While the modeling of the real world problems it have been accepted by many mathematicians that the concept of fractional order derivatives the generalization of classical derivative gives better prediction.
Many definitions of fractional derivative have been introduced by many mathematicians like Riemann-Liouville, Caputo, Laplace, Grnewald-Letnicov, etc. and also definitions have been modified time to time as per requirement, but one could not get a definition which satisfies even the fundamental properties like product rule, quotient rule, chain rule, Rolles theorem, commutative properties of derivative, etc. Even computation of these defined derivatives is too complicated to evaluate. Besides this, there is not a suitable analytic method to solve linear fractional second-order or higher-order differential equations; in fact, one has to use either numerical method or Mittag–Leffler function (series solution) or fractional Laplace transform for fractional initial value problem. It has motivated the authors to try for an analytical method to solve fractional differential equations.
The concept of the conformable fractional derivative has taken place and given by R. Khalil and colleagues which has later generalized by Katugampola. It satisfies most of the properties of derivative which are not fulfilled by previously defined fractional derivatives. Although it has some assumptions of being a differentiable function and restrictions like x > 0, still it is easy to apply and convert a fractional derivative to a classical order derivative. A number of applications [1, 2, 3] as well as various properties [4, 5, 11] have made an interest to researchers to investigate more in this field. We have presented a method of variation of parameters and method of reduction of order to solve a fractional order differential equation using Katugampola’s fractional derivative fractional derivative and its properties We refer [8, 9, 10, 11].
Conformable fractional derivative (CFD) by R. Khalil : For a given conformable fractional derivative of order α is defined as (1)
If g is α-differentiable in some (0,a),a > 0, for t > 0,α∈(0,1) and then we define
Fractional derivative (FD) by Udita Katugampola  : For a given Katugampola fractional derivative of order α is defined as (2)
Fractional derivative at origin defines as above.
Some basic properties of Katugampola fractional derivative 
The presentation in rest of the paper is organized as follows. We introduce the method of reduction of order for conformable fractional differential equation in Section 2 and variation of parameters method with the Katugampola’s fractional order derivative in Section 3. As an applications, we give various examples of the conformable fractional differential equations Section 4. Finally, we conclude the results and give some discussions about it in the Section 5.
2 Method of reduction of order for conformable fractional differential equation
Consider the following linear fractional differential equation:
and P,Q,R are continuous functions of x in [0,1].
Now consider the following cases
- Case I:
If satisfies the homogeneous part, i.e.
We get is the one part of complementary function.
- Case II:
If satisfies the homogeneous part, then
is the one part of the complementary function. Let y = uv be the complete solution of in which u is the solution of homogeneous part, i.e
then we have
Putting these values of derivatives in our equation, we get
and integrating factor (I.F)
using the definition of fractional integral
Hence, from above, it is clear that if one part of complementary function u is known, then the other part will be
3 Method of variation of parameter for conformable fractional differential equation
Consider the following second-order linear fractional differential equation (3)
and P,Q,R are continuous functions of x in in[0,1] Let its complementary function of (3). where u,v are two independent solution of homogeneous part, then we have
Let , where A,B are two undetermined functions, be the complete solution of our equation. Now to determine A,B, we chose an independent condition
putting these values in (3)
Solving with the independent condition, we get
called Wronskian for conformable fractional differential equations. Integrating we get,
and anti-derivative of conformable fractional derivative, hence the complete solution
Consider the following FDE
we have And obviously, so complete solution will be given by z = uv, where is known function.
Now to determine v we have the following differential equation
taking we get
where a,b are constants.
Consider the following FDE
Here we have
The one part of C.F is Then now to determine v we have
Consider the following fractional differential equation
Let us consider as a solution, then auxiliary equation is
Hence, the complete solution
Solve the following FDE
is the one part of the complementary function and will be second independent solution and so complementary function is
Set so that Wronskian
and for particular integral,
Therefore, the complete solution
Consider the following FDE
Let us assume that as a solution of homogeneous part, then the corresponding
hence, the complete solution is
The important properties of the recently developed Udita N.Katugampola fractional derivative were used. The major advantage of the Katugampola’s fractional derivative is that it is limit based rather than defined via a fractional integral as Riemann–Liouville and Caputo derivatives. Moreover, a key property of the Katugampola derivative along with reduction of order and variation of parameters is used for solution of FDE. Using this hybrid approach, we solve various types of CFDE with variable coefficients. One remarkable and interesting fact is that the solution will coincide with the solution of classical differential equation at α = 1.
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About the article
Published Online: 2018-07-24
Published in Print: 2018-09-25