nach oben

Journal of Applied Mathematics and Computing

Erschienen in:

17.11.2021 | Original Research

Numerical framework for the Caputo time-fractional diffusion equation with fourth order derivative in space

verfasst von: Sadia Arshad, Mubashara Wali, Jianfei Huang, Sadia Khalid, Nosheen Akbar

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 5/2022

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A finite difference scheme along with a fourth-order approximation is examined in this article for finding the solution of time-fractional diffusion equation with fourth-order derivative in space subject to homogeneous and non-homogeneous boundary conditions. Caputo fractional derivative is used to describe the time derivative. The time-fractional diffusion equation of order \(0< \gamma < 1\) is transformed into Volterra integral equation which is then approximated by linear interpolation. A novel numerical scheme based on fractional trapezoid formula for time discretization followed by Stephenson’s scheme to discretize the fourth order space derivative is developed for the linear diffusion equation. Afterward, convergence and stability are investigated thoroughly showing that the proposed scheme is unconditionally stable and hold convergence accuracy of order \(O(\tau ^{2}+h^{4})\). The numerical examples are presented in accordance with the theoretical results showing the efficiency and accuracy of the presented numerical technique.

Vorheriger Artikel Computing the Merrifield-Simmons indices of benzenoid chains and double benzenoid chains

Nächster Artikel A coupled system involving nonlinear fractional q-difference stationary Schrödinger equation

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Baskonus, H.M., Mekkaoui, T., Hammouch, Z., Bulut, H.: Active control of a chaotic fractional order economic system. Entropy 17, 5771–5783 (2015)CrossRef

Baskonus, H.M., Bulut, H.: Regarding on the prototype solutions for the nonlinear fractional-order biological population model. In: AIP Conf Proc (2016)

Losada, J., Nieto, J.J.: Fractional integral associated to fractional derivatives with nonsingular Kernels. Progr. Fract. Differ. Appl. 7(3), 137–143 (2021)

Caputo, M., Fabrizio, M.: On the Singular kernels for fractional derivatives. Some applications to partial differential equations. Progr. Fract. Differ. Appl. 7(2), 79–82 (2021)CrossRef

Veeresha, P., Baskonus, H.M., Gao, W.: Strong interacting internal waves in rotating ocean: novel fractional approach. Axioms 10, 123 (2021)CrossRef

Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12(6), 2455–2477 (1975)CrossRef

Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1 (2000)

Rossikhin, Y.A., Shitikova, M.V.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15 (1997)CrossRef

Ren, L., Wang, Y.M.: A fourth-order extrapolated compact difference method for time-fractional convection–reaction–diffusion equations with spatially variable coefficients. Appl. Math. Comput. 312, 1–22 (2017)MathSciNetMATH

10.

Guo, X., Li, Y., Wang, H.: A fourth-order scheme for space fractional diffusion equations. J. Comput. Phys. 373, 410–424 (2018)MathSciNetMATHCrossRef

11.

Xing, Z., Wen, L.: Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations. Appl. Math. Comput. 346, 155–166 (2019)MathSciNetMATH

12.

Ran, M., Luo, T., Zhang, L.: Unconditionally stable compact theta schemes for solving the linear and semi-linear fourth-order diffusion equations. Appl. Math. Comput. 342, 118–129 (2019)MathSciNetMATH

13.

Luo, H., Zhang, Q.: Regularity of global attractor for the fourth-order reaction–diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 17, 3824–3831 (2012)MathSciNetMATHCrossRef

14.

Guo, G., Lu, S.: Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation. Comput. Math. Appl. 71, 1944–1959 (2016)MathSciNetMATHCrossRef

15.

Yang, X., Zhang, H., Xu, D.: Orthogonal spline collocation method for the fourth-order diffusion system. Appl. Math. Comput. 75, 3172–3185 (2018)MathSciNetMATHCrossRef

16.

Soori, Z., Aminataei, A.: A new approximation to Caputo-type fractional diffusion and advection equations on non-uniform meshes. Appl. Numer. Math. 144, 21–41 (2019)MathSciNetMATHCrossRef

17.

Macías-Díaz, J.E., Hendy, A.S., De Staelen, R.H.: A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. Appl. Math. Comput. 325, 1–14 (2018)MathSciNetMATH

18.

Ran, M., Zhang, C.: New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order. Appl. Numer. Math. 129, 58–70 (2018)MathSciNetMATHCrossRef

19.

Cheng, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation. J. Comput. Appl. Math. 362, 574–595 (2019)MathSciNetMATHCrossRef

20.

Zeid, S.S.: Approximation methods for solving fractional equations. Chaos Solitons Fractals 125, 171–193 (2019)MathSciNetMATHCrossRef

21.

Manimaran, J., Shangerganesh, L., Debbouche, A., Antonov, V.: Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion. Front. Phys. 7, 93 (2019)CrossRef

22.

Hu, X., Zhang, L.: A compact finite difference scheme for the fourth-order fractional diffusion-wave system. Comput. Phys. Commun. 182(10), 1645–1650 (2011)MathSciNetMATHCrossRef

23.

Zhong, J., Liao, H., Ji, B., Zhang, L.: A fourth-order compact solver for fractional-in-time fourth-order diffusion equations. arXiv:1907.01708v1 [math.NA] (2019)

24.

Li, X., Wong, P.J.Y.: Numerical solutions of fourth-order fractional sub-diffusion problems via parametric quintic spline (2019). 10.1002/zamm.201800094

25.

Agrawal, O.P.: A general solution for the fourth-order fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 3, 1 (2000)MathSciNetMATH

26.

Agrawal, O.P.: A general solution for a fourth-order fractional diffusion-wave equation defined in a bounded domain. Comput. Struct. 79, 1497 (2001)CrossRef

27.

Jafari, H., Dehghan, M., Sayevand, K.: Solving a fourth-order fractional diffusion-wave equation in a bounded domain by decomposition method. Numer. Methods Partial Differ. Equ. 24, 1115 (2008)MathSciNetMATHCrossRef

28.

Wei, L., He, Y.: Analysis of a fully discrete local discontinuous Galerkin method for time fractional fourth-order problems. Appl. Math. Model. 38, 1511 (2014)MathSciNetMATHCrossRef

29.

Liu, Y., Fang, Z., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703 (2014)MathSciNetMATH

30.

Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92, 1496 (2015)MathSciNetMATHCrossRef

31.

Zhai, S., Feng, X.: Investigations on several compact ADI methods for the 2D time fractional diffusion equation. Numer. Heat Transf. Part B Fundam. 69(4), 364–376 (2015)CrossRef

32.

Chen, C., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous sub-diffusion equations. SIAM J. Sci. Comput. 32, 1740–1760 (2010)MathSciNetMATHCrossRef

33.

Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)MathSciNetMATHCrossRef

34.

Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)MathSciNetMATHCrossRef

35.

Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)MathSciNetMATHCrossRef

36.

Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson-type difference schemes for the sub-diffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)MathSciNetMATHCrossRef

37.

Gao, G.H., Sun, Z.Z., Zhang, H.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)MathSciNetMATHCrossRef

38.

Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)MathSciNetMATHCrossRef

39.

Gao, G.H., Sun, H.W., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain super convergence. J. Comput. Phys. 280, 510–528 (2015)MathSciNetMATHCrossRef

40.

Vong, S., Wang, Z.: High order difference schemes for a time-fractional differential equation with Neumann boundary conditions. East Asian J. Appl. Math. 4, 222–241 (2014)MathSciNetMATHCrossRef

41.

Zhao, L., Deng, W.: A series of high order quasi-compact schemes for space fractional diffusion equations based on the super convergent approximations for fractional derivatives. Numer. Methods Partial Differ. Equ. 31, 1345–1381 (2015)MATHCrossRef

42.

Ji, C.C., Sun, Z.Z.: A high-order compact finite difference scheme for the fractional sub-diffusion equation. J. Sci. Comput. 64, 959–985 (2015)MathSciNetMATHCrossRef

43.

Huang, J., Yang, D.: A unified difference-spectral method for time-space fractional diffusion equations. Int. J. Comput. Math. 94(6), 1172–1184 (2017)MathSciNetMATHCrossRef

44.

Dison, J., Mekee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)MathSciNetCrossRef

45.

Huang, J.F., Tang, Y.F., Väzquez, L.: Convergence analysis of a block-by-block method for fractional differential equations. Numer. Math. Theor. Methods Appl. 5, 229–241 (2012)MathSciNetMATHCrossRef

46.

Cui, M.: Compact difference scheme for time-fractional fourth-order equation with the first Dirichlet boundary conditions. East Asian J. Appl. Math. 9, 45–66 (2019)MathSciNetMATHCrossRef

47.

Ben-Artzi, M., Croisille, J.P., Fishelov, D.: A fast direct solver for the biharmonic problem in a rectangular grid. SIAM J. Sci. Comput. 31, 303–333 (2008)MathSciNetMATHCrossRef

48.

Fishelov, D., Ben-Artzi, M., Croisille, J.-P.: Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equations. J. Sci. Comput. 53, 55–79 (2012)MathSciNetMATHCrossRef

49.

Thomas, J.W.: Numerical Partial Differential Equations (Finite Difference Methods), Texts in Applied Mathematics, vol. 22. Springer, New York (1995)CrossRef

50.

Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)MATHCrossRef

Titel: Numerical framework for the Caputo time-fractional diffusion equation with fourth order derivative in space
verfasst von: Sadia Arshad
Mubashara Wali
Jianfei Huang
Sadia Khalid
Nosheen Akbar
Publikationsdatum: 17.11.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 5/2022
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-021-01635-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 5/2022

A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations

An approximation solution of linear Fredholm integro-differential equation using Collocation and Kantorovich methods

Energy of interval-valued fuzzy graphs and its application in ecological systems

A two-gird method for finite element solution of parabolic integro-differential equations

On the 4-adic complexity of the two-prime quaternary generator

Stochastic analysis of a SIRI epidemic model with double saturated rates and relapse