ISSN 746-7659 England UK Journal of Informaton and Computng Scence Vol. 9 No. 3 4 pp. 69-8 Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalzed Haar Functons Yadollah Ordokhan and Neda Rahm Department of Mathematcs Alzahra Unversty ehran Iran (Receved September 3 accepted February 8 4) Abstract. hs paper presents hybrd of ratonalzed Haar (HRH) functons method for appromate the numercal soluton of the fractonal nonlnear Fredholm ntegro-dfferental equatons (FNFIDEs). he fractonal dervatves are consdered n the sense of Caputo. he fractonal operatonal matr of hybrd of block-pulse and ratonalzed Haar functons are presented. hs matr together wth the dual operatonal matr are used to reduce the computaton of FNFIDEs nto a system of algebrac equatons. Some numercal eamples are gven and the results of applyng ths method demonstrate tme and computatonal are small. Keywords: Fredholm ntegro-dfferental equaton Remann-Louvlle ntegral Caputo fractonal dervatve Fractonal operatonal matr Ratonalzed Haar Hybrd.. Introducton he man purpose of ths paper s to consder the numercal soluton of the FNFIDEs of the types y ( k( D dt f ( () wth the ntal condton and y ( ) Error! Not a vald embedded object. D k( G( ) dt f ( n n N () y ( ) () wth n ntal condtons 3... n. Here D s Caputo s fractonal dervatve s a parameter descrbng the order of fractonal dervatve and s a real known constant. Also f L ([ ]) and k L ([] ) are gven functons s the soluton to be determned and G() s a polynomal of the unknown functon we assume q G ( ) [ ] q N. he fractonal calculus has been appled n many mathematca models. For eample the nonlnear oscllaton of earthquake [] flud-dynamc traffc [] contnuum and statstcal mechancs [3] can be modeled wth fractonal dervatves. here are several methods that are used to solve the fractonal ntegro-dfferental equatons such as Adoman decomposton method [4] collocaton method [5] CAS wavelet method [6] hybrd functons and the collocaton method [7] and second knd Chebyshev Correspondng author. E-mal address: Ordokhan@alzahra.ac.r Publshed by World Academc Press World Academc Unon
7 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons wavelet method [8]. In ths work we report applcaton of HRH functons to solve the FNFIDEs for ths purpose n problems () and () we epandng the hgh order of dervatve by HRH functons wth unknown coeffcents then we can evaluate the unknown coeffcents and obtan an appromate soluton to problems () and (). In ths technc tme and computatonal are small and ths s a good and useful property of the HRH functons method. he artcle s organzed as follows: In secton we ntroduce some necessary fundamentals of the fractonal calculus theory. n secton 3 we present the propertes of HRH functons requred for our subsequent development. In secton 4 we descrbe the soluton of problems () and () by usng HRH functons and n secton 5 we gve some numercal eamples to demonstrate the accuracy of the proposed method.. Fundamentals of fractonal calculus In ths secton we gve some defntons and fundamentals of the fractonal calculus theory. Defnton.. he Remann-Louvlle fractonal ntegral operator of order s defned as [9] I ( ) ( dt whrere (.) s Gamma functon. It has the followng propertes: I ( ) I. ( ) Defnton.. he Caputo defnton of fractonal dervatve operator s gven by [] n n ( n) D I D ( y ( dt ( n ) n n n N. It has followng propertes: D I 3. Propertes of hybrd functons I k ) k! n ( k ) D y (. 3. Hybrd functons of block-pulse and ratonalzed Haar functons he HRH functons h ( n... N R... M M... n r are the order of block-pulse functons and ratonalzed Haar functons respectvely s defned on the nterval [) as [3] n n h ( N n ) h ( N otherwse N (3) JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence Vol. 9 (4) No. 3 pp 69-8 7 In (3) h r ( are the orthogonal set of ratonalzed Haar functons and can be defned on the nterval [) as [4] j u Where J u u. J J h r ( J J (4) otherwse he value of r s defned by two parameters and j as r j h ( ) s defned for j and gven by 3... j 3... h (. snce h ( s the combnaton of ratonalzed Haar functons and block-pulse functons whch are both complete and orthogonal thus the set of hybrd functons are complete orthogonal set. he orthogonalty property of HRH functons s gven by [3] h ( h n r ( d N n n r r oterwse r j r j. 3. Functon appromaton A functon f ( L ([ ]) may be epanded nto HRH functons as [3] cgven by f ( c h ( (6) n r c f h h N f ( h ( d and.. denote the nner product. If the nfnte seres n (6) s truncated then (6) can be wrtten as f ( N M n r c h ( C H( (7) JIC emal for subscrpton: publshng@wau.org.uk
7 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons he HRH functon coeffcent vector C and RH functon vector H( are defned as C c c... c c c... c... c c... c ] (8) [ M M N N NM H H [ H ( H (... H ] (9) ( ( [ h h... h M ]... N. Also we can epand the functon k( L ([] ) nto HRH functon as follow K k vr s an matr such that N k( H ( KH( () k vr H v ( k( H r ( H ( H ( v r v... N r... M. akng the Newton-Cotes nodes as followng [5].... () We have hen from () the square hybrd matr 3 [ H( ) H( )... H( )]. () can be epressed as ( ˆ... ˆ... ˆ dag MM MM MM ) (3) ˆ M M M-square Haar matr ([4]). For eample f M= and N= we have Usng (7) we get From () and (4) we have. 3 f ( ) f ( )... f ( )] C [. (4) K ˆ K (5) JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence Vol. 9 (4) No. 3 pp 69-8 73 Kˆ ˆ ˆ l p k k ( ) p l.... lp lp 3.3 Operatonal matr of the fractonal ntegraton he ntegraton of the vector H( defned n (9) can be defned as [3] P s the operatonal matr of ntegraton. H ( dt PH ( (6) In ths secton we want to derve the HRH functons operatonal matr of the fractonal ntegraton [6]. For ths purpose we consder an m-set of block-pulse functons as b ( m otherwse m m... m. he functon b ( are dsjont and orthogonal. hat s b ( b j ( b ( j j j b ( b j ( j m Ratonalzed Haar functons can be epanded nto an m-set of block-pulse functons [6]. Smlarly HRH functons can be epanded nto ther block-pulse functons as B )] ( [ b ( b (... b m ( and H( B( (7) s matr defned n (). In [7] Klcman and Alzhour have gven the block-pulse operatonal matr of the fractonal ntegraton F as follows: I B( F B( (8) 3 m m m3 F (9) m ( ) JIC emal for subscrpton: publshng@wau.org.uk
74 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons wth m= and ( k ) k ( k ). k Now we obtan the HRH functon operatonal matr of the fractonal ntegraton. Let I H( P H( () P s called the HRH functons operatonal matr of the fractonal ntegraton. Usng (7) and (8) we have from (7) and () we get I H( I B( I B( F B( P H( P B( F B( hen P s gven by P F () Furthermore we have found the operatonal matr of fractonal ntegraton for HRH functons. For eample let N M then we have P 4 ( ) and for. 5 the operatonal matr P.5 3.4 he dual operatonal matr.74.8.. 3 3.5 P can be epressed as followng.8.56...89.747.74.8 he ntegraton of the cross product of two hybrd vector s [3] W s a 3 3.747.49.8.56 W H( H ( d dag ( D D D) () N matr and D s a matr gven by [4] JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence Vol. 9 (4) No. 3 pp 69-8 75 D dag ( ). hen we obtan the matr W that s dual matr of HRH functons. 4. Soluton of FNFIDEs as In ths secton we frst consder the FNFIDEs gven n problem ().o solve for we frst appromate y( y ( C H( H ( C (3) C s the HRH functons coeffcent vector and H( s HRH functons vector. For smplcty we can let that the ntal condton. By usng the propertes of Caputo dervatve we have D Suppose f( and k( can be epressed appromately as I y ( C P H( ) (4) C H( dt C PH ( (5) f ( H ( F k( H ( KH( (6) F and K are gven n (7) and (5) respectvely. Usng () (4) and (6) we have k( D dt H ( KH ( H ( ( P ) Cdt H ( KW ( P ) C (7) wth substtutng (3) (6) and (7) n problem () we obtan therefore I s vector C easly. H ( C H ( KW( P ) C H ( F (8) ( I KW ( P ) ) C F (9) dentty matr. (9) s a system of lnear equatons and can be solved for the unknown In second case we consder the FNFIDEs gven n problem (). we frst assume D C H( H ( C C s the HRH functons coeffcent vector and H( s HRH functons vector. For smplcty we can let that the ntal condtons. So by usng the propertes of Caputo dervatve n secton () and (3) we have Also we let (3) C P H( (3) JIC emal for subscrpton: publshng@wau.org.uk
76 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons z( G( ). (3) Suppose z( f( and k( can be epressed appromately as z( Z H( f ( H ( F k( H ( KH( (33) Z F and K are gven n (7) and (5) respectvely. Usng () (3) and (33) we have k( G( ) dt H ( KH ( Zdt H ( KWZ (34) wth substtutng (3) (33) and (34) n problem () we have H ( C H ( KWZ H ( F (35) then (35) can be wrtten as C KWZ F (36) from (3) (3) and (33) we get Z H( G( C P H( ). (37) In order to construct the appromaton for we collocate (37) n ponts. For a sutable collocaton ponts we choose Newton-Cotes nodes defned n (). By usng (9) () and (3) we have H( ) e e ( ) hen from (36) and (37) we obtan the followng nonlnear systems of algebrac equatons. C KWZ F (38) Z e G( C P e ) 3. herefore (38) can be solved for the unknowns C and Z then requred appromaton to the soluton n problem () s obtaned. 5. Numercal eamples In ths secton we apply the present method and solve some eamples gven n the dfferent papers. All calculatons were performed usng MALAB software. Eample. Consder the followng FNFIDE ([8]) y ( f ( k( D dt ) 4 Where ( ) t 3 3 48 (.5) k t f ( 8 ( ) and the eact solutons 6.5(4.75) 4.5(.5) JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence Vol. 9 (4) No. 3 pp 69-8 77 3 ( 4 y. able shows the absolute error for eample.....3.4.5.6.7.8.9. able Absolute error for dfferent values of M N for eample Peresent method Method of [8] N=3 M=4 M=8 M=6.633 3.57 3 3.59 4.53 3 8.44 4 3.3 4.858 3 7.77 4.4749 4.735 3 4.3479 4.967 4 3.6676 3.47 3 3.68 4 4.7579 5.69 4.5864 4 3.755 3.93 4 8.57 5 5.3464 3.856 4.443 4.5 3.658 3.79 5 9.554 3.563 3 7.96 4 d=3 d= (N=5) (N=5).476.87 3 7.43 7.343 3 We have solved ths eample for N=3 for dfferent M the result n able show that our method s better than method of [8] and computatonal wth our method s small. Eample. Consder the followng FNFIDE ([6-8]) 3 D ( [ ] dt g( 6 g ( ( ) 3 7 7 3. 9 Wth the ntal condtons: y () y (). he eact soluton s. We have solved ths eample for N=3 and M=3 and have compared t wth method of [7]. he comparson JIC emal for subscrpton: publshng@wau.org.uk
78 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons s shown n able. able Comparson of present method and method of [7] of eample Present method Method of [7] Eact Numercal Absolute Numercal Absolute soluton soluton error soluton error..4.6.8.4973 9.75 5.6455.455 4.36653.653 4.64347 3.47 4.4354.354 4.6968.968 3.36563.563 4.64358 3.58 4.4.6.36.64..4 4. 4.33.33 4. CPU 7.47s From able we can see the numercal results that are obtaned wth our method are n a good agreement wth the eact soluton. Although wth present method number of value must be very large but tme and computatonal are small. Eample 3. Consder the followng FNFIDE ([6-8]) wth the ntal condton: D ( [ ] dt 4 y ( ) he only case whch we know the eact soluton for s. In able 3 comparson present method for dfferent N and M wth method of [6] n the case of show the error by usng HRH functons method s smaller than the method of [6]. From able 3 and Fg we conclude that the numercal results s n good agreement wth the eact soluton when. herefore for 3 the cases and that eact soluton s unknown and numercal results are shown n 4 4 able 4 and Fg for N=3 HRH functons method s powerful and relable tool and as numercal results tend to eact soluton of. able 3 Mamum absolute error for eample 3 Present method Method of [6] N=3 M=4 N=3 M=8 (m=) (m=4). k= M= k=3 M= (m =) (m =4) 5.433 4.356 4.733 3 6.579 4 JIC emal for contrbuton: edtor@jc.org.uk
Journal of Informaton and Computng Scence Vol. 9 (4) No. 3 pp 69-8 79....3.4.5.6.7.8.9. able 4 Appromate and eact soluton for dfferent α of eample 3 Eact 4 3 4.6596.366.9454.99998.8678.5536.39896.99995.9595.6578.4554.99988.845.774336.5635.399978.36.88375.67474.499966.387.98667.77473.59995.433.857.87549.699934.5437.846.97394.79993.6544.777.7.89989.7654.377.675.999864 soluton for (α=)....3.4.5.6.7.8.9. 6. Concluson Fg. : Plot of eample 3 for dfferent In the present work HRH functons are used to solve the FNFIDEs. We reduce the FNFIDEs to a system of algebrac equatons by usng the HRH functons together Newton-Cotes nodes. In ths method tme and computatons are small because the matr ntroduces n () contan many zeros and these zeros make HRH functons faster than other methods. Numercal eamples wth satsfactory results are gven to demonstrate that t s relable and useful tool to solve the FNFIDEs. Acknowlegment JIC emal for subscrpton: publshng@wau.org.uk
8 Yadollah Ordokhan et.al. : Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalz ed Haar Functons he work of the frst author was supported by Alzahra Unversty. 7. References [] J. H. He Nonlnear oscllaton wth fractonal dervatve and ts applcatons Internatonal conference on vbratng engneerng 98 Dalan Chna (998) 88-9. [] J. H. He Some applcaton of nonlnear fractonal dffrental equatons and ther appromatons Bulletn of Scence echnology 5 () (999) 86-9. [3] F. Manard Fractals and Fractonal Calculus Contnuum Mechancs Sprnger Verlag New York (997) 9-348. [4] S. Moman M. A. Noor Numercal methods for fourth-order fractonal ntegro-dfferental equatons Appled Mathmatcs and Computaton 8 (6) 754-76. [5] E. A. Rawashdeh Numrcal soluton of fractonal ntegro-dfferental equatons by collocaton method Appled Mathmatcs and Computaton 76 (6) -6. [6] H. Saeed M. M. Moghadam N. Mollahasan G. N. Chuev A CAS wavelet method for solvng nonlnear Fredholm ntegro-dffrental equatons of fractonal odercommuncatons n Nonlnear Scence and Numercal Smulaton 6 () 54-63. [7] J. Hou B. Qn C. Yang Numercal soluton of nonlnear fredholm ntegro dfferental equatons of fractonal order by usng hy brd functons and the collocaton method Journal of Appld Mathematcs 6873 () -9. [8] L. Zhu Q. Fan Solvng fractonal nonlnear Fredholm ntegro-dfferental equatons by the second knd Chebyshev wavelet Communcatons n Nonlnear Scence and Numercal Smulaton 7 (6) () 333-34. [9] I. Podlubny Fractonal Dfferental Equato Academc Press New York (999). [] J. D. Munkhammar Remann-Louvlle Fractonal Dervatves and the aylor-remmann Seres U. U. D. M. Project Report 4. [] A. A. Klbas H. M. Srvastava J. J. rujllo heory and Applcatons of Fractonal Dfferental Equatons North- Holland Mathematc Studes Elsver 4 (6). [] A. Saadatmand M. Dehghan A new operatonal matr for solvng fractonal -order dfferental equatons Computers and Mathematcs wth Applcatons 59 () 36-336. [3] B. Arabzadeh M. Razzagh Y. Ordukhan Numercal soluton of lnear tme-varyng dfferental equatons usng hybrd of block-pulse and ratonalzed Haar functons Journal of Vbraton and Control () (6) 8-9. [4] Y. Ordokhan Soluton of nonlnear Volterra-Fredholm-Hammersten ntegral equatons va ratonalzed Haar functons Appled Mathmatcs and Computaton 87 (6) 436-443. [5] G. M. Phlps P.J. aylor heory and Applcaton of Numercal Analyss Academc Press New York 973. [6] Y. L W. Zhao Haar Wavelet operatonal matr of fractonal order ntegraton and ts applcaton n solvng the fractonal order dfferental equatons Appled Mathmatcs and Computaton 6 () 76-85. [7] A. Klcman Z. A. A. Al Zhour Kronecker operatonal matrces for fractonal calculus and some applcatons Appled Mathematcs and Computaton 87 (7) 5-65. [8] M. F. Al-Jamal E. A. Rawashdehhe appromate soluton of fractonal ntegro-dfferental equatons Int. J. Contemp. Math. Scences 4 () (9) 67-78. JIC emal for contrbuton: edtor@jc.org.uk