A Second-Order Difference Scheme for Generalized Time-Fractional Diffusion Equation with Smooth Solutions

References

[1] O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal. 15 (2012), no. 4, 700–711. 10.2478/s13540-012-0047-7Search in Google Scholar

[2] A. A. Alikhanov, A priori estimates for solutions of boundary value problems for equations of fractional order, Differ. Equ. 46 (2010), 660–666. 10.1134/S0012266110050058Search in Google Scholar

[3] 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

[4] 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. 10.1016/j.amc.2015.06.045Search in Google Scholar

[5] A. A. Alikhanov, A difference method for solving the Steklov nonlocal boundary value problem of second kind for the time-fractional diffusion equation, Comput. Methods Appl. Math. 17 (2017), no. 1, 1–16. 10.1515/cmam-2016-0030Search in Google Scholar

[6] A. A. Alikhanov, A time-fractional diffusion equation with generalized memory kernel in differential and difference settings with smooth solutions, Comput. Methods Appl. Math. 17 (2017), no. 4, 647–660. 10.1515/cmam-2017-0035Search in Google Scholar

[7] A. A. Alikhanov and C. Huang, A high-order L2 type difference scheme for the time-fractional diffusion equation, Appl. Math. Comput. 411 (2021), Paper No. 126545. 10.1016/j.amc.2021.126545Search in Google Scholar

[8] A. A. Alikhanov and C. Huang, A class of time-fractional diffusion equations with generalized fractional derivatives, J. Comput. Appl. Math. 414 (2022), Paper No. 114424. 10.1016/j.cam.2022.114424Search in Google Scholar

[9] R. Du, W. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model. 34 (2010), no. 10, 2998–3007. 10.1016/j.apm.2010.01.008Search in Google Scholar

[10] G.-H. Gao, A. A. Alikhanov and Z.-Z. Sun, The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations, J. Sci. Comput. 73 (2017), no. 1, 93–121. 10.1007/s10915-017-0407-xSearch in Google Scholar

[11] G.-H. Gao and Z.-Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys. 230 (2011), no. 3, 586–595. 10.1016/j.jcp.2010.10.007Search in Google Scholar

[12] J. Hadamard, Essai sur l’étude des fonctions données par leur développement de Taylor, J. Math. Pures. Appl. 8 (1892), 101–186. Search in Google Scholar

[13] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/3779Search in Google Scholar

[14] B. Jin, R. Lazarov, D. Sheen and Z. Zhou, Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal. 19 (2016), no. 1, 69–93. 10.1515/fca-2016-0005Search in Google Scholar

[15] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36 (2016), no. 1, 197–221. 10.1093/imanum/dru063Search in Google Scholar

[16] N. Kedia, A. A. Alikhanov and V. K. Singh, Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel, Appl. Numer. Math. 172 (2022), 546–565. 10.1016/j.apnum.2021.11.006Search in Google Scholar

[17] A. K. Khibiev, Stability and convergence of difference schemes for the multi-term time-fractional diffusion equation with generalized memory kernels, J. Samara State Tech. Univ., Ser. Phys. Math. Sci. 23 (2019), 582–597. 10.14498/vsgtu1690Search in Google Scholar

[18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[19] Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal. 15 (2012), no. 1, 141–160. 10.2478/s13540-012-0010-7Search in Google Scholar

[20] Y. Luchko and J. J. Trujillo, Caputo-type modification of the Erdélyi–Kober fractional derivative, Fract. Calc. Appl. Anal. 10 (2007), no. 3, 249–267. Search in Google Scholar

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

[22] K. B. Oldham and J. Spanier, The Fractional Calculus, Math. Sci. Eng. 111, Academic Press, New York, 1974. Search in Google Scholar

[23] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[24] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[25] A. A. Samarskii, The Theory of Difference Schemes, Monogr. Textb. Pure Appl. Math. 240, Marcel Dekker, New York, 2001. 10.1201/9780203908518Search in Google Scholar

[26] T. Sandev, A. Chechkin, H. Kantz and R. Metzler, Diffusion and Fokker–Planck–Smoluchowski equations with generalized memory kernel, Fract. Calc. Appl. Anal. 18 (2015), no. 4, 1006–1038. 10.1515/fca-2015-0059Search in Google Scholar

[27] M. Stynes, E. O’Riordan and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079. 10.1137/16M1082329Search in Google Scholar

[28] F. I. Taukenova and M. K. Shkhanukov-Lafishev, Difference methods for solving boundary value problems for fractional-order differential equations, Comput. Math. Math. Phys. 46 (2006), 1785–1795. 10.1134/S0965542506100149Search in Google Scholar