Research Article Reproducing Kernel Method for Fractional Riccati Differential Equations

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1 Absrac and Applied Analysis Volume 24, Aricle ID 97967, 6 pages hp://d.doi.org/.55/24/97967 Research Aricle Reproducing Kernel Mehod for Fracional Riccai Differenial Equaions X. Y. Li, B. Y. Wu, 2 and R. T. Wang Deparmen of Mahemaics, Changshu Insiue of Technology, Suzhou, Jiangsu 255, China 2 Deparmen of Mahemaics, Harbin Insiue of Technology, Harbin, Heilongjiang 5, China Correspondence should be addressed o B. Y. Wu; mahwby@sina.com Received 29 December 23; Acceped 7 April 24; Published 27 April 24 Academic Edior: Youyu Wang Copyrigh 24 X. Y. Li 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. This paper is devoed o a new numerical mehod for fracional Riccai differenial equaions. The mehod combines he reproducing kernel mehod and he quasilinearizaion echnique. Is main advanage is ha i can produce good approimaions in a larger inerval, raher han a local viciniy of he iniial posiion. Numerical resuls are compared wih some eising mehods o show he accuracy and effeciveness of he presen mehod.. Inroducion This paper deals wih he numerical soluion of he following fracional Riccai differenial equaion: u α () =p() +q() u () +r() u 2 (), T, <α, u () =, where u α () denoes he Capuo fracional derivaive of order α and u α () = Γ ( α) ( τ) α u (τ) dτ. (2) Riccai differenial equaions arise in many fields []. The problem has araced much aenion and has been sudied by many auhors. However, deriving is analyical soluion in an eplici form seems o be unlikely ecep for cerain special siuaions. Recenly, many numerical mehods [2 9] have been proposed o solve ineger order Riccai differenial equaions. However, he discussion on he numerical mehods for fracional order Riccai differenial equaions is rare. Odiba and Momani [] developed a modified homoopy perurbaion mehod for fracional Riccai differenial equaions. () Li [] presened a numerical mehod for fracional differenial equaions based on Chebyshev waveles. Hosseinnia e al. [2] applied an enhanced homoopy perurbaion mehod for fracional Riccai differenial equaions. Yüzbaşı [3] inroduced a numerical mehod for fracional Riccai differenial equaions using he Bernsein polynomial. Khader [4] developed he fracional Chebyshev finie difference mehod for fracional Riccai differenial equaions. Yang and Baleanu [5], Yang e al. [6], and Baleanu e al. [7]proposed local fracional variaion ieraion for fracional hea conducionandwaveequaions. Recenly, based on he reproducing kernel heory, Cui, Geng, and Lin presened he reproducing kernel mehod (RKM) for linear and nonlinear operaor equaions [8 2]. The mehod has been developed and applied o many problems [22 26]. The aim of his paper is o presen a new mehod for fracional Riccai differenial equaions, based on he RKM and he quasilinearizaion echnique. The res of he paper is organized as follows. In he ne secion, he quasilinearizaion echnique is applied o fracional Riccai differenial equaion. The RKM for reduced linear fracional differenial equaions is inroduced in Secion 3. The numerical eamples are presened in Secion 4. Secion5 ends his paper wih a brief conclusion.

2 2 Absrac and Applied Analysis 2. Quasilinearizaion of Riccai Differenial Equaion () In his secion, he quasilinearizaion echnique is applied o reduce () o a series of linear fracional problems. Define f(, u) = p() + r()u 2. By choosing an appropriae iniial approimaion u () for he funcion u() in f(, u) and epanding f(, u) around u (),ifollowsha f(,u )=f(,u )+(u u ) f +. (3) u u=u Generally, one can wrie for k =,2,...(k = ieraion inde) f(, u k )=f(,u k )+(u k u k ) f +. (4) u u=u k Therefore, he following ieraion formula for () canbe derived: u α k () +a k () u k () =f k (), k=,2,..., (5) u k () =, where a k () = [q() + ( f/ u) u=uk ] = [q() + 2r()u k ()]and f k () = f(, u k ) ( f/ u) u=uk = p() r()u 2 k () and u () is he iniial approimaion. Clearly, o solve (), i suffices for us o solve he series of linear problem (5). 3. Mehod for Solving Linear Fracional Problem (5) To illusrae how o solve (5) we consider he problem of solving LV () = V α () +a() V () =f(), <<T, (6) V () =, where a() and f() are coninuous. By he definiion of Capuo fracional derivaive, (6) is equivalen o he following equaion: Γ ( α) ( τ) α V (τ) dτ + a () V () =f(), V () =. T, If u() C [, T], hen he improper inegral ( τ) α V (τ)dτ eiss and ( τ) α V (τ) dτ = V (τ) V () ( τ) α = α V () α = α V () α dτ + V () ( τ) α dτ V () V (τ) ( τ) α dτ g (τ, V) dτ. (7) (8) Table : Comparison of he numerical soluions wih he oher mehods for α =.75. Ours [] [] Define g (τ, V) = { V () V (τ) { ( τ) α, τ=, {, τ =. This gives us a coninuous funcion on [, ] and hen inegral g(τ, V)dτ also eiss and ((V () V (τ))/( τ) α )dτ = g(τ, V)dτ. Therefore, (6) canbeconveredino ( α V () Γ ( α) α (9) g (τ, V) dτ) + a () V () =f(), V () =. T, () Applying Hermie s quadraure formula o ((V () V (τ))/( τ) α )dτ,oneobains M g (τ, V) dτ = π g( 2M k,u( k )) k 2, () k= where g(, u()) = g((/2)( + ), u((/2)( + ))), M is he number of nodes, and k = cos((2k )/2M), k=,...,m. Then (6) can be furher equivalenly approimaed o Γ ( α) ( α V () α +a() V () =f(), M π g( 2M k, V ( k )) k 2) k= V () =. T, (2) To apply he RKM o (2), i is necessary o consruc he following reproducing kernel Hilber space W 3 [, T]. Definiion. W 3 [, T] = {u() u () is an absoluely coninuous real value funcion, u (3) () L 2 [, T],

3 Absrac and Applied Analysis u().6.4 u() (a) (b).8 u() (c) Figure : The behavior of approimae soluion wih differen values of α ((a) α =.99;(b)α =.75,.99;(c)α =.5,.75,.99). Eac soluion Absolue errors (a) (b) Figure 2: Comparison of approimae soluions wih he eac soluions for α=((a) eac soluion; (b) absolue errors). u() = }. The inner produc and norm in W 3 [, T] are given, respecively, by (u (y), V (y)) 3 =u() V () +u () V () +u () V T () + u (3) V (3) dy, u 3 = (u, u) 3, u,v W 3 [, T]. (3) Theorem 2. W 3 [, T] is a reproducing kernel space and is reproducing kernel is k(,y)={ k (, y), k (y, ), y, y >, (4) where k (, y) = y 2 (7 y 4 y y(4 + y) 2 2 ( 6 + y 3 ))/54.

4 4 Absrac and Applied Analysis Table 2: Comparison of he numerical soluions wih he oher mehods for α =.9. Ours [3] IABMM[2] MHPM[] Table 3: Numerical resuls for α =.99, on [, 4]. Ours (α =.99) Ours(α = ) Eac soluion (α = ) Table 4: Comparison of he numerical soluions wih he oher mehods for α =.75. Ours [3] IABMM[2] MHPM[] Table 5: Comparison of he numerical soluions wih he oher mehods for α =.9. Ours [3] IABMM[2] MHPM[] Definiion 3. W [, T] = {u() u() is an absoluely coninuous real value funcion, u () L 2 [, T]}. Theinner produc and norm in W [, T] are given, respecively, by T (u (y), V (y)) =u() V () + u V dy, u = (u, u), u,v W [, T]. (5) Table 6: Comparison of he numerical soluions wih he oher mehods for α =.. Eac Ours [3] MHPM[] Theorem 4. W [, T] is a reproducing kernel space and is reproducing kernel is Pu LV () = k(,y)={ +y, y, +, y >. ( α V () Γ ( α) α π 2M M k= (6) g( k, V ( k )) k 2 )+a() V (). (7) Clearly, L : W 3 [, T] W [, T] is a bounded linear operaor. Pu φ i () = k(, i ) and ψ i () = L φ i (), where L is he adjoin operaor of L. Theorhonormalsysem {ψ i ()} i= of W3 [, T] can be derived from Gram-Schmid orhogonalizaion process of {ψ i ()} i=, ψ i () = i k= β ik ψ k (), (β ii >,i=,2,...). (8) Theorem 5. If { i } i= is dense on [, T], hen{ψ i()} i= is he complee sysem of W 3 [, T]. Theorem 6. If { i } i= is dense on [, T] and he soluion of (2) is unique, hen he soluion of (2) is i V () = β ik f( k ) ψ i (). (9) i=k= Now, an approimae soluion V N () of (6) canbeobained by he N-erm inercep of he eac soluion V() and N i V N () = β ik f( k ) ψ i (). (2) i=k= Similarly, he approimae soluions u k () can be obained: u k,n () = where A j = j l= β jlf k ( l ). N j= A j ψ j (), (2)

5 Absrac and Applied Analysis 5 4. Numerical Eamples Eample. Consider he following fracional Riccai differenial equaion [ 3]: u α () =+2u() u 2 (), T, <α, u () =. The eac soluion for α=canbeeasilydeerminedobe (22) u () =+ 2 anh ( 2+ log (( + 2) / ( + 2)) ). 2 (23) Applying he proposed mehod, aking T =, k = 3, M = 3, N = 5, he numerical resuls compared wih oher mehods are lised in Tables and 2. TakingT = 4, k = 3, M = 3, N = 5,henumericalresulson[, 4]arelised in Table 3.From Table 3, i is easily found ha he presen approimaions are effecive for a larger inerval, raher han a local viciniy of he iniial posiion. Eample 2. Consider he following fracional Riccai differenial equaion [ 4]: u α () =+u 2 (), u () =. T, <α The eac soluion for α=canbeeasilydeerminedobe (24) u () = e2 e 2 +. (25) According o he presen mehod, aking T =, k = 3, M = 5, N = 5, henumericalresulscompared wih oher mehods are given in Tables 4, 5, and6. Taking T = 4, k = 5, M = 5, N = 8, henumericalresuls on [, 4] areshowninfigures and 2. Fromhesefigures we can conclude ha he obained numerical soluions are in ecellen agreemen wih he eac soluion for a larger inerval. 5. Conclusion In his paper, combining he RKM, he numerical inegral, and quasilinearizaion echniques, a new numerical mehod is proposed for fracional Riccai differenial equaions. The mainadvanageofhismehodishaicanprovideaccurae numerical approimaions on a larger inerval. Numerical resuls compared wih he eising mehods show ha he presen mehod is a powerful mehod for solving fracional Riccai differenial equaions. 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 (Gran nos , 27, and 26222). References [] W. T. Reid, Riccai Differenial Equaions, Academic Press, New York, NY, USA, 972. [2] M.A.El-Tawil,A.A.Bahnasawi,andA.Abdel-Naby, Solving Riccai differenial equaion using Adomian s decomposiion mehod, Applied Mahemaics and Compuaion, vol. 57, no. 2,pp.53 54,24. [3] M. Lakesani and M. Dehghan, Numerical soluion of Riccai equaion using he cubic B-spline scaling funcions and Chebyshev cardinal funcions, Compuer Physics Communicaions, vol. 8, no. 5, pp , 2. [4] F. Z. Geng, Y. Z. Lin, and M. G. Cui, A piecewise variaional ieraion mehod for Riccai differenial equaions, Compuers & Mahemaics wih Applicaions, vol.58,no.-2,pp , 29. [5] B.-Q. Tang and X.-F. Li, A new mehod for deermining he soluion of Riccai differenial equaions, Applied Mahemaics and Compuaion,vol.94,no.2,pp.43 44,27. [6] A. Ghorbani and S. Momani, An effecive variaional ieraion algorihm for solving Riccai differenial equaions, Applied Mahemaics Leers,vol.23,no.8,pp ,2. [7] S. Momani and N. Shawagfeh, Decomposiion mehod for solving fracional Riccai differenial equaions, Applied Mahemaics and Compuaion,vol.82,no.2,pp.83 92,26. [8] S. Abbasbandy, A new applicaion of He s variaional ieraion mehod for quadraic Riccai differenial equaion by using Adomian s polynomials, Compuaional and Applied Mahemaics,vol.27,no.,pp.59 63,27. [9] F. Mohammadi and M. M. Hosseini, A comparaive sudy of numerical mehods for solving quadraic Riccai differenial equaions, he Franklin Insiue. Engineering and Applied Mahemaics,vol.348,no.2,pp.56 64,2. [] Z. Odiba and S. Momani, Modified homoopy perurbaion mehod: applicaion o quadraic Riccai differenial equaion of fracional order, Chaos, Solions & Fracals, vol. 36, no., pp , 28. [] Y. L. Li, Solving a nonlinear fracional differenial equaion using Chebyshev waveles, Communicaions in Nonlinear Science and Numerical Simulaion, vol. 5, no. 9, pp , 2. [2] S. H. Hosseinnia, A. Ranjbar, and S. Momani, Using an enhanced homoopy perurbaion mehod in fracional differenial equaions via deforming he linear par, Compuers & Mahemaics wih Applicaions, vol.56,no.2,pp , 28. [3] S. Yüzbaşı, Numerical soluions of fracional Riccai ype differenial equaions by means of he Bernsein polynomials, Applied Mahemaics and Compuaion, vol.29,no.,pp , 23. [4] M. M. Khader, Numerical reamen for solving fracional Riccai differenial equaion, he Egypian Mahemaical Sociey,vol.2,no.,pp.32 37,23. [5] X. J. Yang and D. Baleanu, Fracal hea conducion problem solved by local fracional variaion ieraion mehod, Thermal Science,vol.7,no.2,pp ,23.

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