Applications Of One Type Of Euler-Lagrange Fractional Differential Equation

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 09 Issue: 09 | Sep 2022 www.irjet.net p-ISSN: 2395-0072

Applications Of One Type Of Euler-Lagrange Fractional Differential Equation

Raharimina L.H.D.L1-Rasolomampiandry G.2-Randimbindrainibe F.3

1PhD Student SCA/ED-STII, Antananarivo University, Madagascar ; ramiarison@moov.mg 2Doctor SCA/ED-STII, Antananarivo University, Madagascar; rasologil@gmail.com 3Professor SCA/ED-STII, Antananarivo University, Madagascar;falimanana@mail.ru ***

Abstract - In this paper, our work is to present some applications of one type of Euler-Lagrange fractional differential equation with a composition of left fractional derivativeofRiemann-Liouvilleof α orderandrightfractional derivative of Caputo of α order ,01   . For these applications, we first considered some harmonic oscillators whose equations of motion are expressed in terms of derivatives of non-fractional order and then we transformed these equations into an equation with fractional derivatives. The expanded form of the fractional differential equation is obtainedusingthefinitedifferencemethodandthedefinitions ofthese two fractional derivatives andthen the translation in theformofamatrixequation, isalsofound.Withtheexamples we have taken, programming in the Matlab language is the method chosen to graphically represent the approximate numerical solutions of this type of equation.

Key Words: fractional derivatives, Euler-Lagrange differentialequation,expandedform, numericalsolutions, MatlabScript.

1.INTRODUCTION

TheEulerLagrangeequationcanbepresentedintwoforms, either in its ordinary form or in fractional form. In its fractionalform,theequationisacompositionoffractional derivatives.

Inordertosolvetheoscillationofaharmonicpendulum,the Euler Lagrange equation composed of the fractional derivativesofRiemann-LiouvilleandCaputoofαorderwas chosenforthiswork

1.1 Presentation of the fractional differential equation to be studied

In recent years, fractional calculations and fractional differential equations have interested many scientists to modeldifferentproblemsinthefieldsofphysics,science,etc. The type of equation that we are going to study, is a composition of left Riemann-Liouville and right Caputo fractional derivatives of α order. It is writing into the followingformEq.1.

0 ()()0 cDDftrft

Where:  0;1 t  ;

f isunknownfunctiontobedetermined;

r isagivenvalue;

01  

(1)

1 0 , cDD

are respectively the fractional derivativesofαorderontherightandontheleftof CaputoandofRiemann-Liouville.

1.2 Presentation of the expanded form and matrix notation of equation (1)

ConcerningthistypeofEuler-Lagrangefractionaldifferential equation, we sought the expanded form of this equation, using the method of finite differences of order 1 and the definitions of the two fractional derivatives. For this, we havesubdividedtheinterval  0;1 into N parts,ofregular pitch 1 h N  , the points are denoted it , 0 iN , with 0 0 t

and 1 Nt

For 11 iN  ,wehavenoted i i tih N  and if the approximatesolutionateachpoint it

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 09 Issue: 09 | Sep 2022 www.irjet.net p-ISSN: 2395-0072

(4) Theapproximatesolutionstobedeterminedunderthe MatlabScriptarethe ,0 i fiN 

2. Transformation of an equation of motion of a usual harmonic oscillator into a fractional harmonic oscillator equation

Werecallthatanequationofmotionofausual,undamped, harmonicoscillator,isanordinarysecond-orderdifferential equationoftheform: 2 "()()0ftft   (5)

Where ω iscalledproperpulsationofmotionand f isthe generalsolutionofthisequation.

We can cite some examples of the following harmonic oscillators:

• thehorizontalelasticpendulum:anobjectofmassµ attachedtoaspringofconstantstiffness k inmotionaround the its equilibrium position and the own pulsation is k    Itsequationofmotioniswritten: "()()0 k xtxt   where x(t) istheextensionofthe spring.

The general solution is of the form: ()sin() ftAt wherethevaluesof A and φ depend onthechosenboundaryconditions.

• theundampedtorsionpendulumwhichperformsa rotationalmovementarounditsequilibriumposition:itisa rotationalharmonicoscillator.

Theequationofmotionis: "()()0 c tt J    where J  isthemomentofinertiaof the solid with respect to the axis of rotation and is the momentoftherestoringtorqueofthetorsionwire.Theown pulsationis c J  

 .

• A circuit LC which is an electrical harmonic oscillatorformedofaperfectcoilwithinductance L anda capacitorwithcapacitance C .Itistheseatoffree,undamped electricaloscillations,ofproperpulsation 1 LC  

Thedifferentialequationgoverningthevariationof the load q inthiscircuit LC isthefollowing:

• Asimplependulum:anobjectofmass  suspended on a wire of length I of negligible mass. This system, in motion,oscillatesarounditsequilibriumpositionandforlow amplitude oscillations 010  , of proper pulsation g l   ,theequationofmotioniswritten: "0 g l  Ifwetransformalltheseordinarydifferential equationsof integer order 2, into an equation with compositions of fractionalderivativesoforder ,01 ,thenweobtain the followingtype ofEuler-Lagrangefractional differential equation:

This last equation is indeed a fractional Euler-Lagrange differentialequationoftheform:

3. ProgrammingunderMatlabandrepresentations of curves for fractional and non-fractional MatlabScriptisthemethodofnumericalresolutionthatwe havechosen.

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functionf=varnalpha(N,alpha)

%N=input('valeurN=') %alpha=input('valeuralpha=') c=zeros(N,N+1); %Calculdec %************************** forj=1:N fork=0:j ifk==0

ta=1/gamma(2-alpha); tb=(1-alpha)*j^(-alpha); tc=(j)^(1-alpha); td=(j-1)^(1-alpha); c(j,k+1)=ta*(tb-tc+td); elseif k<j ta=1/gamma(2-alpha); tb=(j-k+1)^(1-alpha); tc=(j-k)^(1-alpha); td=(j-k-1)^(1-alpha); c(j,k+1)=ta*(tb-2*tc+td); else c(j,k+1)=1/gamma(2-alpha); end end end

%Calculded %********************* d=zeros(N,N); fori=1:N-1 forj=i:N ifj==i&&j~=N d(i,j)=1/gamma(2-alpha); elseif j<N tad=1/gamma(2-alpha); tbd=(j-i+1)^(1-alpha); tcd=(j-i)^(1-alpha); tdd=(j-i-1)^(1-alpha); d(i,j)=tad*(tbd-2*tcd+tdd); else t1d=(N-i-1)^(1-alpha); t2d=(N-i)^(1-alpha); d(i,j)=(t1d-t2d)/gamma(2-alpha); end end end

%Matricedusystèmed'équation %iallantde1àN-1(profondeur) %jindiceligne %kindicecolonne b=1; a=zeros(N+1,N+1); ce=(b/N)^(-2*alpha) r=-0.75/0.50; fori=1:N-1 fork=0:N temp=0;

ifk<i forj=i:N temp=temp+d(i,j)*c(j,k+1); end a(i+1,k+1)=ce*temp; elseifk==i forj=i:N temp=temp+d(i,j)*c(j,k+1); end a(i+1,k+1)=ce*temp+r; else forj=k:N temp=temp+d(i,j)*c(j,k+1); end a(i+1,k+1)=ce*temp; end end end

%Remplissagei=1eti=N+1 %*************************** cc=1; bb=zeros(N+1,1); fori=2:N bb(i,1)=0; end a(1,1)=1; a(N+1,N+1)=1; bb(1,1)=0; bb(N+1,1)=1; f=a\bb;

%calculf=varnalpha(N,alpha);affiche(N,f,color) figure gridminor holdon xlabel('t') ylabel('f(t)') N=1000;

f0=varnalpha(N,.5); affiche(N,f0,'r')

f1=varnalpha(N,.6); affiche(N,f1,'g') f2=varnalpha(N,.8); affiche(N,f2,'b') f3=varnalpha(N,.9); affiche(N,f3,'y') f4=varnalpha(N,1); affiche(N,f4,'c')

%title('\alpha=0,9') title('N=1000') legend('\alpha=0,5','\alpha=0,6','\alpha=0.8','\alpha=0.9',' \alpha=1')

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Wehavetaken k r   inthecaseoftheelasticpendulum forfigure1;and 1 r LC  inthecaseofthecircuit LC for figure2.

Forthefollowinggraphicalrepresentations,letustakethe sameparametersforthenon-fractionalharmonicoscillators, of order 1   and the fractional harmonic oscillators of order ,01 , with the boundary conditions (0)0,(1)1ff (Fig-3)

Fig.3 : Graphicalresult

InFig-3,     1 ()sin1.5 sin1.5 ftt  isthesolutionfor 1   , 0.75,0.50 k   .

Using this programming, we were able to represent these curves. We chose 1000 N  and we took different fractional  values

Fig.1 : Numericalsolutionsfor 0.75 k  and 0.50   for thecaseofthehorizontalelasticpendulum

Wenoticethatthereisadifferencebetweenthegraphical representations in the case where 1   , non-fractional order,andthecaseforthedifferentvaluesof ,01

Itseemsthatthecurvesgetcloserwhentheorder  ,ofthe fractionalderivative,iscloseto1.

4. Conclusion

Duringanexperiment,iftherearevaluesoftheparameters suchthatthecurveapproachesthecurveofthefractional case,itisbetternottobesatisfiedwiththenon-fractional harmonicoscillatorbutitisnecessarytoswitchtothetype offractionaldifferentialequationofEuler-Lagrangethatwe propose.

Our perspective is to continue our research to find other models of fractional differential equations and then apply theminvariousfieldssuchasscience,engineering,etc.

REFERENCES

[1] FrancoisDubois-Ana-Cristina-Galucio:Introductionàla dérivation fractionnaire. Theories et Applications (2009)

Fig.2 : Numericalsolutionsfor 0.1 L  and 100 C  for thecaseofthecircuitLC

[2] Riewe F.: Mechanics with fractional derivatives, Phys. Rev.E.1997

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International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 09 Issue: 09 | Sep 2022 www.irjet.net p-ISSN: 2395-0072

[3] Baleanu D.,Trujillo J.J.: On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dyn.(2008).

[4] Klimek M.: Solutions of Euler-Lagrange equations in fractional mechanics.AIP Conference Proceedings 956.XXVIWorkshoponGeometricalMethodsinPhysics, Eds.P.Kielanowski,A.Odzijewicz,M.Schlichenmaier,T. Voronov,Bialowieza2007

[5] Q,. Yang, F. Liu , I. Turner: Numericals methods for fractional partial differential equations with Riesz space fractionalderivatives.SchoolofMathematicalSciences; UniversityofTechnologyAustralia-2009.

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