Jianping Liu, Xia Li, and Limeng Wu. 1. Introduction

Transcription

1 Mahemaical Problems in Engineering Volume 26, Aricle ID 7268, pages hp://dxdoiorg/55/26/7268 Research Aricle An Operaional Marix of Fracional Differeniaion of he Second Kind of Chebyshev Polynomial for Solving Mulierm Variable Order Fracional Differenial Equaion Jianping Liu, Xia Li, and Limeng Wu HebeiNormalUniversiyofScienceandTechnology,Qinhuangdao,Hebei664,China Correspondence should be addressed o Jianping Liu; liujianping48@26com Received 3 March 26; Revised 29 April 26; Acceped 4 May 26 Academic Edior: Josè A Tenereiro Machado Copyrigh 26 Jianping Liu e al This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied The mulierm fracional differenial equaion has a wide applicaion in engineering problems Therefore, we propose a mehod o solve mulierm variable order fracional differenial equaion based on he second kind of Chebyshev Polynomial The main idea of his mehod is ha we derive a kind of operaional marix of variable order fracional derivaive for he second kind of Chebyshev Polynomial Wih he operaional marices, he equaion is ransformed ino he producs of several dependen marices, which can also be viewed as an algebraic sysem by making use of he collocaion poins By solving he algebraic sysem, he numerical soluion of original equaion is acquired Numerical examples show ha only a small number of he second kinds of Chebyshev Polynomials are needed o obain a saisfacory resul, which demonsraes he validiy of his mehod Inroducion The concep of fracional order derivaive goes back o he 7h cenury, 2 I is only a few decades ago ha i was realized ha he arbirary order derivaive provides an excellen framework for modeling he real-world problems in a variey of disciplines from physics, chemisry, biology, and engineering, such as viscoelasiciy and damping, diffusion and wave propagaion, and chaos 3 6 Orhogonal funcions have received noiceable consideraion for solving fracional differenial equaion (FDE) By using orhogonal funcions, he FDE can be reduced o solve an algebraic sysem, and hen original problems are simplified Ahmadian e al 7 proposed a compuaional mehodbasedonjacobipolynomialsforsolvingfuzzylinear FDE on inerval, Kazem e al 8 consruced a general formulaion for he fracional order Legendre funcions Yüzbaşı 9 gave he numerical soluions of fracional Riccai ype differenial equaions by means of he Bernsein Polynomials Kazem consruced a general formulaion for he Jacobi operaional marix for fracional inegral equaions Tau mehod and collocaion mehod are widely used ools for he soluion of FDE Operaional approach of he au mehod was employed for solving fracional problems A numerical approach was provided for he FDE based on a specral au mehod 2 An efficien mehod based on he shifed Chebyshev-au idea was presened for solving he space fracional diffusion equaions 3 Tau mehod is very effecive for consan coefficien nonlinear problems, bu he mehod is no generally adoped for nonlinear FDE In pracice, since collocaion mehod has he advanages of less compuaion and easy implemenaion, i is more widely applied for solving variable coefficien nonlinear problems The collocaion mehod was used for solving he nonlinear fracional inegrodifferenial equaions 4 The hird kind of Chebyshev waveles collocaion mehod was inroduced for solving he ime fracional convecion diffusion equaions wih variable coefficiens 5 From he lieraures above, we conclude ha many auhors employed au and collocaion mehod for solving differen kinds of FDE based on differen kinds of orhogonal funcions or heir varians However, for he aforemenioned

2 2 Mahemaical Problems in Engineering FDE, he derivaive order is a fixed consan, which does no change spaially and emporally; variable order mulierm FDE is no menioned and solved Therefore, our main moivaion is o give a numerical echnology for solving variable order linear and nonlinear mulierm FDE based on he second kind of Chebyshev Polynomial Wih furher developmen of science research, i is found ha variable order fracional calculus can provide an effecive mahemaical framework for he complex dynamical problems The modeling and applicaion of variable order differenial equaion has been a fron subjec In addiion, he FDE is a special case of variable order ones, so i can also be solved by our proposed echnology Variable order derivaive is proposed by Samko and Ross 6 in 993, and hen Lorenzo and Harley 7, 8 sudied variable order calculus in heory more deeply Coimbra and Diaz 9, 2 used variable order derivaive o research nonlinear dynamics and conrol problems of viscoelasiciy oscillaor Pedro e al 2 researched diffusive-convecive effecs on he oscillaory flow pas a sphere by variable order modeling The developmen of numerical algorihms o solve variable order FDE is necessary Since he kernel of he variable order operaors is very complex for having a variable exponen, i is difficul o gain he soluion of variable order differenial equaion Only a few auhors sudied numerical mehods of variable order fracional differenial equaions Coimbra 9 employed a consisen approximaion wih firs-order accurae for solving variable order differenial equaions Sun e al 22 proposed a second-order Runge-Kua mehod o numerically inegrae he variable order differenial equaion Lin e al 23 sudied he sabiliy and he convergence of an explici finiedifference approximaion for he variable order fracional diffusion equaion wih a nonlinear source erm Chen e al 24, 25 paid heir aenion o Bernsein Polynomials o solve variable order linear cable equaion and variable order ime fracional diffusion equaion A numerical mehod based on he Legendre Polynomials is presened for a class of variable order FDE 26 Chen e al 27 inroduced he numerical soluion for a class of nonlinear variable order FDE wih Legendre waveles To he bes of our knowledge, i is no seen ha operaional marix of variable order derivaive based on he second kind of Chebyshev Polynomial is used o solve mulierm variable order FDE In addiion, for mos lieraures, hey solved variable order FDE defined on he inerval, Accordingly, based on he second kind of Chebyshev Polynomial, we propose a new efficien echnique for solving mulierm variable order FDE defined on he inerval, R The mulierm variable order FDE is given as follows: D α() f () =F(,f(),D β () f (),D β 2() f (),,D β k() f ()), <<R, () where D α() f() and D β i() f() are fracional derivaive in Capuo sense When α() and β i (), i =,2,,k are all consans, () becomes (2); namely, D α f () =F(,f(),D β f (),D β 2 f (),,D β k f ()), <<R Thus, (2) is a special case of () Our proposed mehod can solve boh () and (2) They ofen appear in oscillaory equaions, such as vibraion equaion, fracional Van Der Pol equaion, he Rayleigh equaion wih fracional damping, and fracional Riccai differenial equaion The basic idea of his mehod is ha we derive differenial operaional marices based on he second kind of Chebyshev Polynomial Wih he operaional marices, he equaion is ransformed ino he producs of several dependen marices, which can also be viewed as an algebraic sysem by making use of he collocaion poins By solving he algebraic sysem, he numerical soluion is acquired Since he second kinds of Chebyshev Polynomials are orhogonal o each oher, he operaional marices based on Chebyshev Polynomials grealy reduce he size of compuaional work while accuraely providing he series soluion From some numerical examples, we can see ha our resuls are in good agreemen wih he analyical soluion, which demonsraes he validiy of his mehod Therefore, i has he poenial o uilize wider applicabiliy The paper is organized as follows In Secion 2, some necessary definiions and properies of he variable order fracional derivaives are inroduced The basic definiions ofhesecondkindofchebyshevpolynomialandfuncion approximaionaregiveninsecions3and4,respecivelyin Secion 5, a kind of operaional marix of he second kind of Chebyshev Polynomial is derived, and hen we applied he operaional marices o solve he equaion as given a beginning In Secion 6, we presen some numerical examples o demonsrae he efficiency of he mehod We end he paper wih a few concluding remarks in Secion 7 2 Basic Definiion of Capuo Variable Order Fracional Derivaives Definiion Capuo variable fracional derivaive wih order α() is defined by D α() u () = Γ ( α()) + + ( τ) α() u (τ) dτ (u (+) u( )) α() Γ ( α()) If we assume he saring ime in a perfec siuaion, we can ge Definiion 2 as follows (2) (3)

3 Mahemaical Problems in Engineering 3 Definiion 2 Consider weigh funcion is ω() = R 2 wih,rtheysaisfy he following formulas: D α() u () = Γ ( α()) ( τ) α() u (τ) dτ ( <α() <) By Definiion 2, we can ge he following formula 25: (4) U () =, U () =2( 2 4 )= R R 2, U n+ () =2( 2 R ) U n () U n (), n =, 2, 3,, (7) D α() ( n )= Γ (n+) { Γ (n+ α()) n α(), { n=,2,, {, n = 3 Shifed Second Kind of Chebyshev Polynomial The second kind of Chebyshev Polynomial defined on he inerval I=,is orhogonal based on he weigh funcion ω(x) = x 2 They saisfy he following formulas: U (x) =, U (x) =2x, x 2 U n (x) U m (x) dx = U n+ (x) =2xU n (x) U n (x),, { m =n, { π, { 2 m = n n =, 2,, When,R,lex=2/R ;wecangeshifedsecond kind of Chebyshev Polynomial U n () = U n (2/R ),whose (5) (6) R R 2 U {, m =n, n () U m () d = { π { 8 R2, m = n The shifed second kind of Chebyshev Polynomial U n () can also be expressed as U n (), n =, { = n/2 { ( ) k n 2k (n k)! k! { (n 2k)! (4 R 2), k= n, where n/2 denoes he maximum ineger which is no more han n/2 Le hen Le Ψ () = U (), U (),, U n () T, T () =,,, n T ; If n is an even number, hen (8) (9) Ψ () =AT() () A=BC () B = ( ) ( ) n/2 ( ) ( )!! ( )! ( 4 R ) ( ) (2 )!! (2 )! ( 4 2 R ) ( ) n/2 (n n/2+)! (n/2 )! n 2(n/2 )! ( 4 n 2(n/2 ) R ) ( ) (n )!! (n )! ( 4 n R ) (2 )!! (2 2)! ( R ) (n n/2)! (n/2)! (n 2 n/2)! ( 4 n 2 n/2 R ) (2)

4 4 Mahemaical Problems in Engineering If n is an odd number, hen B ( ) ( )!! ( )! ( 4 R ) = ( ) (2 )!! (2 2)! ( R ) ( ) (2 )!! (2 )! ( 4 2 R ), ( ) (n )/2 (n (n )/2)! ((n )/2)! (n 2 (n )/2)! ( 4 n 2 (n )/2 R ) ( ) (n )!! (n )! ( 4 n R ) ( )( R 2 ) ( )( R 2 ) C= ( 2 2 )( R 2 ) ( 2 2 )( R 2 ) ( )( R 2 ) ( n n )( R 2 ) ( n n )( R 2 ) ( n n 2 2 )( R 2 ) n n n ( n )( R 2 ) (3) Therefore, we can easily gain 4 Funcion Approximaion T n () =A Ψ () (4) Theorem 3 Assume a funcion f(), R be n imes coninuously differeniable Le u n () = n i= λ i U i () = Λ T Ψ n () be he bes square approximaion funcion of f(), whereλ= λ,λ,,λ n T and Ψ n () = U (), U (),, U n () T ;hen f () u n () MSn+ R (n+)! π 8, (5) where M=max,R f (n+) () and S=max{R, } Proof We consider he Taylor Polynomial: f () =f( )+f ( )( )+ +f (n) ( ) ( ) n +f (n+) (η) ( ) n+, n! (n+)!, R, where η is beween and Le p n () =f( )+f ( )( )+ +f (n) ( ) ( ) n ; n! hen (6) (7) f (x) p n (x) = f (n+) (η) ( ) n+ (8) (n+)! Since u n () = n i= λ i U i () = Λ T Ψ n () is he bes square approximaion funcion of f(),we can gain f () u n () 2 f () p n () 2 R = ω () f () p n () 2 d R = ω () f (n+) (η) ( ) n+ d (n+)! = M 2 R (n+)! 2 M 2 R (n+)! 2 Le S=max{R, }; herefore ( ) 2n+2 ω () d ( ) 2n+2 R 2 d f () u n () 2 M2 S 2n+2 R (n+)! 2 R 2 d = M2 S 2n+2 (n+)! 2 πr 2 8 Andbyakinghesquareroos,Theorem3canbeproved 5 Operaional Marices of D α() Ψ n () and D β i() Ψ n () i=,2,,kbased on Shifed Second Kind of Chebyshev Polynomial Consider 2 (9) (2) D α() Ψ n () =D α() AT n () =AD α() n T (2)

5 Mahemaical Problems in Engineering 5 According o (5), we can ge D α() Ψ n () =A Γ (2) Γ (2 α()) α() T Γ (n+) Γ (n+ α()) n α() Γ (2) Γ (2 α()) α() (22) =A Γ (n+) Γ (n+ α()) α() n =AMA Ψ n (), where M Γ (2) Γ (2 α()) α() = Γ (n+) Γ (n+ α()) α() (23) AMA is called he operaional marix of D α() Ψ n () Therefore, In his paper, we use collocaion mehod o solve he coefficien Λ=λ,λ,,λ n T By aking he collocaion poins, (28) will become an algebraic sysem We can gain he soluion Λ = λ,λ,,λ n T by Newon mehod Finally, he numerical soluion u n () = Λ T Ψ n () is gained 6 Numerical Examples and Resuls Analysis In his secion, we verify he efficiency of proposed mehod o suppor he above heoreical discussion For his purpose, we consider linear and nonlinear mulierm variable order FDE and corresponding mulierm FDE For mulierm variable order FDE, we compare our approach wih he analyical soluion For mulierm FDE, we compare our compuaional resuls wih he analyical soluion and soluions in 28 by using oher mehods The resuls indicae ha our mehod is a powerful ool for solving mulierm variable order FDE and mulierm FDE Numerical examples show ha only a small number of he second kinds of Chebyshev Polynomials are needed o obain a saisfacory resul Furhermore, our mehod has higher precision han 28 In his secion, he noaion ε= max i=,,,n f( i) u n ( i ), (29) (2i + ) i =R 2 (n+), i=,,,n, D α() f () D α() (Λ T Ψ n ()) =Λ T D α() Ψ n () Similarly, we can ge =Λ T AMA Ψ n () (24) is used o show he accuracy of our proposed mehod Example (a) Consider he linear FDE wih variable order as follows: where N i D β i() Ψ n () =AN i A Ψ n (), i=,2,,k, (25) Γ (2) Γ(2 β i ()) β i() (26) = Γ (n+) Γ(n+ β i ()) β i() AN i A is called he operaional marix of D β i() Ψ n ()Thus, D β i() f () D β i() (Λ T Ψ n ()) =Λ T D β i() Ψ n () =Λ T AN i A Ψ n () (27) The original equaion () is ransformed ino he form as follows: ad α() f () +b() D β () f () +c() D β 2() f () +e() D β 3() f () +k() f () =g(), y () =2, y () =, where 2 α() f () = a Γ (3 α()) b() 2 β() Γ(3 β ()) c(), R, 2 β 2() Γ(3 β 2 ()) e() 2 β3() Γ(3 β 3 ()) +k() (2 2 2 ) (3) (3) Λ T AMA Ψ n () =F,Λ T Ψ n (),Λ T AN A Ψ n (),,Λ T AN k A Ψ n (),, R (28) The analyical soluion is f() = 2 2 /2 We use our proposed echnology o solve i

6 6 Mahemaical Problems in Engineering 2 R=, n=3 2 R=4, n=6 9 f() 8 f() Analyical soluion Numerical soluion (a) Analyical soluion Numerical soluion (b) Figure : Analyical soluion and numerical soluion of Example (a) for differen R Le f() u n () = Λ T Ψ n (), α() = 2, β () = /3, β 2 () = /4,andβ 3 () = /5; according o (28), we have aλ T AMA Ψ n () +b() Λ T AN A Ψ n () +c() Λ T AN 2 A Ψ n () +e() Λ T AN 3 A Ψ n () +k() Λ T Ψ n () =g() (32) Take he collocaion poins i = R((2i + )/2(n + )), i =,,,n, o process (32), and hen ge aλ T AMA Ψ n ( i )+b( i )Λ T AN A Ψ n ( i ) +c( i )Λ T AN 2 A Ψ n ( i ) +e( i ) Λ T AN 3 A Ψ n ( i ) +k( i ) Λ T Ψ n ( i ) =g( i ), i=,2,,n (33) By solving he algebraic sysem (33), we can gain he vecor Λ=λ,λ,,λ n T Subsequenly,numericalsoluion u n () = Λ T Ψ n () is obained Likely 28, we presen numerical soluion by our mehod for a=, b () =, Table : Values of ε of Example (a) for differen R R n=3 n=4 n=5 n=6 R = 2224e e e 4 R = e e e 4 R = e e e 4 68e 3 In Table,we lis he values of ε a he collocaion poins From Table, we could find ha a small number of Chebyshev Polynomials are needed o reach perfec soluion for differen R Figure shows he analyical soluion and numerical soluion for differen R a collocaion poins We can conclude ha he numerical soluion is very close o he analyical soluion The same rend is observed for oher values of α() and β i (), i =,2,,kAllhevaluesofε are small enough o mee he pracical engineering applicaion Le α() = 2, β () = 234, β 2 () =, β 3 () = 333,and R=as 28; Example (a) becomes a mulierm order FDE, namely, Example (b) This problem has been solved in 28 (b) See 28: ad 2 f () +b() D β f () +c() Df () +e() D β 3 f () +k() f () =g(),,, y () =2, y () =, where (35) c () = /3, e () = /4, k () = /5 (34) f () = a b() e() 2 β c() Γ(3 β ) 2 β 3 Γ(3 β 3 ) +k() (2 2 2 ) (36)

7 Mahemaical Problems in Engineering 7 Table 2: Compuaional resuls of Example (b) for R= Λ ε n=3 8438, 25, 33, T 4449e 6 n=4 8438, 25, 32,, T 4633e 3 n=5 8437, 25, 33,,, T 32743e 2 n=6 8438, 25, 32,,,, T 725e 3 Table 3: Values of ε of Example 2(b) for R=2,4 R n=3 n=4 n=5 n=6 R = e 6 938e e e 4 R = e 6 24e e e 3 The analyical soluion is f() = 2 2 /2 Example (b) is a special case of Example (a), so we sill obain he soluion by our mehod as Example (a) The compuaional resuls are seen in Table 2 We lis he vecor Λ=λ,λ,,λ n T and he values of ε a he collocaion poins As seen from Table 2, he vecor Λ = λ,λ,,λ n T obained is mainly composed of hree erms, namely, λ,λ, λ 2, which is in agreemen wih he analyical soluion f() = 2 2 /2Thevaluesofε are smaller han 28 wih he same size of Chebyshev Polynomials (in 28, he value of ε is e 5 for n=5) In addiion, we exend he inerval from, o, 2 and, 4 Similarly, we also ge he perfec resuls as shown in Table 3, which is no solved in 28 Example 2 (a) As he second example, he nonlinear mulierm variable order FDE D α() f () +D β () f () D β 2() f () +f 2 () =g(), wih g () = Γ (4 α()) 3 α(), R, 36 Γ(4 β ())Γ(4 β 2 ()) 6 β () β 2 (), (37) (38) subjecoheiniialcondiionsf() = f () = f () = is considered The analyical soluion is f() = 3 Le f() = Λ T Ψ(); according o (28), we have aλ T AMA Ψ n +(Λ T AN A Ψ n )(Λ T AN 2 A Ψ n ) +(Λ T Ψ n ) 2 =g() (39) Le α() = 2, β () = sin, andβ 2 () = /4; byaking he collocaion poins, he soluion of Example 2(a) could be gained The values of ε are displayed in Table 4 for differen R From he resul analysis, our mehod could gain saisfacory soluion Figure 2 obviously shows ha he numerical soluion converges o he analyical soluion Table 4: Values of ε of Example 2(a) wih α() = 2, β () = sin, and β 2 () = /4 R n=3 n=4 n=5 n=6 R = 556e e 4 544e 4 227e 3 R = e 5 466e e e 3 R = e 5 32e 4 566e 8 565e If α(), β (), β 2 () are consans, Example 2(a) becomes a mulierm order FDE in 28 This problem for R=has also been solved in 28 (b) See 28: D α f () +D β f () D β 2 f () +f 2 () =g(), wih 2<α<3, <β <2, <β 2 <,,, g () = Γ (4 α) 3 α 36 Γ(4 β )Γ(4 β 2 ) 6 β β 2 (4) (4) The same as 28, we le α = 25, β = 5, andβ 2 = 9 and α = 275, β = 75, andβ 2 = 75 for R=and hen use our mehod o solve hem The compuaional resuls are shown in Tables 5 and 6 As seen from Tables 5 and 6, he vecor Λ=λ,λ,,λ n T obained is mainly composed of four erms, namely, λ,λ,λ 2,λ 3, which is in agreemen wih he analyical soluion f() = 3 Iisevidenhahe numerical soluion obained converges o he analyical soluion for α = 25, β = 5, andβ 2 = 9 and α = 275, β = 75,andβ 2 = 75Thevaluesofε are smaller han 28 wih he same n size In addiion, we exend he inerval from, o, 2 and, 3Similarly,wealsogeheperfecresulsinTables7and 8, bu he problems are no solved in 28 A las, he proposed mehod is used o solve he mulierm iniial value problem wih nonsmooh soluion Example 3 Le us consider he FDE as follows: + 3 Dα y+ 3 Dβ y+y= 2 9 {(2 9 ) 2 +( ) 3 +( ) + 3 }, <α 2, <β, y() = 729, y () =,, 3, (42)

8 8 Mahemaical Problems in Engineering 7 R=, n=3 6 R=4, n= f() 4 3 f() Analyical soluion Numerical soluion (a) Analyical soluion Numerical soluion (b) Figure 2: Analyical soluion and numerical soluion of Example 2(a) wih α() = 2, β () = sin,andβ 2 () = /4 for differen R Table 5: Compuaional resuls of Example 2(b) for R=wih α = 25, β = 5,andβ 2 = 9 Λ ε n=3 288, 287, 937, 56 T 2628e 5 n=4 288, 287, 938, 56, T 59e 4 n=5 288, 287, 937, 56,, T 47362e 3 n=6 288, 287, 937, 56,,, T 28e Table 6: Compuaional resuls of Example 2(b) for R=wih α = 275, β = 75,andβ 2 = 75 Λ ε n=3 287, 288, 938, 56 T 3983e 5 n=4 287, 287, 938, 56, T 76964e 4 n=5 288, 287, 937, 56,, T 42e 2 n=6 288, 288, 938, 56,,, T 8479e Table 7: Values of ε of Example 2(b) for R = 2,3 wih α = 25, β = 5,andβ 2 = 9 R n=3 n=4 n=5 n=6 R = 2 474e 5 358e e e 3 R = e e e e 2 Table 8: Values of ε of Example 2(b) for R=2,3wih α = 275, β = 75,andβ 2 = 75 R n=3 n=4 n=5 n=6 R = e e e e 3 R = 3 374e 4 e e 4 392e in which only for α=2and β=, he analyical soluion is known and given by y= ( 2 /9) 3 By applying he proposed mehod o solve he equaion, wecanobainhahevalueofε is 5853e 3 for n=9the compuaional resuls are shown as Figures 3 and 4 As seen from Figure 3, i is eviden ha he numerical soluion obained converges o he analyical soluion We also plo he absolue error beween he analyical soluion and numerical soluion in Figure 4 I shows ha he absolue error is small, which could mee he needs of general projecs In a word, he proposed mehod possesses simple form, saisfacory accuracy, and wide field of applicaion 7 Conclusion In his paper, we presen an operaional marix echnology basedonhesecondkindofchebyshevpolynomialosolve mulierm FDE and mulierm variable order FDE This echnology reduces he original equaion o a sysem of algebraic

9 Mahemaical Problems in Engineering 9 f() n=9 Acknowledgmens ThisworkisfundedbyheTeachingResearchProjecof Hebei Normal Universiy of Science and Technology (no JYZD243) and by Naural Science Foundaion of Hebei Province, China (Gran no A254763) The work is also funded by Scienific Research Foundaion of Hebei Normal Universiy of Science and Technology 2 References Analyical soluion Numerical soluion Figure 3: Analyical soluion and numerical soluion of Example 3 Absolue error 3 n= Figure 4: Absolue error of he proposed mehod of Example 3 equaions, which grealy simplifies he problem In order o confirm he efficiency of he proposed echniques, several numerical examples are implemened, including linear and nonlinear erms By comparing he numerical soluion wih he analyical soluion and ha of oher mehods in he lieraure, we demonsrae he high accuracy and efficiency of he proposed echniques In addiion, he proposed mehod can be applied by developing for he oher relaed fracional problem, such as variable fracional order inegrodifferenial equaion, variable order ime fracional diffusion equaion, and variable fracional order linear cable equaion This is one possible area of our fuure work Compeing Ineress The auhors declare ha hey have no compeing ineress K Diehelm and N J Ford, Analysis of fracional differenial equaions, Mahemaical Analysis and Applicaions, vol 265, no 2, pp , 22 2 WGGlockleandTFNonnenmacher, Afracionalcalculus approach o self-similar proein dynamics, Biophysical Journal, vol68,no,pp46 53,995 3 Y A Rossikhin and M V Shiikova, Applicaion of fracional derivaives o he analysis of damped vibraions of viscoelasic single mass sysems, Aca Mechanica, vol 2, no, pp 9 25, H G Sun, W Chen, and Y Q Chen, Variable-order fracional differenial operaors in anomalous diffusion modeling, Physica A: Saisical Mechanics and is Applicaions,vol388,no2, pp , 29 5 W Chen, H G Sun, X D Zhang, and D Korošak, Anomalous diffusion modeling by fracal and fracional derivaives, Compuers & Mahemaics wih Applicaions, vol59,no5,pp , 2 6 W Chen, A speculaive sudy of 2/3-order fracional Laplacian modeling of urbulence: some houghs and conjecures, Chaos,vol6,no2,AricleID2326,pp2 26,26 7 A Ahmadian, M Suleiman, S Salahshour, and D Baleanu, A Jacobi operaional marix for solving a fuzzy linear fracional differenial equaion, Advances in Difference Equaions, vol 23, no, aricle 4, pp 29, 23 8 S Kazem, S Abbasbandy, and S Kumar, Fracional-order Legendre funcions for solving fracional-order differenial equaions, Applied Mahemaical Modelling,vol37,no7,pp , 23 9 S Yüzbaşı, Numerical soluions of fracional Riccai ype differenial equaions by means of he Bernsein polynomials, Applied Mahemaics and Compuaion, vol29,no,pp , 23 S Kazem, An inegral operaional marix based on Jacobi polynomials for solving fracional-order differenial equaions, Applied Mahemaical Modelling, vol 37, no 3, pp 26 36, 23 S K Vanani and A Aminaaei, Tau approximae soluion of fracional parial differenial equaions, Compuers & Mahemaics wih Applicaions,vol62,no3,pp75 83,2 2 F Ghoreishi and S Yazdani, An exension of he specral Tau mehod for numerical soluion of muli-order fracional differenial equaions wih convergence analysis, Compuers and Mahemaics wih Applicaions,vol6,no,pp3 43,2 3RFRen,HBLi,WJiang,andMYSong, Anefficien Chebyshev-au mehod for solving he space fracional diffusion equaions, Applied Mahemaics and Compuaion, vol 224, pp , 23

10 Mahemaical Problems in Engineering 4 M R Eslahchi, M Dehghan, and M Parvizi, Applicaion of he collocaion mehod for solving nonlinear fracional inegrodifferenial equaions, Compuaional and Applied Mahemaics,vol257,pp5 28,24 5 F Y Zhou and X Xu, The hird kind Chebyshev waveles collocaion mehod for solving he ime-fracional convecion diffusion equaions wih variable coefficiens, Applied Mahemaics and Compuaion,vol28,pp 29,26 6 S G Samko and B Ross, Inegraion and differeniaion o a variable fracional order, Inegral Transforms and Special Funcions,vol,no4,pp277 3,993 7 C F Lorenzo and T T Harley, Variable order and disribued order fracional operaors, Nonlinear Dynamics, vol29,no, pp 57 98, 22 8 C F Lorenzo and T T Harley, Iniializaion, concepualizaion, and applicaion in he generalized (fracional) calculus, Criical Reviews in Biomedical Engineering, vol35,no6,pp , 27 9 C F M Coimbra, Mechanics wih variable-order differenial operaors, Annals of Physics,vol2,no-2,pp692 73,23 2 GDiazandCFMCoimbra, Nonlineardynamicsandconrol of a variable order oscillaor wih applicaion o he van der pol equaion, Nonlinear Dynamics,vol56,no,pp45 57,29 2 H T C Pedro, M H Kobayashi, J M C Pereira, and C F M Coimbra, Variable order modeling of diffusive-convecive effecs on he oscillaory flow pas a sphere, Vibraion and Conrol, vol 4, no 9-, pp , H G Sun, W Chen, and Y Q Chen, Variable-order fracional differenial operaors in anomalous diffusion modeling, Physica A: Saisical Mechanics and Is Applicaions,vol388,no2, pp , R Lin, F Liu, V Anh, and I Turner, Sabiliy and convergence of a new explici finie-difference approximaion for he variable-order nonlinear fracional diffusion equaion, Applied Mahemaics and Compuaion, vol22,no2,pp , YMChen,LQLiu,BFLi,andYNSun, Numericalsoluion for he variable order linear cable equaion wih Bernsein polynomials, Applied Mahemaics and Compuaion, vol238, no, pp , YMChen,LQLiu,XLi,andYNSun, Numericalsoluion for he variable order ime fracional diffusion equaion wih bernsein polynomials, Compuer Modeling in Engineering and Sciences,vol97,no,pp8,24 26 L F Wang, Y P Ma, and Y Q Yang, Legendre polynomials mehod for solving a class of variable order fracional differenial equaion, Compuer Modeling in Engineering & Sciences, vol, no 2, pp 97, YMChen,YQWei,DYLiu,andHYu, Numericalsoluion for a class of nonlinear variable order fracional differenial equaions wih Legendre waveles, Applied Mahemaics Leers, vol 46, pp 83 88, K Maleknejad, K Nouri, and L Torkzadeh, Operaional marix of fracional inegraion based on he shifed second kind Chebyshev Polynomials for solving fracional differenial equaions, Medierranean Mahemaics,25

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