Numerical Solution of Fractional Variational Problems Using Direct Haar Wavelet Method

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1 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 Numerical Soluion of Fracional Variaional Problems Using Direc Haar Wavele Mehod Osama H. M., Fadhel S. F., Zaid A. M. 3 Deparmen of Mahemaics and Compuer Applicaions, College of Science, Al-Nahrain Universiy, Baghdad, Iraq. Deparmen of Mahemaics and Compuer Applicaions, College of Science, Al-Nahrain Universiy, Baghdad, Iraq. Economic and Adminisraion College, he Iraqi Universiy, Baghdad, Iraq 3. ABSRAC: his paper presens a clear procedure for he fracional variaional soluion via Haar wavele echnique. he fracional derivaive is defined in he Riemann-Liouville sense. he fracional variaional problem is solved by means of he direc mehod using he Haar wavele and he problem will be reduced o he soluion of an algebraic equaions. he numerical soluion for he class of problem considered can be obained direcly from he funcional and here is no need o solve he fracional Euler-Lagrange equaion. he examples are included in order o demonsrae he validiy and applicabiliy of he suggesed approach. KEYWORDS: Haar wavele mehod, Fracional calculus, Calculus of variaion, Fracional calculus of variaion. I. INRODUCION he use of fracional calculus of modeling physical sysem has been widely considered in he las decades, []. Alhough, he concep of he fracional derivaives was inroduced already in he middle of he 9 h cenury by Riemann and Liouville, []. he firs work, devoed exclusively o he subjec of fracional calculus, is he book by Oldham and Spanier [3] published in 974. Afer ha, he number of publicaions abou he fracional calculus has rapidly increased. he reason for his is ha he same physical processes as a anomalous diffusion, complex viscoelasisiy, behaviour of meharonic and biological sysems, rheology, ec. can be described adequaely by classical models, []. A fracional calculus of variaions problem is a subopic of fracional calculus and i is a problem in which eiher he objecive funcional or he consrain equaion or boh conain a leas one fracional derivaive erm, [4]. his occurs naurally in many problems of physics, mechanics and engineering in order o provide more accurae models of physical phenomena (see [5-3]), However, he fracional calculus of variaions is a new field, ;Is saring poin appear o be he references [4], [5] where Riewe developed he nonconcenraive Lagrangian, Hamilonian, and oher conceps of classical mechanics using fracional derivaive, [6]. Paricularly, a fracional calculus of variaion concerns he variaional principles on funcionals involving fracional derivaive as we menion above and his leads o he saemen of fracional Euler-Lagrange equaions (see [4], [4], [7]). Fracional Euler-Lagrange equaions are difficul o solve explicily and consequenly i is of ineres o develop efficien numerical schemes for such dynamical sysems. In his paper, we shall use he direc Haar wavele mehod for a class of fracional variaional problems. Haar wavele heory has been innovaed and applied o various fields in engineering ([8]-[5]), and have proved o be a wonderful mahemaical ool. he idea of his paper is o inroduce Haar waveles, hen presen a direc mehod for solving fracional problems via Haar waveles. he procedure begins by assuming he admissible funcions by Haar waveles wih coefficiens o be deermined, hen esablishing an operaional marix for performing inegraion and finding he necessary condiion for exerimizaion, solving he resuling algebraic equaion yields he Haar coefficiens. his indicaes ha for he class of problems ha will be considered, he numerical soluion can be obained direcly from he funcional, and here is no need o solve he fracional Euler-Lagrange equaions. Copyrigh o IJIRSE 74

2 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 II. FRACIONAL DERIVAIVE AND INEGRAION In his secion, we shall review he basic definiions and properies of fracional inegral and derivaive, which are used furher in his paper, []. Definiion (): he Riemann-Liouville fracional inegral operaor of order >, is defined as: I f() ( ) ( ) ( ) x f x dx, >, x > I f() f(). Definiion (): he Riemann-Liouville fracional derivaive operaor of order >, is defined as: n D d n f() ( ) ( ) ( ) n x f x dx n d where n is an ineger and n < n. Definiion (3): he Capuo fracional derivaive operaor of order, is defined as: c D f() ( ) ( ) ( ) n n d x f x dx n n dx where n is an ineger and n < n. Capuo fracional derivaive has an useful propery: I c D n k ( k) f() f() f ( ) k k! where n is an ineger and n < n. III. HAAR WAVELE he orhogonal se of Haar waveles h i () is a group of square waves wih magniude of ± in cerain inervals and zeros elsewhere. he orhogonal basis {h n ()} of Haar waveles for he Hilber space L [,] consiss of: h n () h ( j k/ j ), n j + k, j, k j, k where: h (), <,.5 h (),.5 each Haar wavele h n has he suppor ( j k, j (k+)), so ha i is zero elsewhere in he inerval [,). Ineresingly, as n inverses, he Haar waveles become more and more localized. herefore, {h n+ ()} form m-local basis. Any funcion f() L [,) can be expanded in Haar series (see []), as: f() cihi ( ), n j + k, j, k () i where he Haar coefficiens c i, i,, ; are given by: c i j f ( ) hi ( ) d Copyrigh o IJIRSE 743

3 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 which are deermined such ha he following inegral square error e is minimized: m e f ( ) cihi ( ) d, m j, j {} i by applying he orhogonal propery of Haar wavele: j, i h ( ) hi ( ) d, i he series in eq.() conains an infinie number of erms. If f() is piecewise consan or may be approximaed as a piecewise consan, hen he sum in eq.() may be deermined afer m-erms: m f() cihi ( ) Cm H m () f ˆ( ) () i where m j, he superscrip indicaes he ransposiion, f ˆ( ) denoes he runcaed sum. he Haar coefficien vecor C m and he Haar funcion vecor H m () are defined as: C m [c, c,, c m ] (3) H m () [h (), h (),, h m ()] aking he collecion poins as follows: (i ) i, i,,, m m we define he m-sequence Haar marix mm as: 3 (m ) mm... H m Hm Hm m m m Correspondingly, we have: ˆ ˆ ˆ 3 ˆ (m )... fm f f f Cm m m m mm Because of he m-square marix mm is an inverible marix, he Haar coefficien vecor C m fˆm mm. IV. OPERAIONAL MARIX OF HE FRACIONAL ORDER INEGRAION C m can be goen by: he inegraion of H m () defined as in eq.(3) can be approximaed by Haar series wih Haar coefficiens P: Hm ( ) d P mm H m () where he m-square marix P is called he Haar wavele operaional marix of inegraion, [6]. Our purpose is o derive he Haar wavele operaional marix of he fracional order inegraion. For his purpose, we recall he definiion () of fracional order inegraion, which is named as Riemann-Liouville fracional inegraion, as following: (I f)() where and f(). ( ) ( ) ( ) f d ( ) *f() is he order of inegraion, () is he gamma funcion and *f() denoes he convoluion produc of Copyrigh o IJIRSE 744

4 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 Now, if f() is expanded in Haar funcions, as shown in eq.(), he Riemann-Liouville fracional order inegraion becomes: (I f)() ( ) *f() C m ( ) { *H m ()} hen if *f() can be inegraed and expanded in Haar funcions, he Reimann-Liouville fracional order inegraion is solved via he Haar funcions. Also, we define an m-se Block Pulse Funcions (BPF) as:, i i b i () m m, oherwise where i,,, m. he funcions b i () are adjoin and orhogonal, ha is:, i b i () b ( ) bi ( ), i, i bi ( ) b ( ) d, m i Because he Haar funcions are piecewise consan, i may be expanded ino m-erm block pulse funcions (BPF) as: H m () mm B m () (4) where B m () [b () b () b i () b m ()]. he block pulse operaional marix of he fracional order inegraion F is defined as follows: (I B m )() F B m () (5) where: m m F m3 m ( ) wih k (k + ) + k + + (k ) +. Nex, we derive he operaional marix of he fracional order inegraion se. mm (I H m )() P H m () (6) where he m-square marix P mm is called he Haar wavele operaional marix of he fracional order inegraion. Using eqs.(4) and (5), we have: (I H m )() (I mm B m )() mm (I B m )() mm F B m () (7) from eqs.(6) and (7), we ge: P mm H m () P mm mm B m () mm F B m () hen, he Haar wavele operaional marix of fracional order of inegraion mm m m P mm F P mm is given by: V. HE APPROACH In his paper, we shall consider he problem of exerimizaion of a funcional J of he form: J[y() ] F [, y( ), D y( )] d (8) Copyrigh o IJIRSE 745

5 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 saisfying he condiion y( ) y,and y( ) is considered o be undeermined where D y( x ) is he Riemann-Liouville fracional derivaive. he regular mehod for solving problem (8) is hrough he Euler equaion [7]: F c F D y D y and F D y c where D is he Capuo fracional derivaive. his paper mainly uses Haar waveles o esablish he direc mehod for fracional variaional problems. Unlike oher direc mehods, beginning wih he assumpion of he variable iself, he mehod we have saed here is like he mehod used by [8] by assuming D y( ) as Haar waveles whose coefficiens are o be deermined: D y( x ) cihi ( ) (9) i aking finie erms as an approximaion, we have: m D y( x ) cihi ( ) = Cm H m () () i Applying I o he boh sides of eq.(), hus y() can be expressed as: y() C m P mm H m () + y( ) () he oher erms in he funcional of eq.(8) are known funcions of he independen variable and can be expanded ino Haar waveles hrough subsiuion, and finally we have: J J(c, c,, c m ) () he original exremiaion of a fracional problem shown in eq.(8) becomes he exremiaion of funcional of a finie se of variables in eq.(). aking parial derivaives of J wih respec o c i, and seing hem equal o zero, we obain: J, i,,, m (3) c i solving for c i, and subsiuing ino eq.(), we have he desired resul. VI. ILLUSRAIVE EXAMPLES In his secion, we shall inroduce some examples in order o confirm he reliabiliy of he proposed mehod. Example (): Consider he funcional: J[y()] ( D y ( )) y ( ) d (4) and he boundary condiion: y() y and y() is unspecified (5) Consider ha < <, and for solving his problem by he direc Haar wavele mehod, we assume ha D y( ) can be expanded in erms of Haar wavele, as follows: m D y( ) cihi ( ) Cm H m () (6) i where: Copyrigh o IJIRSE 746

6 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 C m [c c c m ] H m () [h () h () h m ()] Here, we shall consider m 8 and more accurae resuls may be obained using large m. Now, upon aking he fracional Riemann-Liouville inegraion o he boh sides of eq.(6), hus we ge: y() C m P mm H m () + y (7) he oher condiion according o [7], ha we have is: F ( ) D y and according o our example, we ge: D y( ) which implies ha Cm H m (). herefore: c c c 3 c 7 and his gives c 7 c c c 3 (8) subsiuing eqs.(6), (7) and (8) in eq.(4) yields: J[ y( )] C ( ) ( ) ( ) m H m H m C m C m P m m H m y d C ( ) ( ) ( ) [... ] ( ) H m H m C m C m P m m H m y y y m m H m d herefore: J[y()] C m ( ) ( ) Hm Hm d C m C m P mm m ( ) H dx [y y y ] mm H m ( ) d (9) and i is ineresing o noe ha he definie inegral of h () from o is equal o, while he definie inegral of h, h,, h 7 are equal o zero for m 8, or: h ( ) hi ( ) d, i,,, 7 () Hence, upon using eqs.(8) and () and subsiuing ino eq.(9), we ge: J[y()] Cm K mm C m C m P mm [y y y ] mm () where: K mm Hm ( ) Hm ( ) d Copyrigh o IJIRSE 747

7 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 I I I44 4 If y, hen eq.() becomes: I mm m J[y] Cm K mm C m C m P mm Following able () give he approximae soluion of example () for differen values of and compares he resuls for wih he exac soluion which is given as: y(). ABLE Exac for Example (): Consider he funcional: J[y()] ( ) ( ) D y D y d () and he boundary condiions: y() and y() is unspecified (3) and for solving his example also we le: D y( ) Cm H m () (4) y() Cm P mm H m () + y() (5) here is a variable involved in eq.() explicily and i can be expanded ino Haar series over he inerval [,] Copyrigh o IJIRSE 748

8 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 dm H m () (6) Also, he oher condiion ha we have is: D y( ) which implies ha: Cm H m () and his gives: c 7 c c c 3 + (7) and subsiuing eqs.(4), (6) and (7) ino eq.(), we have: [ ( )] J y C ( ) ( ) mhm Hm C m C ( ) ( ) mhm Hm dm d J[y()] Cm K mm C m + Cm K mm d m (8) Following able () gives he approximae soluion of example () for differen values of and compares he resuls for wih he exac soluion, which was given in [8], as: y() 4 x ABLE Exac for VII. CONCLUSION AND DISCUSSION Direc Haar wavele mehod has been presened for a fracional variaional problem. he procedure considered in his paper can be considered as a generalizaion o he resuls given in [8]. From he illusraive examples, i ca be seen ha his operaional marix approach can obain accurae and saisfying resuls. All compuaional resuls are made by MALAB program. REFERENCES [] I. Podlubny, Fracional Differenial Equaions, Academic press, San Diego, 999. [] Ü. Lepik, Solving fracional inegral equaions by he Haar wavele mehod, Applied Mahemaics and Compuaion, vol.4, pp , 7. [3] K.B. Oldham and J. Spanier, he Fracional Calculus, Academic Press, New York, 974. [4] O. P. Agrawal, "Formulaion of Euler-Lagrange equaions for fracional variaional problems", J. Mah. Anal. Appl., Vol.7, No., pp ,. [5] R. A. El-Nabulsi and D. F. M. orres, Necessary opimaliy condiions for fracional acion-like inegrals of variaional calculus wih Riemann-Liouville derivaives of order (,), Mah. Mehods Appl. Sci., Vol.3, No.5, pp , 7. [6] G. S. F. Frederic and D. F. M. orres, A formulaion of Neoher s heorem of fracional problems of he calculus of vrariaions, J. Mah. Anal. Appl., Vol.334, No., pp , 7. [7] R. A. El-Nabulsi and D. F. M. orres, Fracional acion like variaional problems, J. Mah. Phys., Vol.49, No.5, pp.535-7, 8. Copyrigh o IJIRSE 749

9 ISSN: Engineering and echnology (An ISO 397: 7 Cerified Organizaion) Vol. 3, Issue 5, May 4 [8] G. S. F. Frederico and D. F. M. orres, Fracional conservaion law in opimal conrol heory, Nonlinear Dynam., Vol.53, No.3, pp.5-, 8. [9] P. Almedia, A. B. Malinowska and D. F. M. orres, A fracional calculus of variaion of muliple inegrals wih applicaions o vibraing sring, J. Mah. Phys., Vo.5, No.3, pp.3353-,. [] P. Almedia and D. F. M. orres, Leiwann s direc mehod for fracional opimizaion, Appl. Mah. Compue., Vol.7, No.3, pp ,. [] N. R. O. Bases, R. A. C. Ferreira and D. F. M. orres, Discree-ime fracional variaional problems, Signal Process., Vol.9, No.3, pp.53-54,. [] R. A. C. Ferreira and D. F. M. orres, Fracional h-difference equaion arising from he calculus of variaions, Appl. Anal. Discree Mah., Vol.5, doi:.98/aadm 3F,. [3] D. Mozyrska and D. F. M. orres, Modified opimal energy and iniial memory of fracional coninuous-ime linear sysems, Signal Process., Vol.9, No.3,. [4] F. Riewe, Non concenraive Lagrangian and Hamilonian mechanics, Phys. Rev., Vol.E53, No., pp , 996. [5] F. Riewe, Mechanic wih fracional derivaives, Phys. Rev., Vol.E55, No.3, pp , 997. [6] O. P. Agrawal, A general finie elemen formulaion for fracional variaional problems, J. Mah. Anal. Appl., Vol.337, pp. -, 8. [7] D. Baleanu and S. I. Muslih, Lagrangian formulaion of classical fields wihin Riemann-Liouville fracional derivaive, Physc. Scripa, Vol.7, No.-3, pp. 9-, 5. [8] G. Srang, Waveles and dilaion equaions: a brief inroducion, SIAM Rev., Vol.3, pp , 989. [9] I. Danbechies, he wavele ransform, ime-frequency locaion and signal analysis, IEEE rans. Inform. heory, Vol.36, pp. 96-5, 99. [] C. Chen and C. Hsiao, Haar wavele mehod for solving lumped and disribued-parameer sysems. IEEE Proc-Conrol heory Appl., Vol.44, No., pp , 997. [] C. H. Hsiao and W. J. Wang, Sae analysis and opimal conrol of linear ime varying sysems via Haar waveles, Opimal Conrol Appl. Mehods, Vol.9, pp , 998. [] C. F. Chen and C. H. Hsiao, Waveles approach o opimizing dynamic sysems, IEEE Proc. Conrol heory, Appl., P D46, pp. 3-9, 998. [3] C. H. Hsiao and W. J. Wang, Sae analysis and opimal conrol of ime varying discree sysems via Haar waveles, J. Opimizaion heory Appl., Vol.3, pp , 999. [4] C. H. Hsiao and W. J. Wang, Opimal conrol of linear ime-varying sysems via Haar waveles, J. Opimizaion heory Appl., Vol.3, pp , 999. [5] C. H. Hsiao and W. J.Wang, Sae analysis and parameer esimaion of bilinear sysems via Haar waveles, IEEE rans. Circuis Sys. In Foundaion heory Appl., Vol.47, pp. 46-,. [6] L. Yuanlu and Z. Weiwi, "Haar wavele operaional marix of a fracional order inegraion and is applicaions in solving he facional order differenial equaions''. Appl. Mah. Compu., Vol.6, pp ,. [7] O. P. Agrawal, Fracional variaional calculus and he ransversaliy condiions, J. Phys. A.: Mah. Gen., Vol.39, pp , 6. [8] C. H. Hsiao, Haar wavele direc mehod for solving variaional problems, Mahemaics and Compuers in Simulaion, Vol.64, pp , 4. BIOGRAPHY Ass. Prof. Dr. Osama H. Mohammed has complee his Docorae of Philosophy in 6. He is currenly working as Assisan Professor a he Deparmen of Mahemaics and Compuer Applicaions, College of Science, Al-Nahrain Universiy, Baghdad-Iraq and guiding M.Sc. and Ph.D. sudens Ass. Prof. Dr. Fadhel S. F. is he head of Mahemaics and Compuer Applicaions Deparmen, College of Science, Al-Nahrain Universiy, Baghdad-Iraq. He is received he Ph.D. in 998, were he is he supervisor of more han 4 M.Sc. Suden and 7 Ph.D. Sudens, in which he fields of ineres are hose opics relaed o fuzzy se heory, sochasic differenial equaions and numerical analysis. Mr. Zaid A. is now M.Sc. suden and he complee B.Sc. a 8 from he Deparmen of Mahemaics and Compuer Applicaions, College of Science, Al-Nahrain Universiy, Baghdad- Iraq. His field of ineres is abou finding he numerical soluion of variaional problems of fracional order. Copyrigh o IJIRSE 75