Numerical solutions of ordinary fractional differential equations with singularities

Transcription

1 arxiv: v [mat.na] 8 Jun 28 Numerical solutions of ordinary fractional differential equations wit singularities Yuri Dimitrov, Ivan Dimov, Venelin Todorov Abstract Te solutions of fractional differential equations FDEs) ave a natural singularity at te initial point. Te accuracy of teir numerical solutions is lower tan te accuracy of te numerical solutions of FDEs wose solutions are differentiable functions. In te present paper we propose a metod for improving te accuracy of te numerical solutions of ordinary linear FDEs wit constant coefficients wic uses te fractional Taylor polynomials of te solutions. Te numerical solutions of te two-term and tree-term FDEs are studied in te paper. Introduction In recent years tere is a growing interest in applying FDEs for modeling diffusion processes in biology, engineering and finance [3, 2]. Te two main approaces to fractional differentiation are te Caputo and Riemann- Liouville fractional derivatives. Te Caputo derivative of order α, were <α < is defined as DC α yt)=yα) t)= dα dt α yt)= t y ξ) Γ α) t ξ) α dξ. Te Caputo derivative is a standard coice for a fractional derivative in te models using FDEs. Te finite difference scemes for numerical solution of FDEs involve approximations of te fractional derivative. Let t n = n, were is a small positive number and y n = yt n )=yn). Te L approximation is an important and commonly used approximation of te Caputo derivative. y α) n = α Γ2 α) n σ α) y n + O 2 α), )

2 were σ α) =, σ n α) =n ) α n α and σ α) =+) α 2 α + ) α, =,,n ). In [8] we obtain te second-order approximation te Caputo derivative were δ α) = σ α) y α) n = Γ2 α) α for 2 n and n δ α) y n + O 2), 2) δ α) = σ α) ζα ),δ α) = σ α) + 2ζα ),δ α) 2 = σ α) 2 ζα ). Wen < α < and te function y C 2 [,t n ], te L approximation as an accuracy O 2 α) and approximation 2) as an accuracy O 2). Te zeta function satisfies ζ) = /2. From 2) wit α = we obtain te second-order approximation for te first derivative y n = 3 2 y n 2y n + ) 2 y n 2 + O 2). 3) Te Caputo derivative of order α, were <α < 2 is defined as DC α yt)= yα) t)= dα dt α yt)= t y ξ) dξ. Γ2 α) t ξ) α In [9] we obtain te expansion formula of order 4 α of te L approximation of te Caputo derivative. Wen < α < 2, approximation 2) as an asymptotic expansion Γ2 α) α n δ α) y n = y α) n + D 2 y n α) y ) 2 Γ α)t +α 2 ζα )+ζα 2) y n 3 α + O Γ2 α) min{3,4 α}). Wen <α < 2 approximation 2) as an accuracy O 3 α). In [] we obtain te asymptotic expansion formula ) n y n Γ α) α = +α ζ+α)y n = y α) n ζα) Γ α) y n α ζα ) + 2Γ α) y n 2 α + O 3 α), 2

3 and an approximation of te Caputo derivative were γ α) Γ α) α n γ α) z n = z n α) + O 3 α), 4) = / +α, 3) and γ α) = ζ+α)+ 3 2 ζα) 2 ζα ), γ α) = 2ζα)+ζα ), γ α) 2 = 2 +α + 2 ζα) ζα ). 2 Approximation 4) as an accuracy O 3 α) wen te function z C 3 [,t n ] and satisfies z)=z )=z )=. Te Miller-Ross sequential derivative for te Caputo derivative of order nα is defined as y [nα] t)=d nα yt)= D α CD α C D α Cyt). Te Caputo and Miller-Ross derivatives of order 2α of te function yt) = +t α satisfy y [2α] t) = and y 2α) t)=γ+α)t α /Γ α). Te fractional Taylor polynomials of te function yt) are defined as T α) m t)= m y [α] )t α Γα+). Te fractional Taylor polynomials T m α) t) polyfractonomials) are defined at te initial point of fractional differentiation t = and are polynomials wit respect to t α. An important class of special functions in fractional calculus are te one-parameter and two-parameter Mittag-Leffler functions E α t)= n= t n Γαn+), E α,βt)= n= t n Γαn+β). Te Mittag-Leffler functions generalize te exponential function and appear in te analytical solutions of fractional and integer-order differential equations. Te Miller-Ross derivatives of te function E α t α ) satisfy D nα E α t α )=E α t α ), D nα E α t α ) t= = E α )=. Te two-term equation is called fractional relaxation equation, wen < α <. y α) t)+byt)=ft), y)=y. 5) 3

4 Te exact solution of equation 5) is expressed wit te Mittag-Leffler functions as t yt)=y E α Bt α )+ ξ α E α,α Bξ α )Ft ξ)dξ. 6) Wen te solution of te two-term equation y C 3 [,T], te numerical solutions wic use approximations ), 2) and 4) of te Caputo derivative ave accuracy O 2 α), O 2) and O 3 α). Te numerical solutions of FDEs wic ave smoot solutions ave been studied extensively in te last tree decades. Te finite difference scemes involve approximations of te Caputo derivative related to te constructions of te L approximation and te Grünwald-Letniov difference approximation [, 4,, 2, 9]. Wen Ft)= te two-term equation 5) as te solution yt)=e α Bt α ). Wen <α <, te function E α Bt α ) as a singularity at te initial point, because its derivatives are unbounded at t =. Tis property olds for most ordinary and partial FDEs. Te singularity of te solutions of FDEs adds a significant difficulty to te construction of ig-order numerical solutions [4, 8, 22, 23]. In te present paper we propose a metod for transforming linear FDEs wit constant coeeficients into FDEs wose solutions are smoot functions. In section 2 and section 3 te metod is applied for computing te numerical solutions of te two-term and te tree-term FDEs. 2 Two-term FDE Te numerical and analytical solutions of te two-term ordinary FDE are studied in [5, 6, 7, 9, 3, 5, 7]. Let N be a positive integer and =T/N. Suppose tat α n λ α) y n = y α) n + O βα)) *) is an approximatiom of te Caputo derivative of order βα). Now we drive te numerical solution of two-term equation 5), wic uses approximation *) of te Caputo derivative. By approximating te Caputo derivative of equation 5) at te point t n = n we obtain n α λ α) y n + By n = F n + O βα)). Te numerical solution{u n } N n= of equation 5) is computed wit u = and n α λ α) u n + Bu n = F n, 4

5 u n = λ α) + B α α F n n = λ α) u n ). NS*)) Wen te solution of te two-term equation y C 3 [,T], numerical solutions NS), NS2), NS4) ave accuracy O 2 α),o 2),O 3 α). Wen te solution of equation 5) satisfies y) = y ) = y ) we can coose te initial values of te numerical solution u = u = u 2 =. Te two-term equation y α) t)+byt)=, y)= 7) as te solution yt) = E α Bt α ). Wen <α < te first derivative of te solution is undefined at t =. Numerical solutions NS) and NS2) of equation 7) ave accuracy O α ) [3]. Te numerical results for te error and order of numerical solution NS) wit α =.3,B= and α =.5,B= 2 and NS2) wit α =.7,B= 3 on te interval[,] are presented in Table. Te errors of te numerical metods in Table and te rest of te tables in te paper are computed wit respect to te natural maximum) l norm. Now we transform equation 7) into a two-term equation wose solution as a continuous second derivative. From 7) wit t = we obtain y α) ) = By) = B. By applying fractional differentiation of order α we obtain y [2α] t)+by α) t)=, y [2α] )= By α) )=B 2, y [3α] t)+by [2α] t)=, y [3α] )= By [2α] )= B 3. By induction we obtain te Miller-Ross derivatives of te solution of equation 7) at te initial point t = : Substitute y [nα] )= By [n )α] )= B) n. zt)=yt) T α) m t)=yt) m n= Te function zt) as a Caputo derivative of order α and satisfies te two-term equation Bt α ) n Γαn+). z α) t)=y α) m Bt t)+b α ) n Γαn+), n= z α) t)+bzt)= B)m+ t αm, z)=. 8) Γαm+) Now we use te uniform limit teorem to sow tat wen mα > 2 te solution of equation 8) is a twice continuously differentiable function. 5

6 Lemma. Let ε >,T > and { ) M > max 2B /α Te+2 α, 2 α α +log 2 2B 2/α ε )}. Ten B /α Te αm+ α 2 ) α < 2 ) αm+α 2 and 2B 2/α B /α Te < ε. αm+ α 2 Proof. ) α B /α Te < αm+ α 2 2 B/α Te αm+ α 2 < 2 /α αm+α 2>2B)/α Te. Te first inequality is satisfied wen M > ) 2B /a Te+2 α. α We ave tat ) αm+α 2 B /α Te B = /α Te αm+ α 2 αm+ α 2 2 M+ 2/α > 2B2/α. ε Te second inequality is satisfied for M > 2 α +log 2 ) αm+ 2/α) < 2B 2/α ε ). 2 M+ 2/α < ε 2B 2/α, Teorem 2. Let mα > 2 and z be te solution of equation 8). Ten z C 2 [,T]. Proof. Te solution of equation 8) satisfies zt)=yt) T α) m z t)= t)= n=m+ 6 n=m+ B) n t αn 2 Γαn ). Bt α ) n Γαn+),

7 Denote Z j t)= j n=m+ B) n t αn 2 Γαn ), were j>m. Wen mα > 2 te functions Z j t) are continuous on te interval [,T]. Let ε > and { )} ) M > M > max 2B /α Te+2 α, 2 α α +log 2B 2/α 2. ε Z M t) Z Mt)= From te triangle inequality Z M t) Z Mt) < M n=m+ Te gamma function satisfies [2] M n=m+ B n t αn 2 Γαn ) = B2/α B) n t αn 2 Γαn ). M n=m+ Γ+x) 2/x e 2x+)x+2)> x e) 2, x ) x. Γ+x)> e Ten M B Z M t) Z Mt) <B 2/α /α Te ) αn 2 < B2/α n=m+ αn 2) αn 2 Z M t) Z Mt) <B 2/α B /α Te M αm+ α 2)αM+α 2 n= From Lemma : M n=m+ B /α T ) αn 2 Γαn ). B /α Te ) αn 2 αm+ α 2) αn 2, B /α Te αm+ α 2 Z M t) Z Mt) <B 2/α B /α Te αm+ α 2)αM+α 2 n= 2 n, ) αm+α 2 Z M t) Z Mt) <2B 2/α B /α Te < ε. αm+ α 2 ) αn. From te uniform limit teorem, te sequence of functions Z j t) converges uniformly to te second derivative of te solution of equation 8), wic is a continuous function on te interval [, T]. 7

8 We can sow tat wen mα > 3 ten z C 3 [,T]. Te proof is similar to te proof of Teorem 2. In tis case numerical solutions NS),NS2) and NS4) of equation 8) ave accuracy O 2 α), O 2) and O 3 α). Te numerical results for te maximum error and order of numerical solutions NS),NS2) and NS4) of equation 8) on te inteval[, ] are presented in Table 2, Table 3 and Table 4. 3 Tree-term FDE In tis section we study te numerical solutions of te tree-term equation y [2α] t)+3y α) t)+2yt)=, y)=3,y α) )= 4, 9) were <α <. Substitute wt)=y α) t)+yt). Te function wt) satisfies te two-term equation w α) t)+2wt)=, w)=. Ten wt)= E α 2t α ). Te function yt) satisfies te two-term equation From 6) y α) t)+yt)= E α 2t α ), y)=3. yt)=3e α t α ) t ξ α E α,α ξ α )E α 2t ξ) α )dξ. From.7) in [7] wit γ = α,β =,y=,z= 2 we obtain t ξ α E α,α ξ α )E α 2t ξ) α )dξ =2E α,α+ 2t α ) E α,α+ t α ))t α ) = t 2 α 2t α ) n n= Γαn+α+ ) t α ) n [m=n+] n= Γαn+α+ ) 2t = α ) m m= Γαm+) + t α ) m m= Γαm+) = E α t α ) E α 2t α ). Te tree-term equation 9) as te solution yt)=2e α t α )+E α 2t α ). Wen < α < te first derivative of te solution yt) of equation 9) is undefined at te initial point t =. Te Riemann-Liouville derivative of order α, were n α < n and n is a nonnegative integer is defined as d n DRLyt)= α Γn α) dt n 8 t yξ) dξ. t ξ) +α n

9 Te Caputo, Miller-Ross and Rieman-Lioville derivatives satisfy [5, 7]: DRL 2α yt)=d2α C yt)+ D α yt)=d α C yt)=dα RLyt) y) Γ α)t α, y) Γ 2α)t 2α = y) D2α yt)+ Γ 2α)t 2α + yα) ) Γ α)t α, were te above identity for te Caputo derivative requires tat < α <.5. Tree-term equation 9) is reformulated wit te Riemann-Liouville and Caputo fractional derivatives as DRL 2α yt)+3dα RL yt)+2yt)= 3 Γ 2α)t 2α + 5 Γ α)tα, ) DC 2α yt)+3dα C yt)+2yt)= 4 Γ α)tα,y)=3, <α.5. ) A similar tree-term equation wic involves te Caputo derivative is studied in Example 5. in [22]. Te formulation ) of te tree-term equation wic uses te Caputo derivative as only one initial condition specified and te values of α, were.5<α < are excluded. Te initial conditions of equations ) and ) are inferred from te equations. Formulation 9) of te tree-term equation studied in tis section as te advantages, to te formulations ) and ) wic use te Caputo and Riemann-Lioville derivatives, tat te initial conditions of equation 9) are specified and te analytical solution is obtained wit te metod described in tis section. Now we determine te Miller-Ross derivatives of te solution of equation 9). By applying fractional differentiation of order α we obtain y [n+)α] t)+3y [nα] t)+2y [n )α] t)=. Denote a n = y [nα] ). Te numbers a n are computed recursively wit Substitute a n+ + 3a n + 2a n =, a = 3,a = 4. zt)=yt) T α) m t)=yt) Te function zt) as fractional derivatives z α) t)=y α) t) m n= m n= a n t nα Γαn+). a n t n )α Γαn )+) = yα) t) 9 m n= a n+ t nα Γαn+),

10 z [2α] t)=y [2α] m t) n= a n+ t n )α Γαn )+) = y[2α] t) m 2 n= a n+2 t nα Γαn+). Te function zt) satisfies te condition z) = z α) ) = and its Miller- Ross and Caputo derivatives of order 2α are equal z [2α] t) = z 2α) t). Te function zt) satisfies te tree-term equation were z 2α) t)+3z α) t)+2zt)=ft), z)=,z α) )=, 2) t m )α Ft)= 2a m + 3a m ) Γαm )+) 2a mt mα Γαm+). Now we obtain te numerical solution of tree-term equation 2) wic uses approximation *) of te Caputo derivative. By approximating te Caputo derivatives of equation 2) at te point t n = n we obtain n 2α λ 2α) z n + 3 n α λ α) z n + 2z n = F n + O min{βα),β2α)}). 3) Denote by NS2*) te numerical solution {u n } N n= of equation 2) wic uses approximation *) of te Caputo derivative. From 3) λ 2α) u n +3 α λ α) u n +2 2α n u n + = λ 2α) u n +3 α n Te numbers u n are computed wit u = u = and u n = λ 2α) + 3 α λ α) 2α n F n + 2 2α = λ 2α) = λ α) u n = 2α F n. ) 3 α λ α) u n ). NS2*)) Wen y C 3 [,T] numerical solution NS22), wic uses approximation 2) of te Caputo derivative as an order min{2,3 2α}. Te accuracy of numerical solution NS22) is O 2) wen <α.5 and O 3 2α) for.5 < α <. Te numerical results for te error and te order of numerical solution NS22) of tree-term equation 2) on te interval [, ] wit are presented in Table 5. Numerical solution NS24) wic uses approximation 4) of te Caputo derivative as an accuracy O 3 2α) for all values of α,.5).5, ). Numerical solution NS24) is undefined for α =.5, because te value of Γ 2α) = Γ ) is undefined. Te numerical results for te error and te order of numerical solution NS24) of equation 2)

11 on te interval [, ] are presented in Table 6. Now we obtain te numerical solution NS3*) of equation 2) wit α =.5 wic uses approximation *) for te Caputo derivative. z t)+3z.5) t)+2zt)=ft). 4) By approximating te first derivative wit 3) we obtain 3 2 z n 2z n + ) 2 z n n.5 λ.5) z n + 2z n = F n + O min{2,β.5)}). Te numerical solution{u n } N n= of equation 4) is computed wit u = u = and 3 2 u n 2u n + ) 2 u n n.5 λ.5) u n + 2u n = F n, u n = 3+6λ.5) F n + 4y n y n n = λ.5) u n NS3*)) Te numerical results for te error and te order of numerical solutions NS3), NS32) and NS34) of equation 4) are presented in Table 7. 4 Conclusion In te present paper we propose a metod for improving te numerical solutions of ordinary fractional differential equations. Te metod is based on te computation of te fractional Taylor polynomials of te solution at te initial point and transforming te equations into FDEs wic ave smoot solutions. Te metod is used for computing te numerical solutions of equations 7), and 9) and it can be applied to oter linear fractional differential equations wit constant coefficients. Te fractional Taylor polynomials ave a potential for construction of numerical solutions of linear and nonlinear FDEs wic ave singular solutions. In future wor we are going to generalize te metod discussed in te paper to oter linear FDEs wit constant coefficients and analyze te convergence and te stability of te numerical metods. )

12 5 Acnowledgements Tis wor was supported by te Bulgarian Academy of Sciences troug te Program for Career Development of Young Scientists, Grant DFNP-7-88/27, Project Efficient Numerical Metods wit an Improved Rate of Convergence for Applied Computational Problems, by te Bulgarian National Fund of Science under Project DN 2/4-27 Advanced Analytical and Numerical Metods for Nonlinear Differential Equations wit Applications in Finance and Environmental Pollution and by te Bulgarian National Fund of Science under Project DN 2/5-27 Efficient Stocastic Metods and Algoritms for Large-Scale Computational Problems. References [] Alianov, A.A.: A new difference sceme for te time fractional diffusion equation. Journal of Computational Pysics 28, ) [2] Bastero, J. Galve, F. Peña, A. Romano, M.: Inequalities for te gamma function and estimates for te volume of sections of Bnp, Proc. A.M.S., 3), ) [3] Cartea, A. del Castillo-Negrete, D.: Fractional diffusion models of option prices in marets wit jumps. Pysica A 3742), ) [4] Cen, M. Deng, W.: Fourt order accurate sceme for te space fractional diffusion equations. SIAM Journal on Numerical Analysis 523), ) [5] Dietelm, K.: Te analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer 2) [6] Dietelm, K. Siegmund, S. Tuan, H.T.: Asymptotic beavior of solutions of linear multi-order fractional differential systems. Fractional Calculus and Applied Analysis 25), ) [7] Dimitrov, Y.: Numerical approximations for fractional differential equations. Journal of Fractional Calculus and Applications 53S), 45 24) 2

13 [8] Dimitrov, Y.: A second order approximation for te Caputo fractional derivative. Journal of Fractional Calculus and Applications 72), ) [9] Dimitrov, Y.: Tree-point approximation for te Caputo fractional derivative. Communication on Applied Matematics and Computation 34), ) [] Dimitrov, Y.: Approximations for te Caputo derivative I). Journal of Fractional Calculus and Applications 9), ) [] Ding, H. Li, C.: Hig-order numerical algoritms for Riesz derivatives via constructing new generating functions. Journal of Scientific Computing 72), ) [2] Gao, G.H. Sun, Z.Z. Zang H.W.: A new fractional numerical differentiation formula to approximate te Caputo fractional derivative and its applications. Journal of Computational Pysics 259, ) [3] Jin, B. Lazarov, R. Zou, Z: An analysis of te L sceme for te subdiffusion equation wit nonsmoot data. IMA Journal of Numerical Analysis 36), ) [4] Jin, B. Zou, Z.: A finite element metod wit singularity reconstruction for fractional boundary value problems. ESIM: M2AN 49, ) [5] Li, C. Cen, A. Ye, J.: Numerical approaces to fractional calculus and fractional ordinary differential equation. Journal of Computational Pysics 239), ) [6] Lin, Y. Xu, C.: Finite difference/spectral approximations for te time-fractional diffusion equation. Journal of Computational Pysics 2252), ) [7] Podlubny, I.: Fractional differential equations. Academic Press, San Diego 999) [8] Quintana-Murillo, J. Yuste, S.B.: A finite difference metod wit nonuniform timesteps for fractional diffusion and diffusion-wave equations. Te European Pysical Journal Special Topics 2228), ) 3

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15 Table : Maximum error and order of numerical solutions NS) of equation 7) wit α =.3,α =.5 and NS2) wit α =.7 of order α. α =.3,B= α =.5,B=2 α =.7,B=3 Error Order Error Order Error Order Table 2: Maximum error and order of numerical solution NS) of equation 8) of order 2 α. α =.3,B=,m=7 α =.5,B=2,m=4 α =.7,B=3,m=3 Error Order Error Order Error Order Table 3: Maximum error and order of second-order numerical solution NS2) of equation 8). α =.3,B=,m=8 α =.5,B=2,m=5 α =.7,B=3,m=2 Error Order Error Order Error Order

16 Table 4: Maximum error and order of numerical solution NS4) of equation 8) of order 3 α. α =.3,B=,m=8 α =.5,B=2,m=6 α =.7,B=3,m=5 Error Order Error Order Error Order Table 5: Maximum error and order of numerical solution NS22) of equation 2) of order min{2,3 2α}. α =.3,m=9 α =.4,m=6 α =.7,m=5 Error Order Error Order Error Order Table 6: Maximum error and order of numerical solution NS24) of equation 2) of order 3 2α. α =.3,m=4 α =.4,m= α =.7,m=6 Error Order Error Order Error Order Table 7: Maximum error and order of numerical solution NS3) of equation 4) of order.5 and second-order numerical solutions NS32) ana NS34). NS3),m=4 NS32),m=3 NS34),m=5 Error Order Error Order Error Order

17 Y. Dimitrov Department of Matematics and Pysics, University of Forestry, Sofia 756, Bulgaria I. Dimov Institute of Information and Communication Tecnologies, Bulgarian Academy of Sciences, Department of Parallel Algoritms, Acad. Georgi Boncev Str., Bloc 25 A, 3 Sofia, Bulgaria, ivdimov@bas.bg V. Todorov Institute of Matematics and Informatics, Bulgarian Academy of Sciences, Department of Information Modeling, Acad. Georgi Boncev Str., Bloc 8, 3 Sofia, Bulgaria, vtodorov@mat.bas.bg Institute of Information and Communication Tecnologies, Bulgarian Academy of Sciences, Department of Parallel Algoritms, Acad. Georgi Boncev Str., Bloc 25 A, 3 Sofia, Bulgaria venelin@parallel.bas.bg 7