A Backward Identification Problem for an Axis-Symmetric Fractional Diffusion Equation

Size: px

Start display at page:

Download "A Backward Identification Problem for an Axis-Symmetric Fractional Diffusion Equation"

Aron Palmer
5 years ago
Views:

1 Mathematical Modelling and Analysis Publishe: Taylo&Fancis and VGTU Volume 22 Numbe 3, May 27, ISSN: c Vilnius Gediminas Technical Univesity, 27 eissn: A Backwad Identification Poblem fo an Axis-Symmetic Factional Diffusion Equation Xiangtuan Xiong and Xiaojun Ma Depatment of Mathematics, Nothwest Nomal Univesity Gansu, China (coesp.): xiongxt@gmail.com xiongxt@fudan.edu.cn Received Novembe 4, 26; evised Mach 6, 27; published online May 5, 27 Abstact. We conside a backwad ill-posed poblem fo an axis-symmetic factional diffusion equation which is descibed in pola coodinates. A closed fom solution of the invese poblem is obtained. Howeve, this solution blows up. Fo numeical stability, a geneal egulaization pinciple is pesented fo constucting egulaization methods. Seveal numeical examples ae conducted fo showing the validity and effectiveness of the poposed methods. Keywods: ill-posedness, egulaization, factional diffusion equation, closed fom solution. AMS Subject Classification: 65R35; 47A52. Intoduction Recently, much attention is attacted to the anomalous diffusion phenomena [3]. The fowad poblems fo factional diffusion equations have been widely investigated. Howeve, invese poblems connected with factional diffusion ae not studied enough in spite of the significance. Since Cheng and Yamamoto [] opened up a path to the field of the invese poblem fo factional diffusion equations, the wok on this topic mushoomed. These invese poblems can be divided into seveal categoies: () Numeical diffeentiation poblems,e.g., [, 6, 7], (2) Invese souce poblems, e.g., [4, 24], (3) Backwad time-factional diffusion poblems, e.g., [, 2, 2, 22], (4) Invese coefficient poblems, e.g., [5, 9, 5], (5) Invese ode poblems, e.g., [4, 2], (6) Invese Stum-Liouville poblems, e.g., [6]. The difficulty of invese poblem fo factional diffusion equations lies in the ill-posedness and the factional deivative. Fo moe details on the invese poblems, the eades can efe to a ecent suvey [7] and the efeences theein. Howeve, the woks on invese poblems ae consideed in the Catesian coodinates. Fo backwad time-factional diffusion poblems, Dou and

2 32 X. Xiong and X. Ma Hon [2] pesented a fundamental solution method combined classical egulaization methods, Liu [3, 22] consideed two quasi-evesibility egulaization methods. Yang and Liu [23] poposed a Fouie method. In this aticle, we conside an axis-symmetic factional diffusion equation. Let f(t) C[a, b] be a time-dependant function, < α < is the ode of a factional deivative and the Caputo factional deivative is defined as follows [8] td α u(x, t) = t Γ ( α) u(x, η) η dη (t η) α. A two-paamete Mittag-Leffle function is defined as E α,β = k= z k, α >, β >, z C. Γ (αk + β) The fowad axis-symmetic factional diffusion poblem now can be expessed as follows: td α u(, t) = κ( 2 u(, t) 2 + subject to the following bounday and initial conditions u(, t) ), < < R, < t < T, (.) u(r, t) =, < t < T, lim u(, t) bounded, < t < T, u(, ) = f(), < < R. (.2) Invese Poblem: we want to ecove the solution u(, t) L 2 ((, R), ρ) with t < T fom the final time data u(, T ) := g() L 2 ((, R), ρ) whee L 2 ((, R), ρ) denotes the Hilbet space of squaes Lebesgue measuable functions with weight ρ() = /R 2 defined on (,R). The aim of this wok is to investigate mathematically and numeically the ill-posedness natue of the backwad identification poblem. Unde the setting of a adial diffusion geomety, the solution is explicitly constucted as an infinite seies in tems of the eigenfunctions. Due to the amplifying facto, small petubation of the data can lead to lage eo in the solution. Based on the obsevation on the ill-posedness natue, we give a geneal egulaization pinciple fo numeical stable solution. 2 The closed fom solution Fist we solve the fowad poblem by the sepaation of vaiables. pupose, the analytical solution of poblem (.) (.2) is assumed as Fo this u(, t) = ϕ i ()q i (t), i =, 2,,, (2.) i= whee ϕ i () ae called eigenfunctions. Substituting equation (2.) into equation (.), we can obtain the Bessel diffeential equation and factional diffeential

3 Backwad Identification Poblem 33 equation. In fact, the eigenfunctions ϕ i () ae the solutions of Bessel diffeential equations: d 2 ϕ d 2 + dϕ + λϕ =, d the bounday conditions ae lim ϕ() bounded and ϕ(r) =. And λ is the sepaation constant. Consideing the bounday condition, we can calculate the sepaation constant λ i = ( µi R )2, whee the µ i, i =, 2,, ae the positive zeos of the the zeo-ode Bessel function of the fist kind J (µ), the eigenfunctions ϕ i ae ϕ i () = J (µ i R ) which is a complete othogonal basis in L2 ((, R), ρ). Now the analytical solution (2.) of the poblem is ewitten as u(, t) = i= Fist the othogonality elationship holds i.e., R xj (µ i x)j (µ j x)dx = q i (t)j (µ i ), i =, 2,,. R {, i j, J 2 (µ j )/2, i = j, { R 2 J (µ i R )J, i j, (µ j R )d = J 2 (µ j )/2, i = j. Fom the above equation, we can easily conclude that the function q i (t) satisfies the following factional diffeentiation equation: td α q i (t) = κ ( µ i /R ) 2 qi (t) and the initial condition q i () = f i, whee f i is f i = 2 R 2 J 2 (µ i) R J (µ i R )f()d. Solving the factional diffeentiation equation, we can easily get the solution q i (t) = E α, ( κ ( µ i /R ) 2 t α )f i. Now, finally we have the analytic solution fo the fowad poblem (.) (.2): u(, t) = i= J (µ i R )E α,( κ( µ i R )2 t α )f i, i =, 2,,. (2.2) Now we tun to solving the invese poblem. equality: g() := u(, T ) = i= Fom (2.2), we have the J (µ i R )E α,( κ( µ i R )2 T α )f i, i =, 2,,. (2.3) Math. Model. Anal., 22(3):3 32, 27.

4 34 X. Xiong and X. Ma Using the othogonality elationship, we have g i = E α, ( κ ( µi /R ) 2 T α ) f i, i =, 2,,, 2 R whee g i = R 2 J 2(µi) J (µ i R )g()d. Theefoe, the solution fo the invese poblem is given by u(, t) = J (µ i R ) E α,( κ( µi R )2 t α ) E α, ( κ( µi R )2 T α ) g i, i =, 2,,. (2.4) i= Remak. If α =, in fact E, ( κ( µi R )2 t α ) = exp( κ( µi R )2 t) and E, ( κ( µi R )2 T α ) = exp( κ( µi R )2 T ), then the solution fo the invese poblem is given by u(, t) = i= J (µ i R ) exp(κ(µ i R )2 (T t))g i, i =, 2,,. Since µ i as i, we have exp(κ( µi R )2 (T t)) fo T > t. Theefoe the backwad identification poblem is an ill-posed poblem. Remak 2. If we conside the simila poblem which is fomulated in the case of Catesian coodinate: td α u(x, t) = u xx (x, t), < x < R, < t < T with bounday conditions u(, t) = u(r, t) = and final condition u(x, T ) = g(x). The solution fo the invese poblem is given by u(x, t) = n= sin( nπ R x) E α,( κ(nπ/r) 2 t α ) E α, ( κ(nπ/r) 2 T α ) g n, n =, 2,,, whee g n is the sine Fouie coefficients. Compaing this expession with (2.4), we can see that the poblems in the two cases have almost the same ill-posedness. In this pape, we ae moe inteested in finding the invese solution u(, ) in (2.4). Hee noting that E α, () = we wite the invese solution as u(, ) = i= J (µ i R ) E α, ( κ( µi R )2 T α ) g i, i =, 2,,. (2.5) In ode to show the instability of the solution, we need a lemma fom [3]. Lemma. Assume that < α < α <, then thee exist constants C 2 > C > depending only on α, α, such that C Γ ( α) x E C 2 α,(x), x, Γ ( α) x these estimates ae unifom fo all α [α, α ].

5 Backwad Identification Poblem 35 Fom Lemma, we can conclude that Γ ( α) ( + κ( µ i C 2 R )2 T α ) Because µ i as i, we have E α, ( κ( µi R )2 T α ) E α, ( κ( µi R )2 T α ) = O(κ(µ i R )2 T α ) Γ ( α) ( + κ( µ i C R )2 T α ). with a fixed α which is fa away fom. Theefoe, fom (2.5) we can see that a small petubation in the data g can cause a lage change in the solution u(, ). In the case of factional deivative, the backwad identification poblem is mildly ill-posed. Fom above analysis, we know that the ill-posedness of the backwad identification poblem is caused by the facto. We call K(, µ E α,( κ( µ i R )2 T α ) i) = amplifying E α,( κ( µ i R )2 T α ) facto of the poblem. 3 A egulaization pinciple and its applications Motivated by the filte methods [9], we can appoximate the amplifying facto stably and thus stabilize the ill-posed poblem. By intoducing a egulaization paamete α, we popose a geneal pinciple fo constucting the stabilized amplifying facto K α (, µ i ) which appoaches the amplifying facto K(, µ i ): A geneal pinciple fo constucting K α (, µ i ): () fo evey µ i, lim α K α (, µ i ) = K(, µ i ); (2) fo evey α >, thee exists a constant C( α) which is dependent on α such that K α (, µ i ) C( α) fo all µ i ; (3) thee exists constant c > such that K α (, µ i ) ck(, µ i ) fo all α and µ i. The fist condition guaantees the appoximation popety. The last two conditions guaantee the stabilization of the appoximation poblem. Based on the pinciple, we can constuct seveal specific egulaization methods. Fo examples, we list some of them: Method. The fist stabilized amplifying facto K α (, µ i ) is given by K α (, µ i ) = E α, ( κ(µ i /R) 2 T α ) χ α, whee χ is the chaacteistic function, i.e., χ α = if µ i α α and χ = if α µ i > α. Method 2. The second stabilized amplifying facto K α (, µ i ) is given by K α (, µ i ) = E α,( κ(µ i /R) 2 T α ) E 2 α, ( κ(µ i/r) 2 T α ) + α. Coespondingly, we wite these two egulaization solutions as follows: Math. Model. Anal., 22(3):3 32, 27.

6 36 X. Xiong and X. Ma Solution. The fist egulaization solution u α (, ) is given by u α (, ) = i= J (µ i R ) E α, ( κ(µ i /R) 2 T α ) χ α g i. This method is known as spectal cut-off method. In pactical computation, we use M u M (, ) = J (µ i R ) E α, ( κ(µ i /R) 2 T α ) g i, i= whee M is the egulaization paamete. Solution 2. The second egulaization solution u α (, ) is given by u α (, ) = i= J (µ i R ) E α, ( κ(µ i /R) 2 T α ) E 2 α, ( κ(µ i/r) 2 T α ) + α g i. Remak 3. Motivated by the ecently-developed factional Tikhonov egulaization methods [8], it is inteesting to give some factional egulaization methods, e,g. the factional Tikhonov method. This method is given by modifying the amplifying facto as with γ 2. 4 Numeical examples K α (, µ i ) = E α,( κ(µ i /R) 2 T α ) E γ α, ( κ(µ i/r) 2 T α ) + α Now we give some numeical examples to test the poposed egulaization pinciple. In this section, M and M2 epesent Method, Method 2, espectively. Fist we solve the fowad poblem with some specified function f() to simulate the final value u(, T ) := g() by (2.2), then we add atificial andom noise to g() fo geneating the noisy data g δ (). To avoid the invese cime, we use two diffeent gid configuations of f() fo solving fowad and backwad poblems. Fo the fowad poblem, we specify the value f() at a coase gid { j }, j =,..., M to get g() at the coase gid. Afte geneating the noisy data g δ (), by intepolation, we can get the values of g δ () at a fine gid { j }, j =,..., M 2. Then we solve the backwad poblem by egulaization methods to get the values of f() at the fine gid { j }, j =,..., M 2. In the examples, we take R =, T =, M =, M 2 = 4. We eplace in the solutions with a lage intege M (i.e., we tuncate the seies) in M, M2 and (2.2). Howeve in M2, the intege M plays the ole of egulaization paamete. The discete vesion of noisy data g() is geneated as follows: g δ j = g j + max g j δ and(length(g)) j, j =,..., M 2, whee g j ae the data and max g j denotes the maximum of the data g j, and(length(g)) j is a andom numbe, δ is the noise level.

7 Backwad Identification Poblem 37 Example. Conside the initial data f() = e 2 sin(2π). Figue (a) shows the esults fo the computed g() and Figue (b) shows the computed f() by (2.5) with α =.5 and δ = %. We can see that the solution given by (2.5) is unstable solution fo computation because the ill-posedness of poblem. Theefoe the egulaization method is necessay appoximate g() f() (a) (b) Figue. Example : (a) the input data g(t), (b) the diect computational esult with noisy data When α =.5 and δ = %, we use the egulaization methods M and M2 to constuct the appoximation. The esults ae displayed in Figue 2 (a) and (b). Fom them it is easy to see that the pesented methods ae effective..8.6 appoximate.8.6 appoximate.4.4 f().2 f() (a) (b) Figue 2. Example : (a) esults by M2 with α = 7, (b) Result by M with M =. Example 2. In Example, the solution ae too smooth. In this example, we conside a had example, i.e., the initial data is given by f(t) = {, if.25 t.75,, else. (4.) We find the methods ae still effective. Fist Figue 3 (a) and (b) show the econstuction esults fo M with a fixed factional ode α =., M = and diffeent noise levels. Fom these figues we see that the esults wosen as the noise becomes lage. Math. Model. Anal., 22(3):3 32, 27.

8 38 X. Xiong and X. Ma appoximate f() f() appoximate (a) (b) Figue 3. Example 2. Method with diffeent noisy levels: (a) esult by M with δ =., (b) esult by M with δ =.3. Figue 4 (a), (b) and (c) show the esults with diffeent factional odes α with a fixed noise level δ = %, α = 6. The lage α is, the wose econstuction esult is. This is because the degee of ill-posedness inceases as the factional ode α inceases. And the degee of ill-posedness becomes the lagest at α =..2 appoximate.2 appoximate f() f() (a) (b).2 appoximate.8.6 f() (c) Figue 4. Example 2 (a) M2 with factional ode α =., (b) M2 with diffeent factional ode α =.5, (c) M2 with diffeent factional ode α =.9. Acknowledgements The authos would like to thank Pof. I.Podlubny fo poviding the pogam on Mittag-Leffle function on the web

9 Backwad Identification Poblem 39 The authos would like to thank the eviewes fo thei impotant comments. The eseach was patially suppoted by a gant fom the National Natual Science Foundation of China (No. 6672), and the Natual Science Foundation of Gansu Povince, China (No. 45RJZA37 ). Refeences [] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki. Uniqueness in an invese poblem fo a one-dimensional factional diffusion equation. Invese Poblems, 25():52, 29. [2] F.F. Dou and Y.C. Hon. Numeical computation fo backwad time-factional diffusion equation. Engineeing Analysis with Bounday Elements, 4:38 46, [3] R. Goenflo and F. Mainadi. Random walk models fo space factional diffusion pocesses. Factional Calculus and Applied Analysis, (2):67 9, 998. [4] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto. Detemination of ode in factional diffusion equation. Jounal of Math-fo-Industy, 5(A):5 57, 23. [5] B. Jin and W. Rundell. An invese poblem fo a one-dimensional time-factional diffusion poblem. Invese Poblems, 28(7):75, 22. [6] B. Jin and W. Rundell. An invese Stum-Liouville poblem with a factional deivative. Jounal of Computational Physics, 23(4): , [7] B. Jin and W. Rundell. A tutoial on invese poblems fo anomalous diffusion pocesses. Invese Poblems, 3(3):353, 25. [8] E. Klann and R. Ramlau. Regulaization by factional filte methods and data smoothing. Invese Poblems, 24(2):455, 28. [9] G. Li, D. Zhang, X. Jia and M. Yamamoto. Simultaneous invesion fo the spacedependent diffusion coefficient and the factional ode in the time-factional diffusion equation. Invese Poblems, 29(6):654, 23. [] M. Li, Y. Wang and L. Ling. Numeical Caputo diffeentiation by adial basis functions. Jounal of Scientific Computing, 62():3 35, [] M. Li and X.T. Xiong. On a factional backwad heat conduction poblem: Application to debluing. Computes and Mathematics with Applications, 64(8): , [2] Z. Li and M. Yamamoto. Uniqueness fo invese poblems of detemining odes of multi-tem time-factional deivatives of diffusion equation. Applicable Analysis, 94(3):57 579, [3] J.J. Liu and M. Yamamoto. A backwad poblem fo the timefactional diffusion equation. Applicable Analysis, 89(): , 2. [4] Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo. Uniqueness and econstuction of an unknown semilinea tem in a time-factional eaction-diffusion equation. Invese Poblems, 29(6):659, 23. [5] L. Mille and M. Yamamoto. Coefficient invese poblem fo a factional diffusion equation. Invese Poblems, 29(7):753, 23. Math. Model. Anal., 22(3):3 32, 27.

10 32 X. Xiong and X. Ma [6] D.A. Muio. On the stable numeical evaluation of Caputo factional deivatives. Computes and Mathematics with Applications, 5(9):539 55, [7] L. Pandolfi. A Lavent ev-type appoach to the on-line computation of Caputo factional deivatives. Invese Poblems, 24():54, 28. [8] I. Podlubny. Factional Diffeential Equations. Academic Pess, San Diego, 999. [9] Z. Qian, B.Y.C. Hon and X.T. Xiong. Numeical solution of two-dimensional adially symmetic invese heat conduction poblem. Jounal of Invese and Illposed Poblems, 23(2):2 34, [2] K. Sakamoto and M. Yamamoto. Inital value/boudnay value poblems fo factional diffusion-wave equations and applications to some invese poblems. Jounal of Mathematical Analysis and Applications, 382(): , 2. [2] L. Wang and J. Liu. Total vaiation egulaization fo a backwad time-factional diffusion poblem. Invese Poblems, 29():53, 23. [22] M. Yang and J. Liu. Solving a final value factional diffusion poblem by bounday condition egulaization. Applied Numeical Mathematics, 66:45 58, [23] M. Yang and J. Liu. Fouie egulaization fo a final value timefactional diffusion equation. Applicable Analysis, 94(7):58 526, [24] Y. Zhang and X. Xu. Invese souce poblem fo a factional diffusion equation. Invese Poblems, 27(3):35, 2.

An Exact Solution of Navier Stokes Equation

An Exact Solution of Navier Stokes Equation An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in