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.
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References
Baskonus, H.M., Mekkaoui, T., Hammouch, Z., Bulut, H.: Active control of a chaotic fractional order economic system. Entropy 17, 5771–5783 (2015)
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)
Veeresha, P., Baskonus, H.M., Gao, W.: Strong interacting internal waves in rotating ocean: novel fractional approach. Axioms 10, 123 (2021)
Scher, H., Montroll, E.W.: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12(6), 2455–2477 (1975)
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)
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)
Guo, X., Li, Y., Wang, H.: A fourth-order scheme for space fractional diffusion equations. J. Comput. Phys. 373, 410–424 (2018)
**ng, 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)
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)
Luo, H., Zhang, Q.: Regularity of global attractor for the fourth-order reaction–diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 17, 3824–3831 (2012)
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)
Yang, X., Zhang, H., Xu, D.: Orthogonal spline collocation method for the fourth-order diffusion system. Appl. Math. Comput. 75, 3172–3185 (2018)
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)
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)
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)
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)
Zeid, S.S.: Approximation methods for solving fractional equations. Chaos Solitons Fractals 125, 171–193 (2019)
Manimaran, J., Shangerganesh, L., Debbouche, A., Antonov, V.: Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion. Front. Phys. 7, 93 (2019)
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)
Zhong, J., Liao, H., Ji, B., Zhang, L.: A fourth-order compact solver for fractional-in-time fourth-order diffusion equations. ar**v:1907.01708v1 [math.NA] (2019)
Li, X., Wong, P.J.Y.: Numerical solutions of fourth-order fractional sub-diffusion problems via parametric quintic spline (2019). 10.1002/zamm.201800094
Agrawal, O.P.: A general solution for the fourth-order fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 3, 1 (2000)
Agrawal, O.P.: A general solution for a fourth-order fractional diffusion-wave equation defined in a bounded domain. Comput. Struct. 79, 1497 (2001)
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)
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)
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)
Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92, 1496 (2015)
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)
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)
Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 230, 586–595 (2011)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
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)
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)
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
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)
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)
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)
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)
Huang, J., Yang, D.: A unified difference-spectral method for time-space fractional diffusion equations. Int. J. Comput. Math. 94(6), 1172–1184 (2017)
Dison, J., Mekee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)
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)
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)
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)
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)
Thomas, J.W.: Numerical Partial Differential Equations (Finite Difference Methods), Texts in Applied Mathematics, vol. 22. Springer, New York (1995)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
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This research is supported by the National Natural Science Foundation of China (Grant No. 11701502).
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Arshad, S., Wali, M., Huang, J. et al. Numerical framework for the Caputo time-fractional diffusion equation with fourth order derivative in space. J. Appl. Math. Comput. 68, 3295–3316 (2022). https://doi.org/10.1007/s12190-021-01635-5
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DOI: https://doi.org/10.1007/s12190-021-01635-5
Keywords
- Fractional diffusion equation
- Caputo derivative
- Finite difference method
- Fractional trapezoid formula
- Stephenson’s scheme
- Convergence analysis
- Stability analysis