Home The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
Article
Licensed
Unlicensed Requires Authentication

The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation

  • Anatoly Alikhanov EMAIL logo , Murat Beshtokov and Mani Mehra
Published/Copyright: August 23, 2021

Abstract

In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracy O(h4+τ2α) and O(h4+τ2) . A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems.

Acknowledgements

The authors are very grateful to the anonymous referees and the editors for their valuable suggestions and comments.

The reported study was funded by RFBR according to the Research No 20-51-53007.

References

[1] V.M. Abdullayev, K.R. Aida-zade, Finite-difference methods for solving loaded parabolic equations. Comput. Math. Math. Phys. 56, No 1 (2016), 93–105.10.1134/S0965542516010036Search in Google Scholar

[2] A.A. Alikhanov, A.M. Berezgov, M.Kh. Shkhanukov-Lafishev, Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods. Comput. Math. Math. Phys. 48, No 9 (2008), 1581–1590.10.1134/S096554250809008XSearch in Google Scholar

[3] A.A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equations 46, No 7 (2010), 949–961.Search in Google Scholar

[4] A.A. Alikhanov, Boundary value problems for the diffusion equation of the variable order in differential and difference settings. Appl. Math. Comput. 219 (2012), 3938–3946.Search in Google Scholar

[5] A.A. Alikhanov, Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation. Appl. Math. Comput. 268 (2015), 12–22.Search in Google Scholar

[6] A.A. Alikhanov, A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280 (2015), 424–438.10.1016/j.jcp.2014.09.031Search in Google Scholar

[7] A.A. Alikhanov, A time-fractional diffusion equation with generalized memory kernel in differential and difference settings with smooth solutions. Comput. Methods Appl. Math. 7, No 4 (2017), 647–660.Search in Google Scholar

[8] A. Ashabokov, Z.V. Beshtokova, M.Kh. Shkhanukov-Lafishev, Locally one-dimensional difference scheme for a fractional tracer transport equation. Comput. Math. Math. Phys. 57, No 9 (2017), 1498–1510.10.1134/S0965542517090044Search in Google Scholar

[9] G.I. Barenblat, Yu.P. Zheltov, I.N. Kochina, Basic concept in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 25, No 5 (1960), 852–864.Search in Google Scholar

[10] M.Kh. Beshtokov, Finite-difference method for a nonlocal boundary value problem for a third-order pseudoparabolic equation. Differ. Equations 49, No 9 (2013), 1134–1141.10.1134/S0012266113090085Search in Google Scholar

[11] M.Kh. Beshtokov, A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation. Comput. Math. Math. Phys. 54, No 14 (2014), 1441–1458.10.1134/S096554251409005XSearch in Google Scholar

[12] M.Kh. Beshtokov, On the numerical solution of a nonlocal boundary value problem for a degenerating pseudoparabolic equation. Differ. Equations 52, No 10 (2016), 1341–1354.10.1134/S0012266116100104Search in Google Scholar

[13] M.Kh. Beshtokov, Difference method for solving a nonlocal boundary value problem for a degenerating third-order pseudo-parabolic equation with variable coefficients. Comput. Math. Math. Phys. 56, No 10 (2016), 1763–1777.10.1134/S0965542516100043Search in Google Scholar

[14] M.Kh. Beshtokov, The third boundary value problem for loaded differential Sobolev type equation and grid methods of their numerical implementation. IOP Conf. Ser.: Materials Science and Engineering 158, (2016), 1–7.10.1088/1757-899X/158/1/012019Search in Google Scholar

[15] M.Kh. Beshtokov, Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution. Comput. Math. Math. Phys. 57, No 12 (2017), 1973–1993.10.1134/S0965542517120089Search in Google Scholar

[16] M.Kh. Beshtokov, Boundary value problems for degenerating and nondegenerating Sobolev-type equations with a nonlocal source in differential and difference forms. Differ. Equations 54, No 2 (2018), 250–267.10.1134/S0012266118020118Search in Google Scholar

[17] M.Kh. Beshtokov, To boundary-value problems for degenerating pseudoparabolic equations with Gerasimov-Caputo fractional derivative. Russian Mathematics 62, No 10 (2018), 1–14.10.3103/S1066369X18100018Search in Google Scholar

[18] M.Kh. Beshtokov, Boundary-value problems for loaded pseudoparabolic equations of fractional order and difference methods of their solving. Russian Mathematics 63, No 2 (2019), 3–12.Search in Google Scholar

[19] M.Kh. Beshtokov, Boundary value problems for a pseudoparabolic equation with the Caputo fractional derivative. Differential Equations 55, No 7 (2019), 1–10.Search in Google Scholar

[20] C.M. Chen, F. Liu, V. Anh, I. Turner, Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equations. SIAM J. Sci. Comput. 32, No 4 (2010), 1740–1760; doi:10.1137/090771715.Search in Google Scholar

[21] A.F. Chudnovskii, Thermal Physics of Soils. Nauka, Moscow (1976) [in Russian].Search in Google Scholar

[22] B.D. Coleman, R.J. Duffin, V.J. Mizel, Instability, uniqueness, and nonexistence theorems for the equation ut = uxxuxxt on a strip. Arch. Rat. Mech. Anal. 19, No 2 (1965), 100–116; doi:10.1007/bf00282277.Search in Google Scholar

[23] D.L. Colton, Pseudoparabolic equations in one space variable. J. Differ. Equations 12, (1972), 559–565.10.1016/0022-0396(72)90025-3Search in Google Scholar

[24] D.L. Colton, Integral operators and the first initial-boundary value problems for pseudo-parabolic equations with analytic coefficients. J. Differ. Equations 13 (1973), 506–522.10.1016/0022-0396(73)90009-0Search in Google Scholar

[25] M. Cui, Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228 (2009), 7792–7804.10.1016/j.jcp.2009.07.021Search in Google Scholar

[26] E.S. Dzektser, Equations of motion of free-surface underground water in layered media. Dokl. Akad. Nauk SSSR 220, No 3 (1975), 540–543.Search in Google Scholar

[27] G.H. Gao, H. Sun, Z.Z. Sun, Some temporal second order difference schemes for fractional wave equations. Numer. Methods Partial Diff. Eq. 32 (2016), 970–1001.10.1002/num.22038Search in Google Scholar

[28] M. Hallaire, Le potentiel efficace de leau dans le sol en regime de dessechement. L'Eau et la Production Vegetale. Paris: Institut National de la Recherche Agronomique 9 (1964), 27–62.Search in Google Scholar

[29] I. Karatay, N. Kale, S.R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicolson method. Fract. Calc. Appl.Anal. 16, No 4 (2013), 892–910; DOI:10.2478/s13540-013-0055-2; https://www.degruyter.com/journal/key/FCA/16/4/html.Search in Google Scholar

[30] B.T. Jin, R. Lazarov, Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, No 1 (2016), 197–221.Search in Google Scholar

[31] B.T. Jin, R. Lazarov, D. Sheen, Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19, No 1 (2016), 69–93; DOI:10.1515/fca-2016-0005; https://www.degruyter.com/journal/key/FCA/19/1/html.Search in Google Scholar

[32] B.T. Jin, R. Lazarov, Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview. Comput. Methods Appl. Mech. Engrg. 346 (2019), 332–358; doi:10.1016/j.cma.2018.12.011.Search in Google Scholar

[33] A.I. Kozhanov, On a nonlocal boundary value problem with variable coefficients for the heat equation and the Aller equation. Differ. Equations 40, No 6 (2004), 815–826.10.1023/B:DIEQ.0000046860.84156.f0Search in Google Scholar

[34] S.K. Lele, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, No 1 (1992), 16–42.10.1016/0021-9991(92)90324-RSearch in Google Scholar

[35] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007), 1552–1553.Search in Google Scholar

[36] B.B. Mandelbrojt, The Fractal Geometry Of Nature. Freeman, San-Francisco (1982).Search in Google Scholar

[37] V. Mehandiratta, M. Mehra, G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph. J. Math. Anal. Appl. 447 (2019), 1243–1264.Search in Google Scholar

[38] M. Mehra, K.S. Patel, Algorithm 986舁:舁A suite of compact finite difference schemes. ACM Transactions on Mathematical Software 44, No 2 (2017), 1–31.10.1145/3119905Search in Google Scholar

[39] M. Mehra, K.S. Patel, Fourth-order compact scheme for option pricing under the mertons and kous jump-diffusion models International. J. of Theor. and Appl. Finance. 21, No 4 (2018), 1–26; doi:10.1142/s0219024918500279.Search in Google Scholar

[40] M. Mehra, K.S. Patel, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients. J. of Comput. and Appl. Math. 380 (2020), 1–15; doi:10.1016/j.cam.2020.112963.Search in Google Scholar

[41] A.M. Nakhushev, Fractional Calculus and Its Application. Fizmatlit, Moscow (2003) [in Russian].Search in Google Scholar

[42] K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York (1974).Search in Google Scholar

[43] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar

[44] L.I. Rubinshtein, On heat propagation in heterogeneous media. Izv. Akad. Nauk SSSR. Ser. Geogr. 12, No 1 (1948), 27–45.Search in Google Scholar

[45] W. Rundell, M. Stecher, Maximum principles for pseudoparabolic partial differential equations. J. Math. Anal. Appl. 57, No 1 (1977), 110–118.10.1016/0022-247X(77)90289-XSearch in Google Scholar

[46] A.A. Samarskii, A.V. Gulin, Stability of Finite Difference Schemes. Nauka, Moscow (1973) [in Russian].Search in Google Scholar

[47] A.A. Samarskii, The Theory of Difference Schemes. Nauka, Moscow (1983); Marcel Dekker, New York (2001).Search in Google Scholar

[48] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Minsk (1987); Gordon and Breach, New York (1993).Search in Google Scholar

[49] M.Kh. Shkhanukov, On some boundary value problems for third-order equations arising in the modeling of flows in porous media. Differ. Equations 18, No 4 (1982), 689–699.Search in Google Scholar

[50] M.Kh. Shkhanukov-Lafishev, F.I. Taukenova, Difference methods for solving boundary value problems for fractional differential equations. Comput. Math. Math. Phys. 46, No 10 (2006), 1785–1795.10.1134/S0965542506100149Search in Google Scholar

[51] M.Kh. Shkhanukov-Lafishev, Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions. Comput. Math. Math. Phys. 49, No 7 (2009), 1167–1174.10.1134/S0965542509070094Search in Google Scholar

[52] R.E. Showalter, T. Ting, Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1 (1970), 1–26.10.1137/0501001Search in Google Scholar

[53] Z.Z. Sun, X.N. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006), 193–209.10.1016/j.apnum.2005.03.003Search in Google Scholar

[54] Z.Z. Sun, On the compact difference scheme for heat equation with Neuman boundary conditions. Numer. Methods Partial Diff. Eqns. 25 (2009), 1320–1341.10.1002/num.20402Search in Google Scholar

[55] Z.Z. Sun, G. Gao, Fractional Differential Equations. Finite Difference Methods. China Science Publ. and Media Ltd. and Walter de Gruyter GmbH, Berlin/Boston (2020).10.1515/9783110616064Search in Google Scholar

[56] T.W. Ting, A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45, No 1 (1974), 23–31.10.1016/0022-247X(74)90116-4Search in Google Scholar

[57] A.I. Tolstykh, Compact Difference Schemes and Their Applications to Fluid Dynamics Problems. Nauka, Moscow (1990) [in Russian].Search in Google Scholar

[58] V.V. Uchaikin, Method of Fractional Derivatives. Artishok, Ulyanovsk (2008) [in Russian].Search in Google Scholar

[59] Y.N. Zhang, Z.Z. Sun, H.L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshs. J. Comput. Phys. 265 (2014), 195–210.10.1016/j.jcp.2014.02.008Search in Google Scholar

Received: 2020-10-09
Revised: 2021-07-21
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 24–4–2021)
  4. Research Paper
  5. Three representations of the fractional p-Laplacian: Semigroup, extension and Balakrishnan formulas
  6. Tutorial paper
  7. The bouncing ball and the Grünwald-Letnikov definition of fractional derivative
  8. Research Paper
  9. Fractional diffusion-wave equations: Hidden regularity for weak solutions
  10. Censored stable subordinators and fractional derivatives
  11. Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem
  12. Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H synthesis
  13. Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity
  14. The rate of convergence on fractional power dissipative operator on compact manifolds
  15. Fractional Langevin type equations for white noise distributions
  16. Local existence and non-existence for a fractional reaction–diffusion equation in Lebesgue spaces
  17. Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function
  18. The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation
  19. Robust fractional-order perfect control for non-full rank plants described in the Grünwald-Letnikov IMC framework
  20. Properties of the set of admissible “state control” pair for a class of fractional semilinear evolution control systems
Downloaded on 30.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0053/html
Scroll to top button