via Jumarie fractional derivative, and is also useful in understanding physical systems described by FDE.
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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