دانشگاه تبریز
فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری
پایگاه استنادی علوم جهان اسلام (ISC)

 آمار

تعداد نشریات45
تعداد شماره‌ها1,499
تعداد مقالات18,351
تعداد مشاهده مقاله59,512,877
تعداد دریافت فایل اصل مقاله20,819,074
Rashidinia, Jalil, Mohmedi, Elham. (1400). Numerical solution for solving fractional parabolic partial differential equations. سامانه مدیریت نشریات علمی, 10(1), 121-143. doi: 10.22034/cmde.2021.41150.1787
Jalil Rashidinia; Elham Mohmedi. "Numerical solution for solving fractional parabolic partial differential equations". سامانه مدیریت نشریات علمی, 10, 1, 1400, 121-143. doi: 10.22034/cmde.2021.41150.1787
Rashidinia, Jalil, Mohmedi, Elham. (1400). 'Numerical solution for solving fractional parabolic partial differential equations', سامانه مدیریت نشریات علمی, 10(1), pp. 121-143. doi: 10.22034/cmde.2021.41150.1787
Rashidinia, Jalil, Mohmedi, Elham. Numerical solution for solving fractional parabolic partial differential equations. سامانه مدیریت نشریات علمی, 1400; 10(1): 121-143. doi: 10.22034/cmde.2021.41150.1787

Numerical solution for solving fractional parabolic partial differential equations

Computational Methods for Differential Equations
مقاله 9، دوره 10، شماره 1، فروردین 2022، صفحه 121-143    اصل مقاله (265.77 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2021.41150.1787
نویسندگان
Jalil Rashidinia* ؛ Elham Mohmedi
School of Mathematics, Iran University of Science and Technology, Tehran, Iran.
چکیده
In this paper, a reliable numerical scheme is developed and reviewed in order to obtain an approximate solution of time-fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time-fractional derivative is employed in the Caputo sense. The L2 convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of the spectral method and justification the theoretical analysis and to compare with other existing methods in the literatures.
کلیدواژه‌ها
Time fractional parabolic partial differential equations؛ Caputo derivative؛ Shifted Legendre tau method
مراجع
  • [1]          S. Arshed, Quintic B-spline method for time-fractional super diffusion fourth-order differential equation, Journal of Mathematical Science, 11 (2017), 17-26.
  • [2]          N. Ahmadi, A.R. Vahidi, and T. Allahviranloo, An efficient approach based on radial basis functions for solving stochastic fractional differential equations, Mathematical Sciences, 11 (2017) , 113-118.
  • [3]          K. Burrage, N. Hale, D. kay, and R.A. Khan, An effcient Implicit FEM scheme for fractional-IN-space reaction diffusion equations, SIAM Journal on Computing, 34 (2012), A2145-A2172.
  • [4]          A. H. Bhrawy, M.A. Zaky, and R. A. Van Gorder, A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numerical Algorithms., 71 (2016), 151-180 .
  • [5]          A. H. Bhrawy, E.H.Doha, D. Baleanu, and S. S. Ezz-Eldieng, A spectral tau Algorithm based on Jacobi operational matrix numerical solution of time fractional diffusion-wave equations, Journal of Computational Physics, 293 (2015), 142-156.
  • [6]          C-M. Chen,F. Liu, and V. Anha,  A Fourier  method and an extrapolation technique for Stokes’ rst problem for    a heated generalized second grade fluid with fractional derivative, Computational and Applied Mathematics, 223 (2009), 777-789.
  • [7]          M. Cui, Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, Journal of Computational Physics, 231 (2013), 2621-2633.
  • [8]          M. Cui, Compact finite difference method for the fractional diffusion equation, Journal of Computational Physics, 228 (2009), 7792-7804.
  • [9]          A. Consiglio, Time fractional diffusion equation and its applications in physics, Sessione estiva Anno Academic., (2017).
  • [10]        J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, and C. Liao, The analytical solution and numerical solution of the fractional diffusion-wave equation with damping, Applied Mathematics and Computation, 219 (2012) 1737-1748.
  • [11]        Y. Chen, Y. Wu , Y. Cui, Z. Wang, and D. Jin, Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, Journal of Computational Science, 1 (2010), 146-149.
  • [12]        C. Canuto, M. Y. Hussaain, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, (1989) New York.
  • [13]        C M. Y. Canuto, M. Y. Hussaini,A. Quarteron, and T. A. Zang, T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, (2006) New York.
  • [14]        C-M. Chen, F. Liu, I. Turner, and V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, Journal of Computational Physics 227 (2007), 886-897.
  • [15]        M. Cui,Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients, Journal of Computational Physics, 280 (2015) 143-163.
  • [16]        M. Sh. Dahaghan and H. Hassani, A new optimization method for a class of time fractional convection- diffusion- wave equations, The European Physical Journal Plus (2017) 130-132.
  • [17]        S. Das, Functional Fractional Calculus, (Springer-Verlag, Berlin, Heidelberg, 2011).
  • [18]        S. Das, Functional Fractional Calculus for System Identification and Controls, (Springer-Verlag, Berlin, Heidel-  berg, 2008).
  • [19]        E.H. Doha, A.H. Bhravyy, and S.S.Ezz-Eldien, A new Jacobi operational matrix An application for solving frac- tional differential equations, Applied Mathematical Modelling. 36 (2012), 4931-4943.
  • [20]        N. J. Ford, J. Xiao, and Y. Yan, A finite element method for time fractional partial differential equations, Fractional Calculus and Applied Analysis, 14 (2011), 454-474.
  • [21]        R. Garrappa and M. Popolizio, on the use of matrix functions for fractional partial differential equations, Math- ematics and Computers in Simulation, 81 (2011), 1045-1056.
  • [22]        V. R. Hosseini and E. Shivanian, Local radial point interpolation (MLRPL) method for solving time fractional diffusion-wave equation with damping, Journal of Computational Physics., 312 (2016), 307-332.
  • [23]        J.H. He, Nonlinear oscillationwith fractional derivative and its applications, in Proceedings of the International Conference on Vibrating Engineering 98, Dalian, China, (1988).
  • [24]        J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bulletin  of  Science, Technology and Society, 15 (1999) 86-90.
  • [25]        R. Herrmann, Fractional Calculus: An Introduction for Physicists.World Scientific, Singapore (2011).
  • [26]        R. Hilfer: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).
  • [27]        M.H. Heydari, M.R. Hooshmandsl, F.M. Maalek Ghaini, and C. Cattani, Wavlets method for the time fractional diffusion-wave equation, Journal of physics letters., A 379 (2015), 71-76.
  • [28]        S. Irandoust-Pakchin, M. Dehghan, S. Abdi-mazraeh, and M. Lakestani, Numerical solution for a class of frac- tional convection-diffusion equations using the flatlet oblique multiwavelets, Journal of Vibration and Control., (2013), DOI: 10.1177/1077546312470473.
  • [29]        Y. Jiang and J. Ma High-order finite  element  methods  for  time  fractional  partial  differential  equations,  Journal  of Computational and Applied Mathematics., 235 (2011), 3285-3290 .
  • [30]        M. Javidi and B. Ahmad, Numerical solution of fourth order time fractional partial differential  equations  with variable coefficients, Journal of Applied Analysis and Computation, (2015).
  • [31]        D. Kumar,  J. Singh,  and S. Kumar,  A fractional model of Navier-Stokes equation arising in unsteady ow of a   viscous uid, Journal of the Association of Arab Universities for Basic and Applied Sciences, 17 (2015), 14-19.
  • [32]        I. Karatay and N. Kale, Finite Difference method of fractional parabolic partial differential equations with variable coefficients, International Journal of Contemporary Mathematical Sciences, 9 (2014), 767-776.
  • [33]        I. Karatay, S.R. Bayramoglu and A. Sahin, Implicit difference approximation for the time fractional heat equation with the non local condition, Applied Numerical Mathematics., 61 (2011), 1281-1288.
  • [34]        Z. Kargar, and H. Saeedi, B-Spline wavelet operational method for numerical  solution  of  time-space  fractional  partial differential equations, International Journal of Wavelets, Multiresolution and Information processing, 15 (2017), No, 04 .
  • [35]        C. Li and A. Chen, Numerical methods for fractional partial differential equations, International Journal of Com- puter Mathematics, 95 (2017), 1048-1099.
  • [36]        Z. Li and Y. Yan, Error estimates of high-order numerical methods for solving time fractional partial differential equations, An International journal for theory and applications. 21 Number 3 (2018).
  • [37]        T. A. M. Langlands and B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, Journal of Computational Physics, 205 (2005), 719-736.
  • [38]        C. Li and A. Chen, Numerical methods for fractional partial differential equations, International Journal of Com- puter Mathematics, 95(6-7) (2018), 1048-1099
  • [39]        F. Liu, P. Zhuan, and K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Computers & Mathematics with Applications 64 (2012), 2990-3007.
  • [40]        X. Li and X. Yang, Error estimates of finite  element  methods  for  stochastic  fractional  differential  equations,  Journal of Computational Mathematics, 35 (2017), 346-362.
  • [41]        F. Liu, P. Zhuang, and K. Burrage, Numerical methods and analysis for a class of fractional advection- dispersion models, Computers & Mathematics with Applications, 64 (2012), 2990-3007 .
  • [42]        A. Mohebbi, M. Abaszadeh, and M. Dehghan, Compact finite difference  scheme and RBF meshless approach  for solving 2D Rayleigh-stokes problem for a heated generalized second grade fluid with fractional derivatives, Computer Methods in Applied Mechanics and Engineering, 264 (2013), 163-177.
  • [43]        F. Nasrollahzadeh and S.M.Hosseini, An Implicit Difference-ADI method for the two-dimensional space-time fractional diffusion equation, Iranian Journal of Mathematical Sciences and Informations, 11 (2016), 71-86.
  • [44]        O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, Numerical investigation of fractional nonlinear sine- Gordon and Klein-Gordon models arising in relativistic quantum mechanics, Engineering Analysis with Boundary Elements, 120 (2020), 223-237.
  • [45]        O. Nikan, J. A. Tenreiro Machado, Z. Avazzadeh, and H. Jafari, Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics Journal of Advanced Research, 25 (2020), 205-216.
  • [46]        O. Nikan, J. A. Tenreiro Machado, and A. Golbabai, Numerical solution of time-fractional fourth-order reaction- diffusion model arising in composite environments Applied Mathematical Modelling, 81 (2021), 819-836.
  • [47]        O.  Nikan,  A.  Golbabai,   J.   A.   Tenreiro   Machado,   and   T.   Nikazad,   Numerical   approximation   of   the  time fractional cable equation arising in neuronal dynamics, Engineering with Computers, (2020) 1-19. doi:https://doi.org/10.1007/s00366-020-01033-8.
  • [48]        O. Nikan, H. Jafari, and A. Golbabai, Numerical analysis of the fractional evolution  model  for  heat  flow  in materials with memory, Alexandria Engineering 59 (4) (2020), 2627-2637.
  • [49]        O. Nikan, Z. Avazzadeh, and J. A. Tenreiro Machado, An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model, Journal of King Saud University- Science. 33 (1) (2021), 101243.
  • [50]        K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differntiation and Integration to Arbitrary Order—, Elsevier Academic Press., New York and London. ISBBN: 9780125255509 (1974).
  • [51]        Z. Odibat, Fractional Power Series Solutions of Fractional Differential  Equations  by  Using  Generalized  Taylor  Series, Applied and Computational Mathematics, 19(1) (2020), 47-58.
  • [52]        I. Podlubnyy, Fractional Differential Equations-Mathematics in science and Engineering. Academic Press, San Diego(1999).
  • [53]        M.R. Permoon, J. Rashidinia, A. Parsa, H. Haddadpour, and R. Salehi, Application of radial basis functions and sinc method for solving the forced vibration of fractional viscoelastic beam, Journal of Mechanical Science and ,30 (2016), 3001-3008.
  • [54]        M.R. Permoon, J. Rashidinia, A. Parsa, H. Haddadpour, and R. Salehi, Application of radial basis functions and sinc method for solving the forced vibration of fractional viscoelastic beam, Journal Mechanical science and Technology, 30 (2016) 3001-3008.
  • [55]        J. Rashidinia, A. Parsa, and R. Salehi, Approximation solution of two dimensional Rayleigh-stokes problem for a heated generalized second grade fluid with fractional derivatives, Journal of Zankoy Sulaimani-part A, 18 (2016)  Doi. 10.17656/jzs.10578.
  • [56]        L. Ren, L. Liu, A high-order compact difference method for time fractional Fokker-Planck equations with variable coefficients, Computational and Applied Mathematics, 38(3) (2019), 1-16.
  • [57]        J. Rashidinia and E. Mohmedi, Convergence analysis of tau scheme for the fractional reaction-diffusion equation,  The European Physical Journal Plus, 133 (2018), 402.
  • [58]        P. Rahimkhani and Y.  Ordokhani,  Solving  of  partial  differential  equations  with  distributed  order  in  time  us-  ing fractional-order Bernoulli-Legendre functions, Computational Methods for Differential Equations, DOI: 10.22034/CMDE.2020.36904.1642 (2020).
  • [59]        S. Shen, F. Liu, V. Anh, Numerical approximations and solution techniques  for  the  space-time  Riesz-Caputo fractional advection-diffusion equation, Numerical Algorithms, 56 (2011), 383-404.
  • [60]        F. Soltani Sarvestani, M.  H.  Heydari,  A.  Niknam  ,  and  Z.  Avazzadeh,  A  wavelet  approach  for  the  multi- term time fractional diffusion-wave equation, International Journal of Computer Mathematics., DOI: (2018), 10.1080/00207160.2018.1458097.
  • [61]        E. Sousa, A second order explicit finite difference method for fractional advection diffusion equation, Computers and Mathematics with Applications., 64 (2012), 3141-3152.
  • [62]        A. Saadatmandi and M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method, Mathematical Reports, 17 (2015) 155-166.
  • [63]        S. Shahid-Siddiqi and S. Arshedb, Numerical Solution of Time-Fractional Fourth-Order Partial Differential Equa- tions, International Journal of Computer Mathematics, (2014) 10.1080/00207160.2014.948430.
  • [64]        N. Sweilam and A. M. Nagy, Sinc-Chebyshev collocation method for time-fractional order telegraph equation, Applied and Computational Mathematics, 19(2) (2020) 162-174
  • [65]        V. E.Tarasov, Fractional Dynamics: Applications of Fractional  Calculus  to  Dynamics  of  Particles,  Fields  and Media (Springer-Verlag, Berlin, Heidelberg, 2010).
  • [66]        V.V. Uchaikin, Method of Fractional Derivatives. Artishok, Uljanovsk (2008).
  • [67]        C. Wu, Numerical solution for Stokes’ rst problem for a heated generalized second grade uid with fractional derivative, Applied Numerical Mathematics, (2009) 2571-2583.
  • [68]        Y. M. Wang, A compact finite difference method for a class of time fractional convection-diffusion-wave equations with variable coefficients, Numerical Algorithms., 70 (2015), 625-651.
  • [69]        Y. M. Wang and L. Ren, Efficient compact finite difference methods for  a  class  of  time-fractional  convection-  reaction diffusion equations with variable coefficients, International Journal of Computer Mathematics., (2018) DOI: 10.1080/00207160.2018.1437262.
  • [70]        L. Ren and Y. M. Wang, A fourth-order extrapolated compact difference method for time-fractional convection- reaction-diffusion equations with spatially variable coefficient, Applied Mathematics and Computation, 312 (2017), 1-22.
  • [71]        Q. Wang, J. Liu, C. Gong, X. Tans, G. Fu, and Z. Xing, An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method, Advances in Difference Equations, 1(2016), 1-12.
  • [72]        W. M. Wang, A compact finite difference method for a class of time fractional convection diffusion wave equations with variable coefficients, Numerical Algorithms, 70(3)(2017), 625-651.
  • [73]        W. M. Wang, A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection-diffusion equations, Calcolo, 54 (2017), 733-768.
  • [74]        Y. Xing and Y. Yan, A higher order numerical method for time fractional partial differential equations with nonsmooth data, Journal of Computational Physics, 35 (2018), 305323.
  • [75]        X. Yang, X. Jiang, and H. Zhang, A time-space spectral tau method for the time fractional cable equation and its inverse problem, Applied Numerical Mathematics, 130 (2018) 95-111.
  • [76]        S. B. Yuste and L. Acedo, An explicit finite difference method and a new von numerical-type stability analysis for fractional diffusion equations, SIAM Journal on Numerical Analysis, 42 (2005), 1862-1874.
  • [77]        M. A. Zaky, An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid, Computers & mathematics with Applications, 75(7) (2018), 2243-2258.
  • [78]        G. X. Zhu and Y. F. Nie, On a collocation method for the time-fractional convection-diffusion equation with variable coefficients, Advances in Water Resources, (2016).
  • [79]        J. Zhao, T. Zhang, and R. M. Corless, Convergence of  the  compact  finite  difference  method  for  second-order  elliptic equations, Applied Mathematics and Computation, 182 (2006), 1454-1469.
  • [80]        Y. Zhang, D. A. Benson, and D. M. Reeves, Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications, Advances in Water Resources, 32 (2009), 561-581.
  • [81]        J. Zhang and X. Yang, A class of efficient difference method for time fractional reaction-diffusion equation, Computational and Applied Mathematics, 37 (2018) 4376-4396.
آمار
تعداد مشاهده مقاله: 1,295
تعداد دریافت فایل اصل مقاله: 1,332


سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب