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

Where is any arbitrary number real or complex; and the generalized binomial coefficients are described as follows ^{[1, 16, 17]}

The above formula becomes fractional order integration if we replace by which is Riemann fractional integration formula

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]}

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

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

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

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

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 equationthen 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 A_k’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 thenor we write the following after subtracting from both the sides as demonstrated below

We apply the above result sequentially as demonstrated below

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 A_k’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.

3.1. Particular Integral for

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 caseThe short procedure as follows for Particular Integral that is,

Hence the general solution of equation (3.1) is

3.2. Particular Integral for

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

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 isis 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.

3.3. Evaluation of where

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 andrespectively, 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].

4.1. Use of Method of Un-determinant Coefficient Method to Calculate the Particular Integrals for Different Functional Forms of

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 getwhich 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

4.2. Use of Direct Method to Calculate the Particular Integrals for Different Functional Format of

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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