Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 319 -- 340

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This work is licensed under a Creative Commons Attribution 4.0 International License.

A NOVEL FINITE ELEMENT METHOD WITH ADAPTIVE MESH REFINEMENT FOR NONLINEAR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Bhavyata Patel, Gargi J Trivedi and Trupti P Shah

Abstract. Nonlinear fractional order differential equations (FODEs) model complex phenomena like anomalous diffusion and nonlinear advection, posing computational challenges due to fractional derivatives and nonlinearities. We propose a novel Galerkin finite element method (FEM) that uniquely integrates the L1 scheme with fast convolution (reducing complexity to O(Nt \log Nt) via FFT-based sum-of-exponentials approximation, achieving O(Δ t2-α) accuracy under the assumption that the solution u(t) has sufficient regularity, adaptive mesh refinement (AMR) for spatial accuracy, and adaptive time-stepping for temporal efficiency, addressing nonlinear time-fractional diffusion and Burgers’ equations. The method assumes bounded solutions in L∞(Ω) for Lipschitz continuity of nonlinear terms. Sensitivity analysis via Sobol indices quantifies the impact of fractional order, mesh size, and time step. Extensive numerical experiments, including diverse benchmark problems, demonstrate L2 errors that are up to 50% lower than those of finite difference methods and competitive performance against spectral methods, which may exhibit instability for nonlinear fractional Burgers’ equations . Detailed mathematical derivations and MATLAB-based implementations illustrate the method’s robustness for nonlinear fractional diffusion and Burgers’ equations, with applications to anomalous transport in porous media.

2020 Mathematics Subject Classification: 65M60, 35R11, 65M15, 65N30, 34A08.
Keywords: fractional differential equations, finite element method, adaptive mesh refinement, L1 scheme, fast convolution, sensitivity analysis, nonlinear diffusion, fractional Burgers’ equation.

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References

  1. R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2nd ed., Academic Press, Amsterdam, 2003, pp. 1--320. MR2424078. Zbl 1098.46001.

  2. E. Adel, I. L. El-Kalla, A. Elsaid, M. Sameeh, An adaptive finite element method for Riesz fractional partial integro-differential equations, Mathematical Sciences 18 (2023), 611--624.

  3. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, Germany, 2004. MR2680847. Zbl 1215.34001.

  4. L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, Providence, RI, 2010, pp. 1--662. MR2597943. Zbl 1194.35001.

  5. A. K. Grünwald, Über “bounded” derivations and their application, Zeitschrift für Mathematik und Physik, 12 (1867), 441--480.

  6. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006. MR2218073. Zbl 1092.45003.

  7. L. Kovács, D. A. Drexler, Modeling the physiological phenomena and the effects of therapy on the dynamics of tumor growth, in Modeling of Mass Transport Processes in Biological Media, Academic Press, 2022, pp.~391--403. Zbl 1380.92028.

  8. S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, ACM Transactions on Mathematical Software 43 (2017), no.~4, 1--23.

  9. B. Jin, R. Lazarov, J. Pasciak, Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA Journal of Numerical Analysis, 35(2) (2015), 561--582. MR3335216. Zbl 1321.65142.

  10. M. Li, L. Ge, Adaptive space-time finite element method for time fractional-order reaction-diffusion equations, Journal of Applied Mathematics and Computing 71 (2025), 3153--3178. DOI: 10.1007/s12190-025-02366-7.

  11. Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics 225 (2007), no.~2, 1533--1552. DOI: 10.1016/j.jcp.2007.02.001. MR2349193. Zbl 1126.65121.

  12. Ch. Lubich, A. Schädle, Fast convolution for nonreflecting boundary conditions, SIAM Journal on Scientific Computing 24(1) (2002), 161--179. MR1924419. Zbl 1013.65113.

  13. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. MR2676137.

  14. G. Malviya, J. Sharma, V. Shah, G. Trivedi, Mathematical modelling and controllability analysis of fractional order coal mill pulverizer model, Advances in Differential Equations and Control Processes 32(1) (2025), 1--13. DOI: 10.59400/adecp2075.

  15. R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports 339 (2000), no.~1, 1--77. DOI: 10.1016/S0370-1573(00)00070-3. MR1809268. Zbl 0984.82032.

  16. K. Nedaiasl, R. Dehbozorgi, Galerkin finite element method for nonlinear fractional differential equations, Numerical Algorithms 88 (2021), 113--141. DOI: 10.1007/s11075-020-01032-2. MR4297915. Zbl 1491.65068.

  17. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. MR1658022. Zbl 0924.34008.

  18. A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola, Global Sensitivity Analysis: The Primer, Wiley, Chichester, UK, 2008. MR2382923. Zbl 1161.00304.

  19. T. Sharma, S. Pathak, G. Trivedi, R. C. Sanghvi, Flow modelling in porous medium applying numerical techniques: A comparative analysis, Recent Research Reviews Journal 2(2) (2023), 288--304. DOI: 10.36548/rrrj.2023.2.004.

  20. T. Sharma, S. Pathak, G. Trivedi, Comparative study of Crank–Nicolson and modified Crank–Nicolson numerical methods to solve linear partial differential equations, Indian Journal of Science and Technology 17(10) (2023), 924--931. DOI: 10.17485/IJST/v17i10.1776.

  21. T. Sharma, S. Pathak, G. Trivedi, Numerical modelling through porous media: Modified Crank–Nicolson approach to Burger’s equation, Journal of Advanced Zoology 44(S8) (2023), 363--371. DOI: 10.53555/jaz.v44iS8.4098.

  22. T. Sharma, G. Trivedi, V. Shah, S. Pathak, Numerical solution of one-dimensional dispersion equation in homogeneous porous medium by modified finite element method, Journal of Computational and Engineering Mathematics 11(3) (2024), 16--27. DOI: 10.14529/jcem240302.

  23. I. M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates, Mathematics and Computers in Simulation 55 (2001), no.~1–3, 271--280. DOI: 10.1016/S0378-4754(00)00270-6. MR1823119. Zbl 1005.65004.

  24. T. Q. Tang, Z. Shah, R. Jan, E. Alzahrani, Modeling the dynamics of tumor-immune cells interactions via fractional calculus, The European Physical Journal Plus 137 (2022), no.~3, 367.

  25. G. Trivedi, MDWT: A Modified Discrete Wavelet Transformation-Based Algorithm for Image Fusion, in: Beyond Signals – Exploring Revolutionary Fourier Transform Applications, IntechOpen, 2024, Chapter 6, pp. 1--20. ISBN: 978-1-83634-241-0. DOI: 10.5772/intechopen.1006477.

  26. G. Trivedi, MNSCT – A Novel Modified NSCT-Based Algorithm for Enhanced Medical Image Fusion, Medical Imaging Process & Technology 8(1) (February 2025), 1--15. ISSN: 2578-160X. DOI: 10.24294/mipt10655.

  27. G. Trivedi, Mathematical modeling and numerical analysis of diffusion processes in image processing, Journal of Applied Mathematics 3(2) (2025), 1--15. DOI: 10.59400/jam2736.

  28. G. J. Trivedi, R. C. Sanghvi, A new approach for multimodal medical image fusion using PDE-based technique, Suranaree Journal of Science and Technology 30(4) (2023), Article ID 030132, 1--7. DOI: 10.55766/sujst-2023-04-e0843.

  29. G. Trivedi, R. C. Sanghvi, Mosaicfusion: Merging modalities with partial differential equation and discrete cosine transformation, Journal of Applied and Pure Mathematics 5(5–6) (2023), 389--406. DOI: 10.23091/japm.2023.389.

  30. G. Trivedi, R. C. Sanghvi, Novel algorithm for multifocus image fusion: Integration of convolutional neural network and partial differential equation, Surveys in Mathematics and its Applications 19 (2024), 179--195. MR4739097. Zbl 7890974.

  31. G. Trivedi, V. Shah, J. Sharma, R. C. Sanghvi, On solution of non-instantaneous impulsive Hilfer fractional integro-differential evolution system, Mathematica Applicanda 51(1) (2023), 33--50. DOI: 10.14708/ma.v51i1.7167. MR4636519. Zbl 1549.34220.

  32. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley–Teubner, Stuttgart, 1996, pp. 1--256.

  33. X. Zhao, X. Hu, W. Cai, G. E. Karniadakis, Adaptive finite element method for fractional differential equations using hierarchical matrices, Computer Methods in Applied Mechanics and Engineering 325 (2017), 56--76. DOI: 10.1016/j.cma.2017.06.017. MR3693419. Zbl 1439.65091.

  34. M. Zayernouri, G. E. Karniadakis, Fractional spectral collocation method, SIAM Journal on Scientific Computing 36(1) (2014), A40--A62. MR3150177. Zbl 1294.65097.




Bhavyata Patel
Department of Applied Mathematics,
Faculty of Technology and Engineering,
The Maharaja Sayajirao University of Baroda, Vadodara, India.
e-mail: bhavyatapatel11@gmail.com


Dr. Gargi Trivedi - Corresponding author
Department of Applied Mathematics,
Faculty of Technology and Engineering,
The Maharaja Sayajirao University of Baroda, Vadodara, India.
e-mail: gargi1488@gmail.com, gargi.t-appmath@msubaroda.ac.in


Dr. Trupti Shah
Department of Applied Mathematics,
Faculty of Technology and Engineering,
The Maharaja Sayajirao University of Baroda, Vadodara, India.
e-mail: trupti.p.shah-appmath@msubaroda.ac.in

https://www.utgjiu.ro/math/sma