On the Kolmogorov forward equations within Caputo and Riemann-Liouville fractions derivatives

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Preston Dixon
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DOI: 1.1515/auom-216-45 An. Şt. Univ. Ovidius Constanţa Vol.

1 DOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 24(3,216, 5 2 On the Kolmogoov fowad equations within Caputo and Riemann-Liouville factions deivatives Mohsen Alipou and Dumitu Baleanu Abstact In this wok, we focus on the factional vesions of the well-known Kolmogoov fowad equations. We conside the poblem in two cases. In case 1, we apply the left Caputo factional deivatives fo (, 1] and in case 2, we use the ight Riemann-Liouville factional deivatives on R +, fo (1, +. The exact solutions ae obtained fo the both cases by Laplace tansfoms and stable subodinatos. 1 Intoduction and peliminaies In the last decades, attention of scientists has been attacted to genealizations of classical pocesses and diffeential equations by the factional ode fo deivatives. Fo instance, existence and uniqueness of the factional diffeential equations [1], the factional intego-diffeential equations [2], the factional diffusions [3, 4, 5, 6], the factional telegaph equation [7] and factional Poisson pocesses [8, 9, 1, 11]. In this wok we conside genealization of the well-known Kolmogoov fowad equations [12] in two cases: in the fist case, the left Caputo factional deivative is applied and in the second case the ight factional Riemann-Liouville deivative is used. These models in special cases educe to the well-known factional elaxation equations [13] and discete vesion of the factional maste equation [14]. Key Wods: Caputo factional deivative, Riemann-Liouville factional deivative, Mittag- Leffle functions, Factional Kolmogoov fowad equations, Stable subodinato. 21 Mathematics Subject Classification: 6G52, 34A8, 33E12, 26A33. Received: Accepted: Download Date 2/1/18 12:59 PM

2 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 6 We conside hee the factional Kolmogoov fowad equations (FKFE: d { λb dt B k (t = 1 (t, k = 1, λbk (t + i=1 λb ibk i (t, k {2, 3,...}, (1 fo t, >, with initial conditions { Bk 1 k = 1, ( = k {2, 3,...}, (2 whee λ R and b i, i {1, 2,...} ae the distibution of some compounding andom vaiable with values in i {1, 2,...} and d dt is factional deivativ. The exact solutions to (1-(2 will be obtained fo difeent values of (, +. The poblem (1-(2 is analyzed with two the diffeent factional deivative opeatos that be defined as follows (efe [15, 16, 17, 18] to see these definitions and popeties of them: 1. The left Caputo factional deivative of ode : C Dt 1 f(t := Γ (1 t (t τ f (τdτ, t >, (, 1]. 2. The ight Riemann-Liouville factional deivative on R + of ode : ( RL t D+ 1 f(t := d n f(τ Γ (n dt n+1 dτ, t >, (1, + (τ t t whee n =. Note that, fo = 1, we have C D t = d dt and RL t D + = d dt. Remak 1.1. Fo = 1, equation (1 educes to the well-known Kolmogoov fowad equations [1]. Fo k = 1, the equation (1 coincides with the wellknown factional elaxation equations [11]. Fo k > 1, they can be seen as a discete vesion of the factional maste equation [12]. 2 Oiginal esults In this section, we use some notations as follows: i L [f(t; s] := e s t f(t dt. ii E,β (t = j= Leffle function. t j Γ(j+β, x R,, β C, R(, R(β > (The Mittag- Download Date 2/1/18 12:59 PM

3 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 7 Theoem 2.1. The exact solution of the FKFE { C Dt Bk λ1 B1 (t, k = 1, (t = λbk (t + i=1 λb ibk i (t, k {2, 3,...}, (3 fo t >, < 1 and with initial conditions { Bk 1 k = 1, ( = k {2, 3,...}, (4 fo λ R and b i (i = 1, 2,...that ae the distibution of some compounding andom vaiable with values in i {1, 2,...}, is given by E, 1 ( λt, k = 1 ( ( Bk (t = Π (λt =1 (i 1, i 2,..., i W b ( ij! E, 1 ( λ t, k = 2, 3,... (5 whee W k = (i 1, i 2,..., i i j {1, 2,... k + 1} and i j = k. (6 Poof. We use induction method to pove the esult (5. We stat by k = 1, so we have Then, fom the Laplace tansfom C D t B 1 (t = λb 1 (t, B 1 ( = 1. s L[B 1 (t; s] s 1 B 1 ( = λ L[B 1 (t; s] L[B 1 (t; s] = s 1 s + λ. Since [ ] L t n E (n, 1 ( λt ; s = n!s 1 (s n+1, n =, 1,..., (7 + λ Download Date 2/1/18 12:59 PM

4 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 8 we have Fo k = 2, equation (3 becomes B 1 (t = E, 1 ( λ t. C D t B 2 (t = λb 2 (t + λb 1 B 1 (t, B 2 ( =. The Laplace tansfom gives s 1 s L[B2 (t; s] s 1 B2 ( = λ L[B2 (t; s] + λb 1 s + λ L[B 2 (t; s] = λb 1 s 1 (s + λ 2. So, the invese Laplace s Tansfom applied in (7 give B 2 (t = b 1 λ t E (1, 1 ( λ t. Theefoe, the equation (3 is satisfied fo k = 2. Fo k = 3, equation (3 becomes C D t B 3 (t = λb 3 (t + λb 1 B 2 (t + λb 2 B 1 (t, B 3 ( =. By the Laplace tansfom, we have s L [B 3 (t; s] s 1 B 3 ( = λl[b 3 (t; s] + λb 1 λb 1 s 1 So, we get L [B 3 (t; s] = λ2 b 2 1 s 1 (s + λ 3 + λb 2s 1 (s + λ 2. Now, we apply the invese Laplace s Tansfom with (7, thus B 3 (t = b 2 λt E (1, 1 ( λ t + b 2 1 By the same way fo k = 4, we can wite (λt 2 2 s 1 (s + λ 2 + λb 2 (s + λ. E (2, 1 ( λ t. B4 (t = b 3 λt E (1, 1 ( λ (λt 2 t +2b 1 b 2 E (2, 1 2 ( λ t +b 3 1 (λt 3 6 E (3, 1 ( λ t. Download Date 2/1/18 12:59 PM

5 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 9 So, we have shown that (5 is satisfied fo k = 1, 2, 3, 4. Now, fom supposition of induction, we assume the (5 is tue fo B i (t, i {1,..., k 1}. Then we will pove that (5 holds fo B k (t. We apply the Laplace tansfom on (3 with k 2, so we have s L[Bk (t; s] s 1 Bk ( = λl[bk (t; s] + λ b i L[Bk i(t; s]. Theefoe by (5 and (7 we have and (s + λ L [Bk (t; s] = λ k 2 i=1 b i +λb s 1 s +λ. L [Bk (t; s] = k 2 i=1 b i ( k i 1 =1 ( k i 1 s +λb 1. (s +λ 2 =1 ( k i 1 (i1,..., i W ( k i 1 (i1,..., i W ( Π ( Π i=1 λ s 1 (s +λ +1 λ +1 s 1 (s +λ +2 Now, by evesing of tansfom (7, the above equation is educed to: B k (t = ( k 2 k i 1 i=1 bi =1 +λt b E (1, 1 ( λ t. ( k i 1 (i1,..., i W ( Π By eplacing the fist sigma to the second sigma, we have (λt +1 E (+1 (+1!, 1 ( λ t B k (t = k 2 =1 (( k 1 +λt b E (1, 1 ( λ t. i=1 b i ( (i 1,..., i W k i 1 Note that fo = 1, 2,..., k 2 k 1 i=1 b i (i 1,..., i W k i 1 ( Π ( Π = (i 1,..., i +1 W +1 (λt +1 E (+1 (+1!, 1 ( λ t ( +1 Π. Download Date 2/1/18 12:59 PM

6 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 1 So, Bk (t = (( k 2 =1 (i 1,..., i +1 W +λt b E (1, 1 ( λ t. Now, by eplacing + 1 to we have Bk (t = (( =2 (i 1,..., i W +λt b E (1, 1 ( λ t. +1 Finally, we get the esult (5 as follows Bk (t = =1 (i 1,..., i W Theefoe, the poof is completed. ( Π ( +1 Π ( Π (λt (λt +1 (+1! E (+1, 1 ( λ t! (λt! E (, 1 ( λ t, 1 ( λ t. Now, we conside b i = (1 ρρ i 1 as distibution of compounding andom vaiable with values in i {1, 2,...}. So, we can see the behavios of exact solutions Bk (t fo the poblem (3 and (4 fo ρ =.5, λ =.5, k = 1, 2, 3 and =.7,.8,.9, 1 in figues 1-3. E ( Remak 2.1. Fo = 1 and by witing B 1 k = B k in Theoem 2.1, the solution of the following equations { d dt B 1(t = λb 1 (t d dt B k (t = λb k (t + λ i=1 b ib k i (t, k {2, 3,...}, with t >, (8 is given by B k ( = { 1 k = 1, k {2, 3,...}, (9 Download Date 2/1/18 12:59 PM

7 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE B 1 Α t Α 1 Α.9 Α.8 Α t Figue 1: Plot of B 1 (t fo ρ =.5, λ =.5 and =.7,.8,.9, B 2 Α t Α 1.1 Α.9 Α.8 Α t Figue 2: Plot of B 2 (t fo ρ =.5, λ =.5 and =.7,.8,.9, 1. Download Date 2/1/18 12:59 PM

8 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE Α 1 B 3 Α t Α.9 Α.8 Α t Figue 3: Plot of B 3 (t fo ρ =.5, λ =.5 and =.7,.8,.9, 1. e λ t k = 1 ( B k (t = =1 (i 1, i 2,..., i W (Π b (λt ij! e λ t k {2, 3,...} (1 Using the notations of [19], we show the stable subodinato of index by A (t. Futhemoe its invese (o hitting time pocess is defined as L (t = inf {z z >, A (z > t} fo all t. Theoem 2.2. Let{ N (t be the pocess defined as N 1 (H (t, t, L (t, (, 1, whee H (t = A 1 and H (t = t fo = 1, (t, (1, +, unde the assumption that N 1 and H ae independent. Then the poblem (1-(2 is satisfied by the distibution Bk (t = P {N (t = k}, k 1, fo any >. Poof. By noting that the definitions l (z, t and h 1 (z, t a the densities of L (t and A 1 (t, espectively, we have Download Date 2/1/18 12:59 PM

9 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 13 Bk (t = B k (z l (z, tdz, < 1, B k (z h 1 (z, tdz, > 1, Now, we do the poof in two cases (, 1] and (1, + ]. In the fist case, fo (, 1] we define 1( k = 1, j = 1, a k = (i 1, i 2,..., i W (Π b λ ij! k 2, j = 1,..., k, so the (5 can be educed Bk (t = k =1 ak t E (,1 ( λt. Let F (ν, t = k=1 νk Bk (t be the pobability geneating function, then we can wite. e st F (ν, tdt = e st k=1 νk Bk (tdt = e st k=1 νk k =1 ak t = = ( + k=1 νk k =1 ak ( E,1!s 1 = (s +λ +1 ( λt dt k=1 νk s 1 k =1 ak e µλ µ ds s 1 e µs k=1 νk k = s 1 e µs k=1 νk Bk 1(µdµ = = e st F 1 (ν, µl (µ, tdµ dt, =1 ak s 1 e µs F 1 (ν, µdµ since µ 1 e µs = e s t l (µ, tdt(see [2]. Thus, we get F (ν, t = F 1 (ν, µ l (µ, tdµ B k (t = In the second case fo (1, +, we have B k (t = B k (τ h 1 (τ, tdτ. So, we can wite e µ(s +λ µ dµ B 1 k(µ l (µ, tdµ. RL t D+ B k (t = B k (τ RL t D+ h 1 (τ, tdτ = B k (τ [ ] + = B k (τ h 1 (τ, t d dτ B k(τ h 1 (τ, tdτ z= = ( + λb k (τ + λ i=1 b ib k i (τ h 1 (τ, tdτ = λ B k (τ h 1 (τ, tdτ λ i=1 = λb k (t λ i=1 b ib k i (t. τ h 1 (τ, tdτ b i B k i (τh 1 (τ, tdτ Download Date 2/1/18 12:59 PM

10 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 14 Fom [21], we have lim the law of A 1 ν (t: h 1 τ (τ, t =. Also the following equation goven on with RL t D+ h 1 (τ, t = τ h 1 (τ, t, τ, t >, (1, +, h 1 (, t =, h 1 (τ, = δ(τ. In [22] and [23], this esult is poved fo = n N and > 1, espectively. So, since d dt := RL t D+ fo (1, +, the poof is complete. Now, we focus on the exact solution to (1-(2 fo (1, +. Theoem 2.3. The exact solution of FKFE (1-(2 fo (1, + is Bk (t = 1 e tλ =1 ( (i 1, i 2,..., i W k 2. ( Π k = 1, ( λ d! ( 1 dλ e tλ 1 Poof. by Theoem 2.2 and (1 fo (1, +, we obtain Bk (t = B 1 k (τ h 1 (τ, tdτ = = e λ z h 1 (τ, tdτ k = 1, ( ( Π λ =1 (i 1, i 2,..., i W b + ij! e λ τ τ h 1 (τ, tdτ k 2, 1 e tλ =1 ( (i 1, i 2,..., i W k 2, ( Π k = 1, ( λ d! ( 1 dλ e tλ 1 Download Date 2/1/18 12:59 PM

11 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE B 1 Α t.8.6 Α 1.4 Α 1.3 Α Α t Figue 4: Plot of B 1 (t fo ρ =.5, λ =.5 and = 1, 1.3, 1.6, 1.9. because fom [8], we know 1 + e λ τ h 1 (τ, t dτ = e tλ. Also since h 1 (z, = δ(z, fo the initial conditions in (2 we can wite: Bk ( = B 1 k (τh 1 (τ, dτ { 1, k = 1 = B1 k ( =, k {2, 3,...}. Similaly, let b i = (1 ρρ i 1. Then, we show plots of exact solutions Bk (t fo the poblem (1 and (2 fo ρ =.5, λ =.5, k = 1, 2, 3 and = 1, 1.3, 1.6, 1.9 in figues Conclusion In this wok, by Laplace tansfoms and stable subodinatos, we have obtained the analytical solutions of factional Kolmogoov fowad equations Download Date 2/1/18 12:59 PM

12 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE Α 1 B 2 Α t.1 Α 1.3 Α Α t Figue 5: Plot of B 2 (t fo ρ =.5, λ =.5 and = 1, 1.3, 1.6, B 3 Α t Α 1 Α 1.3 Α Α t Figue 6: Plot of B 3 (t fo ρ =.5, λ =.5 and = 1, 1.3, 1.6, 1.9. Download Date 2/1/18 12:59 PM

13 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 17 with the left Caputo factional deivative fo (, 1] and the ight Riemann- Liouville factional deivative on R + fo (1, +. These poblems cove the well-known Kolmogoov fowad equations, factional elaxation equations and discete vesion of the factional maste equation. Refeences [1] Anbe A., Belabi S., Dahmani Z. (213, New existence and uniqueness esults fo factional Diffeential equations, An. St. Univ. Ovidius Constanta, 21, n.3, [2] Cenea A. (215, Some emaks on a factional intego-diffeential inclusion with bounday conditions, An. St. Univ. Ovidius Constanta, 23, n.1, [3] Mainadi F. (1996, The fundamental solutions fo the factional diffusion-wave equation, Appl. Math. Lett., 9, n.6, [4] Mainadi F. (1996, Factional elaxation-oscillation and factional diffusion-wave phenomena, Chaos, Solitons and Factals, 7, n.9, [5] Angulo J.M., Ruiz-Medina, M.D., Anh V.V., Geckosch W. (2, Factional diffusion and factional heat equation, Adv. in Appl. Pobab., 32, [6] Osinghe E., Beghin L. (29, Factional diffusion equations and pocesses with andomly-vaying time, Ann. Pobab., 37, n.1, [7] Osinghe, E., Beghin, L. (24, Time-factional equations and telegaph pocesses with Bownian time, Pobability Theoy and Related Fields, 128, [8] Laskin, N. (23, Factional Poisson pocess, Communications in Nonlinea Science and Numeical Simulation,, 8, [9] Mainadi F., Goenflo R., Scalas E. (24, A factional genealization of the Poisson pocesses, Vietnam J. Math., 32, Download Date 2/1/18 12:59 PM

14 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 18 [1] Beghin L., Osinghe E. (29, Factional Poisson pocesses and elated plana andom motions, Electon. J. Pobab., 14, n.61, [11] Meeschaet M. M. Nane, E., Veillaisamy P. (211, The factional Poisson pocess and the invese stable subodinato. Electon. J. Pobab., 59, [12] Kalin, S., Taylo, H.M. (1981, A Second Couse in Stochastic Pocesses. Academic Pess, London. [13] Beghin. L. (212,Factional elaxation equations and Bownian cossing pobabilities of a andom bounday. Adv. in Appl. Pobab., 44, [14] Hilfe, R., Anton, L. (1995, Factional maste equations and factal andom walks. Phys. Rev. E [15] Kilbas A.A., Sivastava H.M., Tujillo J.J. (26, Theoy and Applications of Factional Diffeential Equations, vol. 24 of Noth-Holland Mathematics Studies, Elsevie Science B.V., Amstedam. [16] Podlubny I. (1999, Factional Diffeential Equations, San Diego:Academic Pess. [17] Baleanu D, Diethelm K, Scalas E, Tujillo J. (212, Factional calculus models and numeical methods. Singapoe: Wold Scientific. [18] G. Samko, A. Kilbas, O. Maichev, (1993, Factional integals and deivatives: Theoy and applications, Godon and Beach. [19] Samoodnitsky G., Taqqu M.S. (1994, Stable Non-Gaussian Random Pocesses. Chapman Hall, New Yok. [2] Hahn M.G., Kobayashi K., Umaov S. (21, Fokke-Planck-Kolmogoov equations associated with time-changed factional Bownian motion, Poc. Ame. Math. Soc., 139, n.2, [21] Uchaikin V.V., Zolotaev V.M. (1999, Chance and Stability: Stable Distibutions and thei Applications, VSP, Utecht. [22] D Ovidio M. (211, On the factional countepat of the highe-ode equations, Statist. Pobab. Lett., 81, Download Date 2/1/18 12:59 PM

15 RIEMANN-LIOUVILE FRACTIONAL DERIVATIVE 19 [23] Beghin L., Macci C. (212, Altenative foms of compound factional Poisson pocesses, Abst. Appl. Anal., 212, 1-3. Mohsen Alipou, Depatment of Mathematics, Faculty of Basic Science, Babol Noshivani Univesity of Technology, P.O. Box , Babol, Ian. Dumitu Baleanu, Depatment of Mathematics, Cankaya Univesity, 653 Ankaa, Tukey Institute of Space Sciences, P.O. Box MG-23, Maguele-Buchaest, Romania Download Date 2/1/18 12:59 PM

On the Poisson Approximation to the Negative Hypergeometric Distribution

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