Trigonometric cubic B-spline collocation method for time-fractional diffusion equation

Kammappa, Zainab; Awasthi, Ashish

doi:10.22034/cmde.2025.66393.3103

Trigonometric cubic B-spline collocation method for time-fractional diffusion equation

Document Type : Research Paper

Authors

Department of Mathematics, National Institute of Technology Calicut, 673601, Kerala, India.

10.22034/cmde.2025.66393.3103

Abstract

This paper presents a numerical approach based on Cubic Trigonometric B-spline (CuTBS) interpolation for solving Time-Fractional Diffusion Equations (TFDEs) involving the Caputo-Fabrizio fractional time derivative. The CuTBS-based scheme effectively combines accurate spatial interpolations with a robust finite-difference discretization for the fractional derivative, ensuring high precision in both temporal and spatial domains. The method is unconditionally stable and demonstrates second-order convergence in time and space. Numerical experiments are conducted to validate the applicability and feasibility of the technique.

Keywords

Main Subjects

Fractional partial differential equations

References

[1] A. Alshabanat, M. Jleli, S. Kumar, and B. Samet, Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits, Frontiers in Physics, 8 (2020), 64.
[2] M. Amin, M. Abbas, M. K. Iqbal, and D. Baleanu, Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations, Advances in Difference Equations, 2019 (2019), 1–22.
[3] A. Atangana and R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional deriv-ative and application to groundwater pollution equation, Advances in Difference Equations, 2016 (2016), 1–13.
[4] B. Baeumer, S. Kurita, and M. M. Meerschaert, Inhomogeneous fractional diffusion equations, Fractional Calculus and Applied Analysis, 8(4) (2005), 371–386.
[5] A. Benzahi, N. Abada, N. Arar, S. A. Idris, M. S. Abdo, and W. Shatanawi, Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results, Alexandria Engineering Journal, 87 (2024), 186–200.
[6] W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary Differential Equations, John Wiley & Sons, USA, 2017.
[7] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation & Applications, 1(2) (2015), 73–85.
[8] M. Cui, A high-order compact exponential scheme for the fractional convection–diffusion equation, Journal of Computational and Applied Mathematics, 255 (2014), 404–416.
[9] M. Du, Z. Wang, and H. Hu, Measuring memory with the order of fractional derivative, Scientific reports, 3 (2013), 3431.
[10] S. Esmaeili and R. Garrappa, A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, International Journal of Computer Mathematics, 92(5) (2015), 980–994.
[11] R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dynamics, 29 (2002), 129–143.
[12] L. He, Y. Zhu, and Z. Lu, A cubic spline interpolation based numerical method for fractional differential equations, Journal of Industrial and Management Optimization, 20(8) (2024), 2770–2794.
[13] J. Hristov, Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, Frontiers in Fractional Calculus, Bentham Science Publishers, (2018), 269–341.
[14] N. Khalid, M. Abbas, and M. K. Iqbal, Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms, Applied Mathematics and Computation, 349 (2019), 393–407.
[15] C. Lakshmi and A. Awasthi, Numerical simulation of Burgers’ equation using cubic B-splines, Nonlinear Engi-neering, 6(1) (2017), 61–77.
[16] C. Lakshmi and A. Awasthi, Robust numerical scheme for nonlinear modified Burgers equation, International Journal of Computer Mathematics, 95(9) (2018), 1910–1926.
[17] H. Liu, A. Cheng, H. Yan, Z. Liu, and H. Wang, A fast compact finite difference method for quasilinear time frac-tional parabolic equation without singular kernel, International Journal of Computer Mathematics, 96(7) (2019), 1444–1460.
[18] H. Liu, X. C. Zheng, and H. F. Fu, Analysis of a multi-term variable-order time-fractional diffusion equation and its Galerkin finite element approximation, Journal of Computational Mathematics, 40 (2022), 814–834.
[19] F. Mainardi, P. Paradisi, and R. Gorenflo, Probability distributions generated by fractional diffusion equations, arXiv preprint arXiv:0704.0320, (2007).
[20] M. M. Meerschaert, Fractional calculus, anomalous diffusion, and probability, Fractional Dynamics: Recent Ad-vances, World Scientific, (2012), 265–284.
[21] R. Metzler, W. G. Gl¨ockle, and T. F. Nonnenmacher, Fractional model equation for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 211(1) (1994), 13–24.
[22] I. A. Mirza and D. Vieru, Fundamental solutions to advection–diffusion equation with time-fractional Ca-puto–Fabrizio derivative, Computers & Mathematics with Applications, 73(1) (2017), 1–10.
[23] S. T. Mohyud-Din, T. Akram, M. Abbas, A. I. Ismail, and N. H. M. Ali, A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation, Advances in Difference Equations, 2018 (2018), 1–17.
[24] K. Mustapha, B. Abdallah, and K. M. Furati, A discontinuous Petrov–Galerkin method for time-fractional diffu-sion equations, SIAM Journal on Numerical Analysis, 52(5) (2014), 2512–2529.
[25] K. Oldham and J. Spanier, Fractional Calculus, Academic Press, London/New York, 1973.
[26] M. Onitsuka and I. I. El-Fassi, Generalized Caputo–Fabrizio fractional differential equation, Journal of Applied Analysis and Computation, 14(2) (2024), 964–975.
[27] R. Pakhira, U. Ghosh, and S. Sarkar, Study of memory effects in an inventory model using fractional calculus, Applied Mathematical Sciences, 12(17) (2018), 797–824.
[28] P. Pandey, J. F. G´omez-Aguilar, M. K. A. Kaabar, Z. Siri, and A. M. Abd Allah, Mathematical modeling of COVID-19 pandemic in India using Caputo–Fabrizio fractional derivative, Computers in Biology and Medicine, 145 (2022), 105518.
[29] S. Poojitha and A. Awasthi, Eloquent numerical approach for solving generalized time fractional convection-diffusion-reaction problems, Physica Scripta, 99(12) (2024), 125277.
[30] S. Poojitha and A. Awasthi, Operational matrix based numerical scheme for the solution of time fractional diffusion equations, Fractional Calculus and Applied Analysis, 27(2) (2024), 877–895.
[31] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, Netherlands, 1998.
[32] P. Roul, A fourth order numerical method based on B-spline functions for pricing Asian options, Computers & Mathematics with Applications, 80(3) (2020), 504–521.
[33] P. Roul and V. P. Goura, A high order numerical scheme for solving a class of non-homogeneous time-fractional reaction diffusion equation, Numerical Methods for Partial Differential Equations, 37(2) (2021), 1506–1534.
[34] P. Roul and T. Kumari, A quartic trigonometric B-spline collocation method for a general class of nonlinear singular boundary value problems, Journal of Mathematical Chemistry, 0 (2022), 1–17.
[35] P. Roul and V. P. Goura, A sixth order optimal B-spline collocation method for solving Bratu’s problem, Journal of Mathematical Chemistry, 58 (2020), 967–988.
[36] P. Roul, V. M. K. P. Goura, and R. Agarwal, A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation, Journal of Mathematical Chemistry, 61(10) (2023), 2146–2175.
[37] P. Roul, V. M. K. P. Goura, and R. Cavoretto, A numerical technique based on B-spline for a class of time-fractional diffusion equation, Numerical Methods for Partial Differential Equations, 39(1) (2023), 45–64.
[38] K. M. Saad, New fractional derivative with non-singular kernel for deriving Legendre spectral collocation method, Alexandria Engineering Journal, 59(4) (2020), 1909–1917.
[39] M. Shafiq, F. A. Abdullah, M. Abbas, A. S. Alzaidi, and M. B. Riaz, Memory effect analysis using piecewise cubic B-spline of time fractional diffusion equation, Fractals, 30(8) (2022), 2240270.
[40] A. Shaikh, A. Tassaddiq, K. S. Nisar, and D. Baleanu, Analysis of differential equations involving Caputo–Fabrizio fractional operator and its applications to reaction–diffusion equations, Advances in Difference Equations, 2019(1) (2019), 1–14.
[41] R. Singh, J. Mishra, and V. K. Gupta, The dynamical analysis of a tumor growth model under the effect of fractal fractional Caputo–Fabrizio derivative, International Journal of Mathematics and Computer in Engineering, 2023.
[42] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, USA, 2004.
[43] N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, Crank-Nicolson finite difference method for solving time-fractional diffusion equation, Journal of Fractional Calculus and Applications, 2(2) (2012), 1–9.
[44] H. Sun, W. Chen, and K. Sze, A semi-discrete finite element method for a class of time-fractional diffusion equations, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371(1990) (2013), 20120268.
[45] A. Taghavi, A. Babaei, and A. Mohammadpour, On the stable implicit finite differences approximation of diffusion equation with the time fractional derivative without singular kernel, Asian-European Journal of Mathematics, 13(6) (2020), 2050111.
[46] N. H. Tuan, L. V. C. Hoan, and S. Tatar, An inverse problem for an inhomogeneous time-fractional diffusion equation: a regularization method and error estimate, Computational and Applied Mathematics, 38 (2019), 1–22.
[47] M. Yaseen, M. Abbas, A. I. Ismail, and T. Nazir, A cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Applied Mathematics and Computation, 293 (2017), 311–319.
[48] M. Yaseen, M. Abbas, T. Nazir, and D. Baleanu, A finite difference scheme based on cubic trigonometric B-splines for a time-fractional diffusion-wave equation, Advances in Difference Equations, 2017 (2017), 1–18.
[49] M. Yaseen and M. Abbas, An efficient computational technique based on cubic trigonometric B-splines for time fractional Burgers’ equation, International Journal of Computer Mathematics, 97(3) (2020), 725–738.
[50] S. M. Zin, A. A. Majid, A. I. M. Ismail, and M. Abbas, Cubic trigonometric B-spline approach to numerical solution of wave equation, International Journal of Mathematical, Computational, Physical and Quantum Engineering, 8 (2014), 1212–1216.
[51] P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, Journal of Applied Mathematics and Computing, 22 (2006), 87–99.