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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

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Volume 19, Issue 6


Hybrid Method for Solution of Fractional Order Linear Differential Equation with Variable Coefficients

Amit Ujlayan / Ajay Dixit
Published Online: 2018-07-24 | DOI: https://doi.org/10.1515/ijnsns-2017-0167


In this paper, we proposed a new analytical hybrid methods for the solution of conformable fractional differential equations (CFDE), which are based on the recently proposed conformable fractional derivative (CFD) in R. Khalil, M. Al Horani, A. Yusuf and M. Sababhed, A New definition of fractional derivative, J. Comput. Appl. 264 (2014). Moreover, we use the method of variation of parameters and reduction of order based on CFD, for the CFDE. Furthermore, to show the efficiency of the proposed analytical hybrid method, some examples are also presented.

Keywords: fractional derivative; conformable fractional derivative; method of variation of parameters; method of reduction of order

MSC 2010: 34A08; 26A33

1 Introduction

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].

To study the brief history of fractional calculus, we refer to the reader [1, 6, 7] and for more details about conformable.

Conformable fractional derivative (CFD) by R. Khalil [8]: For a given g:[0,)R, conformable fractional derivative of order α is defined as Dα(g)(t)=limϵ0g(t+ϵt1α)g(t)ϵ;t>0,0<α1.(1)

If g is α-differentiable in some (0,a),a > 0, for t > 0,α∈(0,1) and limt0+g(α)(t)exist1 then we define gα(0)=limt0+g(α)(t)

Fractional derivative (FD) by Udita Katugampola [9] : For a given g:[0,)R, Katugampola fractional derivative of order α is defined as Dα(g)(t)=limϵ0g(teϵtα)g(t)ϵ;t>0,0<α1.(2)

Fractional derivative at origin defines as above.

Some basic properties of Katugampola fractional derivative [9]

  • Dα[af+bg]=aDα[f]+bDα[g](Linearity)

  • Dα[fg)]=fDα[g]+gDα[f](Productrule)

  • Dα[fg]=Dα[f(g)]Dα[g](Chainrule)

  • Dα[f(t)]=t1αf(t)

where f is differentiable function and α∈(0,1]. For more details of the properties, we refer to the reader [8, 9]

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: (DαDα+αPDα+α2Q)y=R,

where Dαy=dαydxα=x1αdydx;0<α1

and P,Q,R are continuous functions of x in [0,1].

Now consider the following cases

  • Case I:

    If y=emxα satisfies the homogeneous part, i.e.

    DαDα+αPDαy+α2Qy=0 We get m2+Pm+Q=0emxα is the one part of complementary function.

  • Case II:

    If y=xmα satisfies the homogeneous part, then

    m(m1)+Pmx+Qx2=0xmα 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 dαydxα=vdαudxα+dαvdxαu.

Putting these values of derivatives in our equation, we get vdαdxαdαudxα+dαdxαdαvdxαu+2dαudxαdαvdxα+αPvdαudxα+dαvdxαu+α2Quv=R u(DαDαv+αPDαv)+2uDαuDαv=R, DαDαv+αPDαv+2DαuDαv=R/u DαDαv+αP+2uDαuDαv=R/u, Dαv=qDαq+αP+2uDαuq=R/u,

and integrating factor (I.F) I.F.=eααP+2uuαdxα=Iα, q.Iα=αRuIαdxα+c1dαvdxα=1IααRuIαdxα+1Iαc1, v=α1IααRuIαdxα+1Iαc1dxα+c2,

and so y=uv=α1IααRuIαdxα+1Iαc1+c2u.

where Iα=eααP+2uuαdxα=eααPdxαexα12x1αududxdx =eααPdxαe2logu=u2eααPdxα

using the definition of fractional integral αg(x)dxα=xα1g(x)dx.

Hence, from above, it is clear that if one part of complementary function u is known, then the other part will be uαu2eααPdxαdxα

3 Method of variation of parameter for conformable fractional differential equation

Consider the following second-order linear fractional differential equation (DαDα+αPDα+α2Q)y=R(x),(3)

where Dαy=yα=dαydxα=x1αdydx;0<α1

and P,Q,R are continuous functions of x in in[0,1] Let its complementary function of (3). y=au+bv, where u,v are two independent solution of homogeneous part, then we have (DαDα+αPDα+α2Q)u=0,(DαDα+αPDα+α2Q)v=0.

Let y=Au+Bv , where A,B are two undetermined functions, be the complete solution of our equation. Now to determine A,B, we chose an independent condition Aαu+vBα=0,

since y=Au+Bvyα=Auα+Bvα+(uAα+vBα)=Auα+Bvα.

Also, Dαyα=ADαuα+Bαvα+uαAα+DαvαB,

putting these values in (3) ADαuα+Bαvα+uαAα+DαvαB+αP(Auα+Bvα)+α2Q(au+bv)=RA(Dαuα+αPuα+α2Qu)+B(Dαvα+αPvα+α2Qv)+Auα+Bvα=RAαuα+Bαvα=R.

Solving with the independent condition, we get Aα=dαAdxα=vRuvαvuα=vRWα

And Bα=dαAdxα=uRuvαvuα=uRWα,

where, Wα=uvuαvα=uvαvuα

called Wronskian for conformable fractional differential equations. Integrating we get, A=X+c1,B=Y+c2

where X=αvRWαdxα

and Y=αuRWαdxα

and αf(x)dxα=f(x)xα1dx= anti-derivative of conformable fractional derivative, hence the complete solution y=c1u+c2v+Xu+Yv

4 Applications

Example 4.1:

Consider the following FDE 4xD1/2D1/2z2(x+2x)D1/2z+(x+2)z=4xxex,

we have P=(1+2/x),Q=1x+2x,R=xex And P+Qx1/2=0 obviously, so complete solution will be given by z = uv, where u=x is known function.

Now to determine v we have the following differential equation D1/2D1/2v+12P+2uD1/2uD1/2v=R/u, D1/2D1/2v+121x+2xD1/2xD1/2v=ex, D1/2D1/2v+121x+1xD1/2v=ex,

taking q=D1/2v, we get d1/2qdx1/212q=exI.F=e1/212dx1/2=e12xdx=ex,

So qex=1/2exexdx1/2+aq=d1/2vdx1/2=ex1xdx+aex,

And therefore, v=ex2x+a1xdx+b=2exdx+2ax+b

Hence, z=(2exdx+2ax+b)ex

where a,b are constants.

Example 4.2:

Consider the following FDE D1/3y1/313cotx1/3y1/319(1cotx1/3)y=ex1/3sinx1/3.

Here we have P=cotx1/3,Q=cotx1/31,R=ex1/3sinx1/3 1+P+Q=0ex1/3

The one part of C.F is ex1/3, Then now to determine v we have u=ex1/3d1/3dx1/3d1/3vdx1/3+13P+2ud1/3udx1/3d1/3vdx1/3=sinx1/3, d1/3dx1/3d1/3vdx1/3+13cotx1/3+2ex1/3d1/3udx1/3d1/3vdx1/3=sinx1/3, d1/3dx1/3d1/3vdx1/3+13cotx1/3+2ex1/3x2/3dudxd1/3vdx1/3=sinx1/3, d1/3qdx1/3+13cotx1/3+23q=sinx1/3,q=d1/3vdx1/3, I.F.=e1/3(2cotx1/3)3dx1/3=e(2cotx1/3)3x2/3dx=e2x1/3sinx1/3, qe2x1/3sinx1/3=3e2x1/32+cq=d1/3vdx1/3=3sinx1/32+ce2x1/3sinx1/3 v=3sinx1/32+ce2x1/3sinx1/3x2/3dx+c =9cosx1/3235ce2x1/3(2sinx1/3+cosx1/3)+c

Hence, y=9cosx1/3235ce2x1/3(2sinx1/3+cosx1/3)+cex1/3

Example 4.3:

Consider the following fractional differential equation DαDαy3αDαy+2α2y=exα1+exα,0<α1

Let us consider y=emxα as a solution, then auxiliary equation is m23m+2=0m=1,2.

So C.F=c1exα+c2e2xα.

Take u=exαv=e2xα

Then Wronskian, Wα=uvuαvα=exαe2xααexα2αe2xα=αe3xα P.I=Xu+Yv

where X=αvRWαdxα=αe2xαexααe3xα(1+exα)dxα=xα1α(1+exα)dx; xα=txα1dx=dxαetα2(1+et)dt=1α2log(1+exα)

and Y=αuRWαdxα=αexαexααe3xα(1+exα)dxα=xα1αexα(1+exα)dx =1α2et(1+et)dt=1α2logexα1+exα

Hence, the complete solution y=c1exα+c2e2xα+1α2log(1+exα)exα+1α2logexα1+exαe2xα

Example 4.4:

Solve the following FDE (1xβ)Dβyβ+βxβyββ2y=2(xβ1)2exβ;0<β1.

We have Dβyβ+βxβ1xβyββ211xβy=2(1xβ)exβ

so that P=xβ1xβ,Q=11xβ,R=21xβexβ.

Now, 1+P+Q=0exβ

is the one part of the complementary function and P+Qxβ=0xβ will be second independent solution and so complementary function is c1exβ+c2xβ

Set u=exβ,v=xβ, so that Wronskian Wβ=uvuβvβ=exβxββexββ=β(1xβ)exβ

and for particular integral, yp=Xu+Yv X=βvRWβdxβ=2(1xβ)xβ1xββe2xβ(1xβ)dx=2xβxβ1e2xβdx xβ=txβ1dx=dtβ=2te2tdt=1βe2xβ12+xβ

and Y=βuRWβdxβ=2(1xβ)xβ1exββe2xβ(1xβ)dx=2xβ1exβdx=2exββ

Therefore, the complete solution y=(c1+X)exβ+(c2+Y)xβ

Example 4.5:

Consider the following FDE D1/2D1/2y+y=tan2x.

Let us assume that y=emx1/2as a solution of homogeneous part, then the corresponding

A.E. is m2+4=0m=±2i C.F=asin2x+bcos2x,

choose u=sin2x,v=cos2x W1/2=uvu1/2v1/2=cos2xsin2xsin2xcos2x=1 X=αvRWαdxα=1/2tan2xcos2x1dx1/2=x1/2sin2xdx=cos2x,

and Y=αuRWαdxα=1/2tan2xsin2x1dx1/2 =x1/2sin22xcos2xdx ={sin2xlog(sec2x+tan2x)},

hence, the complete solution is y=asin2x+bcos2xlog(sec2x+tan2x)cos2x

5 Conclusion

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

Received: 2017-08-06

Accepted: 2018-05-20

Published Online: 2018-07-24

Published in Print: 2018-09-25

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 6, Pages 621–626, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0167.

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