ENOC-8, Saint Petesbug, ussia, June, 3 July, 4 8 AXIS-SYMMETIC FACTIONAL DIFFUSION-WAVE POBLEM: PAT I-ANALYSIS N. Özdemi Depatment of Mathematics, Balikesi Univesity Balikesi, TUKEY nozdemi@balikesi.edu.t O. P. Agawal Mechanical Engineeing, Southen Illinois Univesity Cabondale, IL, USA om@eng.siu.edu D. Kaadeniz and B. B. İskende Depatment of Mathematics, Balikesi Univesity Balikesi, TUKEY mat okyanus@hotmail.com beyzabillu@hotmail.com Abstact: This is pat I of a two pat pape on an axis-symmetic factional diffusion-wave poblem. In this pat we focus on the esponse of the system subjected to extenal excitation. We define the poblem in tems of iemann-liouville factional deivatives and use modal analysis appoach to educe the continuum poblem to a countable infinite degees-of-feedom poblem fo which solution could be found in closed fom. Hee we use Günwald-Letnikov appoximation to find a numeical solution to the poblem. This will allow us to solve axis-symmetic factional optimal contol poblems which could not be solved in closed fom. We validate the scheme by compaing the numeical esults with the analytical solutions. The fomulation and the appoach pesented hee extends ou ealie wok on factional diffusion in -dimension to axis symmetic case. Key wods Factional Calculus, Factional deivatives, Factional diffusion-wave equation, Axis symmetic, diffusion-wave equation, Günwald-Letnikov scheme, Bessel function. Intoduction Factional deivatives ae genealizations of odinay diffeentiations and integations to non-intege odes. Some of the definitions of factional deivatives poposed include iemann-liouville, Günwald-Letnikov, Weyl, Caputo, iesz and Machaud deivatives ([Oldham and Spanie, 974], Milleoss93, [Samko, Kilbas and Maichev, 993], [Podlubny, 999]. Factional deivatives aise in many physical poblems, fo example fequency dependent damping behavio of mateials, elaxation functions fo viscoelastic mateials, factional PI λ D µ contol of dynamical systems, motion of a plate in a Newtonian fluid, and phenomena in electomagnetics and acoustics ([Podlubny, 999]. In this pape we focus on an axis-symmetic Factional Diffusion Wave Equations (FDWEs. The FDWEs ae obtained by eplacing the intege ode (time and/o space deivatives in odinay diffusion/wave equations with the factional deivatives. The FDWEs have been consideed by the seveal investigatos in ecent yeas. Oldham and Spanie [974] consideed a factional diffusion equation that contained fist-ode deivative in space and half-ode deivative in time. Wyss [986] and Schneide and Wyss [989] pesented the solutions of the timefactional diffusion and wave equations in tems of Fox functions. Giona, Cebelli and oman [99] pesented factional diffusion equations descibing the elaxation phenomena in complex viscoelastic mateials. Giona and oman [99a] pesented factional diffusion equations fo tanspot phenomena in andom media. Giona and oman [99b] and oman and Giona [99] used factional diffusion equations to descibe one and thee dimensional cases of anomalous diffusions on factals without extenal foces. Thei wok extends the expession of Oldham and Spanie [97]. Mainadi ([996a], [996b], [997] used a Laplace tansfom method to obtain fundamental solutions fo a FDWE and the solutions fo factional elaxationoscillations. Mainadi [997] and Mainadi and Paadisi [997] showed that as the ode of factional deivative in a FDWE inceases fom to, the pocess changes fom slow diffusion to classical diffusion to diffusionwave to classical wave pocesses. Agawal [] pesented fundamental solutions fo an FDWE whee the diffusion equation contained a fouth ode space deivative and a factional ode time deivative. Zou, en and Qiu [4] consideed a factional diffusion equation of highe dimension to descibe anomalous diffusion pocesses involving extenal foce fields by using Giona and oman s heuistic agument. Hilfe [] poposed the closed fom solution of a factional diffusion poblem in tems of H-functions. Agawal ([], [] used modal/integal tansfom methods to find solutions of FDWEs defined
in finite domains. The method is geneal and can be used to find closed fom solutions to many poblems in vibation analysis of factional systems whee modal/integal tansfom methods could be applied. Given that vibation of continuous system is a vast field, these papes open multitude of possibilities and new poblems in the field of factional diffusion-wave. Modal/integal tansfom methods have ecently been used to solve factional genealization of Navie-Stokes equations in El-Shahed and Salem [El-Shahed and Salem, 4] and a time factional adial diffusion in a sphee [Povstenko, 7]. Agawal [3] pesented stochastic analysis of FDWEs defined in -dimension. It should be pointed out that vey little wok has been done in the aea of stochastic analysis of factional ode engineeing systems. Since the appoach pesented hee also uses modal/integal tansfom methods, it could be extended to all factional stochastic poblems in multi-dimensions whee the tansfom methods could be applied. The exact solution of a factional diffusion equation with an absobent tem and a linea extenal foce appeas in [Schot, et al, 7]. Exact solutions of genealized nonlinea factional diffusion equations with extenal foce and absoption ae pesented in [Liang, et al 7]. This bief eview of fomulations and methods fo FD- WEs is by no means complete. Othe fomulations, methods, and solutions could be found, among othes, in [West, Bologna and Gigolini 3] and the efeences thee in. In this pape, we pesent a numeical scheme fo an axissymmetic factional diffusion-wave poblem. We define the poblem in tems of iemann-liouville factional deivatives and use modal/integal tansfom method pesented in Agawal ([], [] to educe the continuum poblem to a countable infinite degees-of-feedom poblem fo which solution could be found in closed fom. Poblems simila to the one consideed hee have also been solved using simila techniques in [El-Shahed and Salem, 4] and [Povstenko, 7]. Howeve, the fomulation hee diffes with Agawal ([], [], [El-Shahed and Salem, 4] and [Povstenko, 7] in two espects. Fist, Agawal ([], [] consides one dimensional poblems wheeas this pape consides an axis-symmetic poblem. Second, Agawal ([], [], [El-Shahed and Salem, 4] and [Povstenko, 7] find only closed fom solutions, wheeas this pape also finds numeical solutions using Günwald-Letnikov appoximation. In the sequel, we will pesent a fomulation fo an axis-symmetic factional optimal contol poblem fo which a closed fom solutions could not be found. Ou numeical scheme used hee will allow us to find numeical solutions to the poblem. In the next section, we pesent the fomulation and the analytical solution to an axis-symmetic FDWE. Axis-symmetic FDWE and its analytical solution In this section, we pesent an axis-symmetic FDWE in tems of the iemann-liouville Factional Deivative (LFD, and povide its analytical solution. Howeve, much of it can also be applied to fomulations defined in tems of othe factional deivatives. We begin with the LFD which is defined as [Podlubny, 999], ad α t f(t = ( n t d (t τ n α f(τdτ Γ(n α dt ( whee f(t is a function, α, (n < α < n, is the ode of the deivative, t is the time vaiable, and n is an intege. In case α is an intege, the factional deivative is eplaced with an odinay deivative. Futhemoe, if f is dependent on two o moe vaiables, then the odinay deivative in Eq. (is eplaced with a patial deivative. The axis-symmetic factional diffusion-wave poblem can now be defined as follows: Find the esponse of the system α ( w t α = β w + a w + u (,t, ( subjected to the following bounday and initial conditions, and w (,t =, t >, (3 w (, = ẇ (, =, (4 whee is the adial space coodinate, β is a constant which depends on the physical popeties of the system, u(,t is the extenal souce tem, and is the bounday of the domain of. The second condition in Eq. (4 is consideed only if α >. In the case of heat tansfe, u(, t epesents the ate of heat geneation, and in the case of membane vibation, it epesents the extenal focing function. Using the method of sepaation of vaiables, it can be demonstated that the eigenfunctions φ j ( fo this poblem ae (see, e.g. [Keyszig, 6], ( φ j ( = J µ j, j =,,, (5 whee J ( is the zeo-ode Bessel function of the fist kind, and µ j, j =,,,, ae the positive oots of the equation J (µ =. (6 To find the solution of the poblem defined by Eqs. ( to (4, assume that can be given as w (,t = ( q j (tj µ j. (7 j=
Using Eqs. (, (3, (4, and (7, and the othogonality conditions we obtain and {, i j xj (µ i x J (µ j x dx = J (µj, i = j. (8 d α q k (t dt α = β ( µk qk (t + f k (t (9 q k ( = q k ( = ( whee J ( is the fist-ode Bessel function of the fist kind, and f k ( is given as f k (t = ( J (µ J µ k u (,t d. ( k Once again, the second condition in Eq. ( is consideed when α >. By applying the Laplace tansfom to Eq. (9, using Eq. (, and then taking invese Laplace tansfom, we get whee q k (t = t Q k (t = L { Q k (t τ f k (τ dτ ( s α + β ( µ k }, (3 is the factional Geen s function, which can be witten in closed fom as [Podlubny, 999], Q k (t = t α E α,α ( β ( µk t α. (4 Hee L is the invese Laplace tansfom opeato, and E α,β is the two-paamete Mittag-Leffle function (see [Podlubny, 999]. Substituting Eq. ( into Eq. (7, we obtain the closed fom solution of the axis-symmetic factional diffusion wave equation defined by Eqs. ( to (4 as w (,t = ( t J µ k k= Q k (t τ f k (τ dτ. (5 Thus, w(, t can be obtained povided u(, t is known. It will be shown in the second pat of this pape that in the case of factional optimal contol u(, t is not known a pioi, and it is solved along with othe vaiables. Fo this case, a numeical scheme is necessay. In the next section, we pesent a numeical algoithm to solve the factional diffeential equations defined by Eqs. (9 and (. 3 Numeical Algoithm The numeical algoithm pesented hee is based on an algoithm given in [Podlubny, 999], and it elies on the Günwald-Letnikov appoximation of the factional deivative. Fo simplicity in the discussion to follow, we dop the subscipt k fom Eqs. (9 and (, and ewite them as and d α q (t dt α = cq (t + f (t (6 q( = q( = (7 whee c = β (µ k /. The algoithm can now be descibed as follows:. Divide the time inteval into seveal subintevals of equal size h (also called the step size.. Appoximate d α q/dt α at node i using the Günwald- Letnikov fomula as [Podlubny, 999], d α q dt α = h α i j= w (α j q (i j, (8 whee q (j is the numeically computed value of q at node j, and w (α j ae the coefficients defined as [Podlubny, 999], ( w (α = ; w (α j = α + w (α j j, j =, (9 3. Appoximate Eq. (6 at node i, and solve fo q (i to obtain q (i = i + ch α (hα f(t i j= w (α j q (i j, i = i α, ( whee i α = if i α, and i α = if i α >. In case α is geate than, q ( is detemined using Eq. (7 and taking linea appoximation between q ( and q ( (see, [Podlubny, 999]. 4. Use Eq. ( to find q (i at all nodes.
Thus, the numeical solution of Eqs. (6 and (7 is obtained. To obtain the solution of Eqs. ( to (4, we eplace in Eq. (7 with an intege M (i.e. we tuncate the seies, solve Eqs. (9 and ( fo k fom to M, and substitute the esults in Eq. (7. Numeical studies show that only a small M is sufficient to find accuate esults. 4 Numeical Example In this section, we pesent some simulation esults fo the Diffusion-Wave poblem defined by Eqs. ( to (4 fo t >, < α, [,]. Fo simulation pupose, we take = β = u(,t =, and vay M and h. We compute f k (t using Eq. ( and solve Eqs. (9 and ( fo k =,,,M using the algoithm discussed in Sec. 3. Finally, we use Eq. (7 to find the esponse. We also find the analytical solution using Eq. (5 fo compaison pupose. Fo computation pupose, this seies is tuncated afte M tems. The esults of this study ae as follows: Figues and show the analytical and numeical esults fo fo α = and α =, espectively. In this study, we take =.5, M = 5, and h =.. Figue shows that the diffusion eaches a steady state in a vey shot time, and Figue shows the undamped vibational chaacteistics of the system. Note that in both cases, the analytical and numeical esults ae vey close. est of the figues show numeical esults fo vaious M, h, and α. Figue 3 shows at =.5 fo α =.5 and M = 5,, and. All thee cuves ae vey close indicating that only few tems ae necessay to compute esponse of the system. Note that the diffusion pocess is slow. Figue 4 shows at =.5 fo α =., M = 5 and h =,.,., and.. Note that the solutions convege as the step size is educed, which indicates that the algoithm may be stable. Figue 5 shows fo α =.5,.7,.9 and, and Figue 6 shows the same fo α =.5,.7,.9 and. In both cases, the following values ae used: =.5, M = 5 and h =.. Both figues show that as α appoaches an intege value, the solution fo the intege ode system is ecoveed. These two figues also show that as α changes fom.5 to, the esponse changes fom sub-diffusion to diffusion to diffusion-wave to wave solution. Figues 7, 8 and 9 show the whole field esponse fo α =.5,.5 and, espectively. Fo these simulations, we took M = 5 and h =.. These figues also show the changing behavio of as α changes fom.5 to. Note that eaches a steady state fo α =.5 and.5, but it continue to oscillate fo α =.. Futhe, in all thee cases, w/ is at =, as expected. 5 Conclusions An axis-symmetic factional diffusion-wave poblem was defined in tems of the iemann-liouville factional deivative, and a modal/integal tansfom method was pesented to educe the continuum poblem to a countable infinite degees-of-feedom poblem. A Laplace tansfom based technique was used to find closed fom solu-..8.6.4...8.6.4. analytical solution numeical solution.5.5 Figue. Compaison of the analytical and the numeical solution of fo α =, =.5, M = 5 and h =..4.35.3.5..5..5 analytical solution numeical solution.5 3 4 5 Figue. Compaison of the analytical and the numeical solution of fo α =, =.5, M = 5 and h =..6.4...8.6.4. M=5 M= M=.5..5..5.3.35.4 Figue 3. The solution of fo α =.5, =.5. and M = 5,, tions. The Günwald-Letnikov appoximation was used to develop an algoithm fo numeical solution of the poblem. esults show that both the analytical and the numeical esults agee well. As the time step size is educed, the solutions convege, and only few tems in the seies ae necessay to find a solution close to the exact solution. The esponse of the system changes fom subdiffusion to diffusion to diffusion-wave to wave solutions as α changes fom.5 to. The numeical algoithm developed hee will allow us to find numeical solutions fo axis-symmetic factional optimal contol poblems.
..8.6.4...8.4.3...6.4. h= h=. h=. h=. 3 4 5 Figue 4. The solution of fo α =, =.5, M = 5. and h =,.,.,. 6 4 (adius..4 Figue 8. Thee dimensional figue of fo α =.5, h =. and M = 5..6.8..8.6.4...8.6.4. α=.5 α=.7 α=.9 α=.5.5 Figue 5. The solution of fo =.5, h =., M = 5. and α =.5,.7,.9,.6.4.. 6 4 (adius..4 Figue 9. Thee dimensional figue of fo α =, h =. and M = 5..6.8.4.35.3.5..5..5 α=.5 α=.7 α=.9 α=.5 3 4 5 Figue 6. The solution of fo =.5, h =., M = 5. and α =.5,.7,.9,.3.5..5..5 6 4 Figue 7. Thee dimensional figue of fo α =.5, h =. and M = 5...4.6.8 efeences Agawal, O.P. (. A geneal solution fo the fouthode factional diffusion-wave equation. Fact Calc Appl Anal. 3, -. Agawal, O.P. ( A geneal solution fo a fouthode factional diffusion-wave equation defined in a bounded domain. Computes & Stuctues. 79, pp. 497-5. Agawal, O. P. (. Solution fo a factional diffusionwave equation in a bounded domain. Jounal of Nonlinea Dynamics. 9, pp. 45-55. Agawal, O.P. (3. esponse of a diffusion-wave system subjected to deteministic and stochastic fields. ZAMM, Jounal of Applied Mathematics and Mechanics. 83, pp. 65-74. El-Shahed M. and Salem A. (4. On the genealized Navie-Stokes equations. Applied Mathematics and Computation. 56, pp. 87-93. Giona, M. and oman, H.E. (99a. Factional diffusion equation fo tanspot phenomena in andom media Physica A: Statistical and Theoetical Physics. 85, pp. 87-97. Giona, M. and oman, H.E. (99b. Factional diffusion equation on factals: one-dimensional case and asymptotic behaviou. J Phys A. 5, pp. 93-5. Giona, M., Cebelli, S. and oman, H.E. (99. Factional diffusion equation and elaxation in complex vis-
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