Research Article Numerical Algorithm to Solve a Class of Variable Order Fractional Integral-Differential Equation Based on Chebyshev Polynomials

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1 Mahemaical Problems in Engineering Volume 25, Aricle ID 926, pages hp://dxdoiorg/55/25/926 Research Aricle Numerical Algorihm o Solve a Class of Variable Order Fracional Inegral-Differenial Equaion Based on Chebyshev Polynomials Kangwen Sun and Ming Zhu School of Aeronauic Science and Technology, Beihang Universiy, Beijing 9, China Correspondence should be addressed o Ming Zhu; zhuming99@63com Received 24 May 25; Revised 3 Augus 25; Acceped 3 Augus 25 Academic Edior: Kishin Sadarangani Copyrigh 25 K Sun and M Zhu 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 purpose of his paper is o sudy he Chebyshev polynomials for he soluion of a class of variable order fracional inegraldifferenial equaion The properies of Chebyshev polynomials ogeher wih he four kinds of operaional marixes of Chebyshev polynomials are used o reduce he problem o he soluion of a sysem of algebraic equaions By solving he algebraic equaions, he numerical soluions are acquired Furher some numerical examples are shown o illusrae he accuracy and reliabiliy of he proposed approach and he resuls have been compared wih he exac soluion Inroducion Fracional calculus has araced increasing aenion for decades since i plays a vial role in differen disciplines of science and engineering 3 Compared wih ineger order differenial equaion, fracional differenial equaion has he advanage ha i can beer describe some naural physics processes and dynamic sysem processes, because he fracional order differenial operaors are nonlocal operaors Many physics, chemisry, and engineering sysems can be eleganly modeled wih he help of he FDEs, such as dielecric polarizaion, viscoelasic sysems, conrol heory, chaoic behavior, and elecrolye-elecrolye polarizaion 4 6 Since is remendous applicaions in several disciplines, considerable aenion has been given o he exac and he numerical soluions of fracional differenial equaions and fracional inegral equaions Even numerical approximaion of fracional differeniaion of rough funcions is no easy as i is an ill-posed problem Oher han modeling aspecs of hese differenial equaions, he soluion echniques and heir reliabiliy are raher more significan In order o obain he goal of highly accurae andreliablesoluions,severalmehodshavebeenproposed o solve he fracional order differenial and fracional order inegral equaions The mos commonly used mehods are Variaional Ieraion Mehod 7, Adomian Decomposiion Mehod 8, 9, Generalized Differenial Transform Mehod,, and Wavele Mehod 2, 3 Recenly, more and more physiciss and mahemaicians arefindinghanumerousimporandynamicalproblems exhibi fracional order behavior which can vary wih space and ime This fac indicaes ha variable order calculus provides an effecive mahemaical framework for he descripion of complex dynamical problems The concep of a variable order operaor is a much more recen developmen, which is a new orienaion in engineering Many researchers have proposed differen definiions of variable order differenial operaors, each of hese wih a specific meaning o ge desired goals The variable order operaor definiions recenly proposed in he engineering include he Riemann- Liouville definiion, Marchaud definiion, Grünwald definiion, Capuo definiion, and Coimbra definiion 4, 5 In his paper, he main objecive is o inroduce he Chebyshev polynomials mehod o solve he variable order fracional inegral-differenial equaion The mehod is based on reducing he equaion o a sysem of algebraic equaions by expanding he soluion as Chebyshev polynomials wih unknown coefficiens The main characerisic of an operaional mehod is o conver he inegral-differenial equaion

2 2 Mahemaical Problems in Engineering ino an algebraic one I no only simplifies he problem bu also speeds up he compuaion Oursudyfocusesonaclassofvariableorderfracional inegral-differenial equaion as follows: D α() u (x, ) g (x, )+ u (x, ) + u (x, T) k (x, T) dt = f (x, ) subjec o he iniial condiions u (x, ) =g(x) x,, u (, ) =h(),, + u (x, T) dt D α(x) (u(x, )g(x, )) is fracional derivaive of Capuo sense; when g(x, ) = u(x, ), heiniialproblemischanged o nonlinear equaion f(x, ), g(x, ), u(x, ), andk(x, ) are assumed o be casual funcions of ime and space on he secion (x,),,,f(x, ), g(x, ),andk(x, ) are known and u(x, ) is he unknown, < α(x) 2 Chebyshev Polynomials and Their Properies The well-known Chebyshev polynomials are defined on he inerval, and can be deermined wih he recurrence formula 6 () (2) n/2 denoes he ineger par of n/2 and n denoes posiive ineger The orhogonaliy condiion is T i (x) T j (x) x 2 dx = π, i = j = { π 2 { {, i =j, i = j = In order o use hese polynomials on he inerval,, we define he shifed Chebyshev polynomials by inroducing he change of variable x=2 Therefore,heshifedChebyshev polynomials are defined as T n () = T n (2 ) Theanalyic form of he shifed Chebyshev polynomials T n () of degree n is given by Le T n () =n n k= (5) ( ) n k 2 2k (n+k )! (2k)! (n k)! k, n=,2, (6) Φ () = T (),T (),,T n ()T (7) T n+ (x) =2x T n (x) T n (x), T (x) =, T (x) =x, n=,2, (3) TheChebyshevpolynomialsgivenby(6)canbeexpressedin he marix form The analyic form of he Chebyshev polynomials T n (x) of degree n is given by n/2 T n (x) =n i= ( ) i 2 n 2i (n i )! (i)! (n 2i)! xn 2i, (4) Φ () = AT n (), (8) 2 A = 2 ( ) 2! 2 ( ) 2 2 2! 2 ( ) 2 4 3! 2! 2! 4!, d n ( ) n (n )! n ( ) n 2 2 n! n ( ) n (n+)! n( ) 2 2n (2n )! n! 2! (n )! 4! (n 2)! (2n)! Τ n () = n (9)

3 Mahemaical Problems in Engineering 3 Obviously Τ n () = A Φ () () Afuncionu() L 2 (, ) can be expressed in erms of he ChebyshevbasisInpracice,onlyhefirs(n + ) erm of Chebyshev polynomials is considered Hence u () n i= c i T i () = c T Φ (), () c = c,c,,c n T, c i (i =,,2,,n) are called Chebyshev coefficiens and c = Q (u, Φ())Thedimension of Q is (n + ) (n + ); i is called he inner produc marix which is given by Q = Φ () Φ T () dx = (AT n ())(AT n ()) T d = A ( T n () T T n () d) A T = AHA T, H = 2 n+ (2) 2 n+ 3 n+2 d (3) n+2 2n + For he funcion u(x, ) L 2 (,, ),wecanalsoobain is approximaion by using Chebyshev polynomials n n u (x, ) u ij T i () T j () = ΦT (x) UΦ (), (4) i= j= u u u n u u u n U = d (5) u n u n u nn Theorem (see 6) The error in approximaing u() by he sum of is firs n erms is bounded by he sum of he absolue values of all he negleced coefficiens If hen u n () = n i= E T (m) = u () u n () for all u(),alln,andall, c i T i () (6) c k (7) k=n+ Theorem 2 (see 7) The Capuo fracional derivaive of order α for he shifed Chebyshev polynomials can be expressed inermsofheshifedchebyshevpolynomialshemselvesinhe following form: θ i,j,k = D α (T i ()) = i k α k= α j= θ i,j,k T j (), (8) ( ) i k 2i (i+k )!Γ (k α+/2) Γ (k+/2)(i k)!γ(k α j+)γ(k α+j+), j=,, (9) Theorem 3 The error E T (m) = Dα (u()) D α (u n ()) in approximaing D α (u()) by D α (u n ()) is bounded by E T (m) i=n+ c i( i k= α k α j= θ i,j,k ) Proof A combinaion of (7) and (25) leads o E T (m) = Dα (u ()) D α (u n ()) = i=n+ bu T j,sowecange E T (m) c i ( i=n+ i k α k= α j= c i ( i k= α θ i,j,k T j ()), k α j= θ i,j,k ) (2), (2) subracing he runcaed series from he infinie series, bounding each erm in he difference, summing he bounds, and hence compleing he proof of he heorem 3 Operaional Marix of he Chebyshev Polynomials 3 Fracional Calculus Before we inroduce he Chebyshev polynomials operaional marix of he fracional inegraion, we firs review some basic definiions of fracional calculus, which have been given in 8 Definiion 4 The Riemann-Liouville fracional inegral operaor of order α(): I α() a+ u () = Γ (α ()) ( T) α() u (T) dt, a+ (22) > Re (α ()) > Definiion 5 Riemann-Liouville fracional derivae wih order α(): D α() a+ u () = Γ (m α()) d m d m a u (τ) α() m+ dτ ( τ) (23) (m α() <m)

4 4 Mahemaical Problems in Engineering Definiion 6 Capuo s fracional derivae wih order α(), ( < α() ): D α() u () = Γ ( α()) + + ( τ) α() u (τ) dτ (u (+) u( )) α() Γ ( α()) (24) If we assume he saring ime in a perfec siuaion, we can obain he definiion as follows: D α() u () = Γ ( α()) + ( τ) α() u (τ) dτ ( <α() <) (25) Generally, we adop (25) as he definiion of fracional derivae in Capuo sense Wih he definiion above, we can obain he following formula ( < α() ): Define he (n + ) (n) marix V (n+) n and vecor T n () as V (n+) n = 2 d, n T n () = n Equaion(28)mayhenberesaedas (n ) (29) Φ () = AV (n+) n T n () (3) Now we expand vecor T n () in erms of Φ()From(),we ge D α() c= D α() x β { β = = Γ(β+) { { Γ(β+ α()) xβ a() β =, 2, 3, (26) T n () = B Φ (), (3) B =A, A 2,,A n T (32) A k is kh row of A, k=,2,,n Then we have 32 The Operaional Marix of he Secion as u(x, )/ in erms of Chebyshev Polynomials The differeniaion of vecor Φ() in (7) can be given by Φ () = DΦ (), (27) D is he (n+) (n+) operaional marix of derivaives for Chebyshev polynomials From (8) we have Φ () = A n n (28) Φ () = AV (n+) n B Φ () (33) Therefore we obain he operaional marix of he secion as u(x, )/ as follows: u (x, ) = Φ T (x) UΦ () = Φ T (x) UAV (n+) n B Φ () (34) 33 The Operaional Marix of he Secion as D α() (u(x, )g(x, )) in erms of Chebyshev Polynomials If we approximae he funcions u(x, ), g(x, ) wih Chebyshev polynomials, hey can be wrien as u(x, ) = Φ T (x)uφ() and g(x, ) = Φ T (x)gφ(),u is unknown and G is known Thenwehave D α() u (x, ) g (x, ) =D α() Φ T (x) UΦ () Φ T () GΦ (x) =Φ T (x) UD α() Φ () Φ T () GΦ (x) = Φ T (x) UD α() AT n () (AT n ())T A T GΦ (x) = Φ T (x) UAD α() T n () (T n ())T A T GΦ (x) n = Φ T (x) UAD α() ( n ) A T GΦ (x) = Φ T (x) UAD α() 2 n+ d A T GΦ (x) n n 2n 2n

5 Mahemaical Problems in Engineering 5 = Φ T (x) UA Γ (n+) Γ (n+ α()) n α() = Φ T (x) UAMA T GΦ (x) Γ (2) Γ (2 α()) α() Γ (2) Γ (2 α()) α() Γ (3) Γ (3 α()) 2 α() d Γ (n+2) Γ (n+2 α()) n+ α() Γ (n+) Γ (n+ α()) n α() Γ (n+2) Γ (n+2 α()) n+ α() A T GΦ (x) Γ (2n + ) Γ (2n+ α()) 2n α() (35) Now we define M = Γ (n+) Γ (n+ α()) n α() Γ (2) Γ (2 α()) α() Γ (2) Γ (2 α()) α() Γ (3) Γ (3 α()) 2 α() d Γ (n+2) Γ (n+2 α()) n+ α() Γ (n+) Γ (n+ α()) n α() Γ (n+2) Γ (n+2 α()) n+ α() (36) Γ (2n + ) Γ (2n+ α()) 2n α() M is called he operaional marix of he secion as D α() (u(x, )g(x, )) wih Chebyshev polynomials So we have D α() u (x, ) g (x, ) =Φ T (x) UAMA T GΦ (x) (37) 34 The Operaional Marix of he Secion as u(x, T)dT in ermsofchebyshevpolynomials The inegraion of he vecor Φ() in (7) can be expressed as Φ () dt = PΦ (), (38) P is he (n+) (n+) operaional marix of inegraion for Chebyshev polynomials So we have Φ (T) dt = AT n (T) dt = A T n (T) dt c T = A dt = A T n = A p T p, 2 2 n+ n+ (39) A p is an (n + ) (n + ) marix: A p = A 2 d, n+ 2 T p = n+ (4) Now we approximae he elemens of vecor T p in erms of Φ()By(),henwehave k = A k+ Φ (), k=,,n, (4) A k+ is he k+h row of A for k =,,n We jus need o approximae n+ = c T n+φ() Byusingc = Q (u, Φ()),wehave

6 6 Mahemaical Problems in Engineering c n+ = Q n+ Φ () d We define n+ B,n () d = Q n+ B,n () d n+ B n,n () d 2n + 2 ( n n ) ( ) ( 2n+ n+ ) ( 2n+ n+2 ) ( = Q P = A 2 A 3 A n+ c n+ n n ) ( 2n+ 2n+ ) T (42) (43) Then we can ge T p = P Φ() Therefore we have he operaional marix of inegraion as follows: So we have P = A p P (44) u (x, T) dt = Φ T (x) UΦ (T) dt = Φ T (x) U Φ (T) dt = Φ T (x) UPΦ () = Φ T (x) UA p P Φ () (45) 35 The Operaional Marix of he Secion as u(x, T)k(x, T)dT in erms of Chebyshev Polynomials Firsly, we approximae he funcion k(x, ) wih Chebyshev polynomials; i can be wrien as k(x, ) = Φ T (x)kφ(), andk is known So we have u (x, T) k (x, ) dt = (Φ T (x) UΦ (T) Φ T (T) KΦ (x))dt=φ T (x) U (Φ (T) Φ T (T))dTKΦ(x) = Φ T (x) UA T T n T T 2 T n+ d dta T KΦ (x) = Φ T (x) T n T n+ T 2n 2 2 UA d n+ n+ n+2 n+2 = Φ T (x) UARA T KΦ (x) n+ n+ n+2 n+2 A T KΦ (x) 2n + 2n+ (46) We define 2 2 n+ n+ R = n+2 n+2 (47) d n+ n+ n+2 n+2 2n + 2n+ R is called he operaional marix of he secion as u(x, T)k(x, T)dT in erms of Chebyshev polynomials Therefore he iniial equaion is ransformed ino he producs of several dependen marixes as follows: Φ T (x) UAMA T GΦ (x) + Φ T (x) UPΦ () + Φ T (x) UAV (n+) n B Φ () + Φ T (x) UARA T KΦ (x) =f(x, ) (48) Dispersing (48) wih (x i, j ) (i =,2,,n; j =,2,,n), by using a symbolic sofware such as Mahemaica, we can ge U So he numerical soluion of he original problem is obained ulimaely 4 Numerical Examples To demonsrae he efficiency and he pracicabiliy of he proposed mehod based on Chebyshev polynomials mehod, we presen some examples and find heir soluion via he mehod described in he previous secion Example Consider D /3 u (x, )(x++) + u (x, ) + u (x, T) dt + u (x, T)(x+) dt = f (x, ) u (x, ) =x 2 u (, ) = 2 x,,,, (49)

7 Mahemaical Problems in Engineering The absolue error beween he exac soluion and he numerical soluion is displayed in Figure Taking n=3,dispersingx i =k i /4 /8, j =k j /4 /8 (k i =,2,3,4; k j =,2,3,4),wecangehemarixU as follows: 5 x 2 Figure : The absolue error beween he numerical soluion and he exac soluion when n=2 f (x, ) = x 3 +x2 + 2 x 2 2 +x3 3 /3 6 (9+8) 6( 9 + ) x ( 9 + )( 6 + )( 3 + ) Γ ( /3) (5) 3 3 U = (52) The absolue error beween he exac soluion and he numerical soluion is displayed in Figure 2 Example 2 Consider The exac soluion of he above equaion is u(x, ) = x Taking n=2,dispersingx i =k i /3 /6, j =k j /3 /6 (k i =,2,3; k j =,2,3),wecangehemarixU as follows: U = (5) D sin /3 u (x, )(x) + u (x, ) + u (x, T) dt + u (x, T)(x+) dt = f (x, ) u (x, ) = (+x) 2 u (, ) = (+) 2 x,,,, (53) f (x, ) =2(+x+) x+32 x+ 3 x+3x x 2 +x sin /3 x54(++x) 2 + (+x) sin ( 3 (5+4+5x) + (+x) sin ) Γ ( sin /3)( 9 + sin )( 6 + sin )( 3 + sin ) (54) The exac soluion of he above problem is u(x, ) = (+x+) 2 Taking n=2,dispersingx i =k i /3 /6, j =k j /3 /6 (i =, 2, 3; j =, 2, 3), we can obain he marix U as follows: 2 4 U = (55) U = (56) The absolue error beween he exac soluion and he numerical soluion is displayed in Table Taking n=3,dispersingx i =k i /3 /6, j =k j /3 /6 (i =, 2, 3; j =, 2, 3), hemarixu is displayed as follows: The absolue errorbeween he exac soluion and he numerical soluion is displayed in Table 2 Taking n = 4,dispersingx i = k i /5 /, j = k j /5 / (i =,2,,5; j =,2,,5),hemarixU is displayed as follows:

8 8 Mahemaical Problems in Engineering Table : The absolue error beween he numerical soluion and he exac soluion when n=2 = = 3 = 5 = 7 = 9 x = x = 2257e 4 486e e e 4 578e 4 x = e e e e e 4 x = e e e 3 79e e 4 x = e 4 645e 3 775e e 4 48e 4 x = e 4 66e e e e 4 x = e 4 487e 3 63e e e 4 x = 7 565e e e 3 862e 4 295e 4 x = e e 3 2e e e 4 x = 9 386e e e e 4 322e 5 x = 8274e e e e 4 923e 4 Table 2: The absolue error beween he numerical soluion and he exac soluion when n=3 = = 3 = 5 = 7 = 9 x = x = 87636e e e e e 5 x = e e e e e 5 x = e e e e e 5 x = e e e e e 5 x = e e e e e 5 x = e e e e e 5 x = e e e e e 5 x = e e e e 6 x = e e e e e 5 x = 836e e e e e U = (57) The absolue error beween he exac soluion and he numerical soluion is displayed in Table 3 When g(x, ) = u(x, ), he iniial equaion becomes nonlinear equaion Example 3 describes he siuaion Example 3 Consider 5 x Figure 2: The absolue error beween he numerical soluion and he exac soluion when n=3 D /3 u 2 (x, ) +D /4 u (x, ) + 2 u (x, ) 2 =f(x, ) u (x, ) =x 2, u (, ) = 2 (x, ),,, f (x, ) =2 (58) 36 2 /3 ( ( 2 + )( 9 + ) x 2 ) + ( 2 + )( 9 + )( 6 + )( 3 + ) Γ ( /3) 32 2 /4 + ( )Γ( /4) (59) The exac soluion of he above equaion is u(x, ) = x 2 + 2

9 Mahemaical Problems in Engineering 9 Table 3: The absolue error beween he numerical soluion and he exac soluion when n=4 = = 3 = 5 = 7 = 9 x = x = e e e e e 4 x = e e e e e 5 x = e e e 5 856e e 4 x = e e e e e 4 x = e e e e 5 4e 4 x = e e e e 5 433e 4 x = e e e e e 4 x = e e e e e 4 x = e e e e e 4 x = 78553e e e e 4 u (x, ) u (x, ) x 5 x Figure 3: The numerical soluion for Example 3 of n=2 Figure 4: The exac soluion for Example 3 This is a nonlinear variable order fracional differenial equaion; he numerical soluion can also be gained wih he mehod proposed in Secion 3 when n 2 Taking n=2,dispersingx i =k i /2 /4, j =k j /2 /4 (k i =,2; k j =,2), we can obain he marix U as follows: U = (6) 2 The numerical soluion obained by our mehod and he exac soluion are shown in Figures 3 and 4 The absolue error beween he exac soluion and he numerical soluion is displayed in Figure 5 When n 3, he compuaion is very large and geing he numerical soluion is a very difficul hing From Figures 5, Tables 3, we can see ha he absolue errors are very small and only a small number of Chebyshev polynomials are needed Compared wih he oher mehods proposedin9,2,hemehodinhispaperhassignifican advanages The calculaing resuls also show ha combined wihchebyshevpolynomialshemehodinhispapercan 5 x Figure 5: The absolue error for Example 3 of n=2 be effecively used in he numerical soluion of he fracional equaion From he above resuls, he numerical soluions are in good agreemen wih he exac soluion

10 Mahemaical Problems in Engineering 5 Conclusion Inhepresenpaper,heapplicaionandscopeofheChebyshev polynomials have been exended o a class of variable order fracional inegral-differenial equaion successfully Acually we derive four kinds of operaional marixes using Chebyshevpolynomialsanduseheseosolvehevariable order fracional inegral-differenial equaion numerically By solving he sysem of algebraic equaions, numerical soluions are obained Numerical examples illusrae he powerfulness of he proposed mehod The soluions obained using he suggesed mehod show ha numerical soluions are in very good coincidence wih he exac soluion The mehod can be applied by developing for he oher fracional problem Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper Acknowledgmen This work was suppored by he Naional Naural Science Foundaion of China under Gran no 5374 References J C Wang, Realizaions of generalized Warburg impedance wih RC ladder neworks and ransmission lines, he Elecrochemical Sociey,vol34,no8,pp95 92,987 2 F J Valdes-Parada, J A Ochoa-Tapia, and J Alvarez-Ramirez, Effecive medium equaions for fracional Fick s law in porous media, Physica A: Saisical Mechanics and is Applicaions,vol 373, pp , 27 3 H Sun, W Chen, C Li, and Y Chen, Fracional differenial models for anomalous diffusion, PhysicaA:SaisicalMechanicsandIsApplicaions,vol389,no4,pp ,2 4 M Ichise, Y Nagayanagi, and T Kojima, An analog simulaion of non-ineger order ransfer funcions for analysis of elecrode processes, Elecroanalyical Chemisry, vol 33, no 2, pp , 97 5 H H Sun, A A Abdelwahab, and B Onaral, Linear approximaion of ransfer funcion wih a pole of fracional order, IEEE Transacions on Auomaic Conrol, vol 29, no 5, pp , R C Koeller, Applicaions of fracional calculus o he heory of viscoelasiciy, JournalofAppliedMechanics,vol5,no2,pp , Z M Odiba, A sudy on he convergence of variaional ieraion mehod, Mahemaical and Compuer Modelling, vol 5, no 9-, pp 8 92, 2 8 I L El-Kalla, Convergence of he Adomian mehod applied o a class of nonlinear inegral equaions, Applied Mahemaics Leers,vol2,no4,pp ,28 9 M M Hosseini, Adomian decomposiion mehod for soluion of nonlinear differenial algebraic equaions, Applied Mahemaics and Compuaion,vol8,no2,pp ,26 S Momani, Z Odiba, and V S Erurk, Generalized differenial ransform mehod for solving a space- and ime-fracional diffusion-wave equaion, Physics Leers A,vol37,no5-6,pp , 27 Z Odiba, S Momani, and V S Erurk, Generalized differenial ransform mehod: applicaion o differenial equaions of fracional order, Applied Mahemaics and Compuaion, vol 97, no 2, pp , 28 2 YMChen,MXYi,andCXYu, Erroranalysisfornumerical soluion of fracional differenial equaion by Haar waveles mehod, Compuaional Science,vol3,no5,pp , 22 3 J L Wu, A wavele operaional mehod for solving fracional parial differenial equaions numerically, Applied Mahemaics and Compuaion,vol24,no,pp3 4,29 4 C F Lorenzo and T T Harley, Variable order and disribued order fracional operaors, Nonlinear Dynamics,vol29,no 4, pp 57 98, 22 5 C F Coimbra, Mechanics wih variable-order differenial operaors, Annalen der Physik, vol 2, no -2, pp , 23 6 M A Snyder, Chebyshev Mehods in Numerical Approximaion, Prenice-Hall, Englewood Cliffs, NJ, USA, E H Doha, A H Bhrawy, and S S Ezz-Eldien, A Chebyshev specral mehod based on operaional marix for iniial and boundary value problems of fracional order, Compuers and Mahemaics wih Applicaions, vol62,no5,pp , 2 8 S G Samko, Fracional inegraion and differeniaion of variable order, Analysis Mahemaica,vol2,no3,pp23 236, 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 , 29 2 PZhuang,FLiu,VAnh,andITurner, Numericalmehodsfor he variable-order fracional advecion-diffusion equaion wih anonlinearsourceerm, SIAM Journal on Numerical Analysis, vol47,no3,pp76 78,29

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