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A time-fractional diffusion equation with space-time dependent hidden-memory variable order: analysis and approximation

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Abstract

A time-fractional partial differential equation with a hidden-memory space-time dependent variable order is studied. The well-posedness and high-order regularity estimates of the problem are proved via the resolvent estimates. Accordingly, an error estimate of the L-1 discretization without any artificial regularity assumption of the true solution is analyzed. Numerical experiments are performed to substantiate the theoretical results and to study the behavior of the variable-order fractional partial differential equations.

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References

  1. Adams, R.A., Fournier, J.J.: Sobolev spaces. Elsevier, NY (2003)

    MATH  Google Scholar 

  2. Akrivis, G., Li, B., Lubich, C.: Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp. 86, 1527–1552 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Baveye, P., Vandevivere, P., Hoyle, B.L., DeLeo, P.C., de Lozada, D.S.: Environmental impact and mechanisms of the biological clogging of saturated soils and aquifer materials. Critic. Rev. Environ. Sci. Technol. 28, 123–191 (2006)

    Google Scholar 

  4. Diethelm, K., Ford, N.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Du, R., Alikhanov, A., Sun, Z.: Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations. Comput. Math. Appl. 79, 2952–2972 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Embrechts, P., Maejima, M.: Selfsimilar Processes Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2002)

    MATH  Google Scholar 

  7. Ervin, V., Heuer, N., Roop, J.: Regularity of the solution to 1-d fractional order diffusion equations. Math. Comp. 87, 2273–2294 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Fu, Z., Reutskiy, S., Sun, H., Ma, J., Khan, M.: A robust kernel-based solver for variable-order time fractional pdes under 2d/3d irregular domains. Appl. Math. Lett. 94, 105–111 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Gandossi, L., Estorff, U.V.: An overview of hydraulic fracturing and other formation stimulation technologies for shale gas production, Scientific and Technical Research Reports. Joint Research Centre of the European Commission; Publications Office of the European Union, (2015). https://doi.org/10.2790/379646

  10. Garrappa, R., Moret, I., Popolizio, M.: On the time-fractional schrödinger equation: theoretical analysis and numerical solution by matrix mittag-leffler functions. Comput. Math. Appl. 74, 977–992 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Glöckle, W., Nonnenmacher, T.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)

    Google Scholar 

  12. Gracia, J., O’Riordan, E., Stynes, M.: Convergence analysis of a finite difference scheme for a two-point boundary value problem with a riemann-liouville-caputo fractional derivative. BIT Numer. Math. (2019). https://doi.org/10.1007/s10543-019-00777-0

    Article  MATH  Google Scholar 

  13. Gunzburger, M., Li, B., Wang, J.: Sharp convergence rates of time discretization for stochastic time-fractional pdes subject to additive space-time white noise. Math. Comp. 88, 1715–1741 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, A3129–A3152 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56, 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Jin, B., Li, B., Zhou, Z.: Subdiffusion with a time-dependent coefficient: analysis and numerical solution. Math. Comp. 88, 2157–2186 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Kian, Y., Soccorsi, E., Yamamoto, M.: On time-fractional diffusion equations with space-dependent variable order. Ann. Henri Poincaré 19, 3855–3881 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Kopteva, N.: Error analysis of the l1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comp. 88, 2135–2155 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Le, K., McLean, W., Stynes, M.: Existence, uniqueness and regularity of the solution of the time-fractional fokker-planck equation with general forcing. Commun. Pure Appl. Anal. 18, 2765–2787 (2019)

    MathSciNet  Google Scholar 

  20. Li, B., Wang, H., Wang, J.: Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order. ESAIM: M2AN (2020). https://doi.org/10.1051/m2an/2020072

    Article  Google Scholar 

  21. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Li, Z., Wang, H., Xiao, R., Yang, S.: A variable-order fractional differential equation model of shape memory polymers. Chaos Solit. Fract. 102, 473–485 (2017)

    MathSciNet  Google Scholar 

  23. Liang, H., Brunner, H.: The convergence of collocation solutions in continuous piecewise polynomial spaces for weakly singular volterra integral equations. SIAM J. Numer. Anal. 57, 1875–1896 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dynam. 29, 57–98 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Lubich, C.: Convolution quadrature and discretized operational calculus. I and II. Numer. Math. 52, 129–145 (1988)

    MathSciNet  MATH  Google Scholar 

  27. Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65, 1–17 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Luchko, Y., Yamamoto, M.: On the maximum principle for a time-fractional diffusion equation. Fract. Calc. Appl. Anal. 20, 1131–1145 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Lv, C., Xu, C.: Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38, A2699–A2724 (2016)

    MathSciNet  MATH  Google Scholar 

  30. McLean, W., Mustapha, K., Ali, R., Knio, O.: Well-posedness of time-fractional advection-diffusion-reaction equations. Fract. Calc. Appl. Anal. 22, 918–944 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. De Gruyter Stud. Math., De Gruyter, Berlin (2011)

  32. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    MathSciNet  MATH  Google Scholar 

  33. Mustapha, K.: Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130, 497–516 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Mustapha, K., McLean, W.: Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation. IMA J. Numer. Anal. 32, 906–925 (2011)

    MathSciNet  MATH  Google Scholar 

  35. Ouhabaz, E.M.: Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123, 1465–1474 (1995)

    MathSciNet  MATH  Google Scholar 

  36. Podlubny, I.: Fractional Differential Equations. Academic Press, NY (1999)

    MATH  Google Scholar 

  37. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math. Anal. Appl. 382, 426–447 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Samko, S.: Fractional integration and differentiation of variable order: an overview. Nonlinear Dynam. 71, 653–662 (2013)

    MathSciNet  MATH  Google Scholar 

  39. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39, 1–12 (2003)

    Google Scholar 

  40. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Sun, H., Chang, A., Zhang, Y., Chen, W.: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22, 27–59 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  43. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  44. Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51, 1088–1107 (2013)

    MathSciNet  MATH  Google Scholar 

  45. Wang, H., Zheng, X.: Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. Appl. 475, 1778–1802 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Wang, X., Xu, S., Zhou, S., Xu, W., Leary, M., Choong, P., Qian, M., Brandt, M., Xie, Y.M.: Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: A review. Biomaterials 83, 127–141 (2016)

    Google Scholar 

  47. Weis, L.: A new approach to maximal \(L_p\)-regularity. In: Herrenalb, Bad (ed.) Evolution Equations And Their Applications In Physical And Life Sciences. Lecture Notes in Pure and Appl. Math., vol. 215, pp. 195–214. Dekker, New York (2001)

    Google Scholar 

  48. Ye, H., Gao, J., Ding, Y.: A generalized gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    MathSciNet  MATH  Google Scholar 

  49. Zeng, F., Zhang, Z., Karniadakis, G.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM J. Sci. Comput. 37, A2710–A2732 (2015)

    MathSciNet  MATH  Google Scholar 

  50. Zheng, X., Wang, H.: An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation. SIAM J. Numer. Anal. 58, 2492–2514 (2020)

    MathSciNet  MATH  Google Scholar 

  51. Zheng, X., Wang, H.: Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/draa013

    Article  Google Scholar 

  52. Zheng, X., Wang, H.: Wellposedness and smoothing properties of history-state based variable-order time-fractional diffusion equations. ZAMP 71, 34 (2020)

    MathSciNet  MATH  Google Scholar 

  53. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. All data generated or analyzed during this study are included in this article.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiangcheng Zheng

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

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  1. Xiangcheng Zheng
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Correspondence to Hong Wang.

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Communicated by Mihaly Kovacs.

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This work was partially funded by the ARO MURI Grant W911NF-15-1-0562 and the National Science Foundation under Grant DMS-2012291.

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Zheng, X., Wang, H. A time-fractional diffusion equation with space-time dependent hidden-memory variable order: analysis and approximation. Bit Numer Math 61, 1453–1481 (2021). https://doi.org/10.1007/s10543-021-00861-4

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