Abstract
We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.
Acknowledgements
The authors thank the University of New South Wales (Faculty Research Grant “Efficient numerical simulation of anomalous transport phenomena”), the King Fahd University of Petroleum and Minerals (project No. KAUST005) and the King Abdullah University of Science and Technology.
References
[1] E. A. Abdel-Rehim, Implicit difference scheme of the space-time fractional advection diffusion equation. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1452–1469; 10.1515/fca-2015-0084; https://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Search in Google Scholar
[2] A. A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46 (2010), 660–666; 10.1134/S0012266110050058.Search in Google Scholar
[3] A. A. Alikhanov, Boundary value problems for the diffusion equation of the variable order in differential and difference settings. Appl. Math. Comput. 219 (2012), 3938–3946; 10.1016/j.amc.2012.10.029.Search in Google Scholar
[4] C. N. Angstmann, B. I. Henry, B. A. Jacobs, and A. V. McGann, A time-fractional generalised advection equation from a stochastic process. Chaos, Solitons and Fractals102 (2017), 175–183; 10.1016/j.chaos.2017.04.040.Search in Google Scholar
[5] L. C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations. Nonlinear Anal. 74 (2011), 1892–1912; 10.1016/j.na.2010.10.060.Search in Google Scholar
[6] H. Brunner, Volterra Integral Equations: an Introduction to Theory and Applications. Cambridge University Press (2017); 10.1017/9781316162491.Search in Google Scholar
[7] H. Brunner, A. Pedas, and G. Vainikko, The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations. Math. Comp. 68 (1999), 1079–1095; 10.1090/S0025-5718-99-01073-X.Search in Google Scholar
[8] J. Cao, Ch. Li, and Y.-Q. Chen, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (ii). Fract. Calc. Appl. Anal. 18, No 3 (2015), 735–761; 10.1515/fca-2015-0045; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.Search in Google Scholar
[9] J. Dixon and S. McKee, Weakly singular Gronwall inequalities. ZAMM Z. Angew. Math. Mech. 66 (1986), 535–544; 10.1002/zamm.19860661107.Search in Google Scholar
[10] B. I. Henry, T. A. M. Langlands, and P. Straka, Fractional Fokker–Planck equations for subdiffusion with space- and time-dependent forces. Phys. Rev. Lett. 105 (2010), 170602; 10.1103/PhysRevLett.105.170602.Search in Google Scholar PubMed
[11] B. I. Henry, T. A. M. Langlands, and S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E74 (2006), 031116; 10.1103/PhysRevE.74.031116.Search in Google Scholar
[12] B. I. Henry and S. L. Wearne, Fractional reaction-diffusion. Phys. A276 (2000), 448–455; 10.1016/S0378-4371(99)00469-0.Search in Google Scholar
[13] B. Jin, B. Li, and Z. Zhou, Discrete maximal regularity of time-stepping schemes for fractional evolution equations. Numer. Math. 138 (2018), 101–131; 10.1007/s00211-017-0904-8.Search in Google Scholar PubMed PubMed Central
[14] S. Karaa and A. K. Pani, Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data. ESAIM: M2AN52, No 2 (2018), 773–801; 10.1051/m2an/2018029.Search in Google Scholar
[15] J. Kemppainen, Existence and uniqueness of the solution for a time-fractional diffusion equation. Fract. Calc. Appl. Anal. 14, No 3 (2011), 411–417; 10.2478/s13540-011-0025-5; https://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml.Search in Google Scholar
[16] J. Klafter and I. M. Sokolov, First Steps in Random Walks. Oxford University Press (2011); 10.1093/acprof:oso/9780199234868.001.0001.Search in Google Scholar
[17] A. Kubica and M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fract. Calc. Appl. Anal. 21, No 2 (2018), 276–311; 10.1515/fca-2018-0018; https://www.degruyter.com/view/j/fca.2018.21.issue-2/issue-files/fca.2018.21.issue-2.xml.Search in Google Scholar
[18] T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, No 2 (2005), 719–736; 10.1016/j.jcp.2004.11.025.Search in Google Scholar
[19] T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells. SIAM J. Appl. Math. 71, No 4 (2011), 1168–1203; 10.1137/090775920.Search in Google Scholar
[20] Kim Ngan Le, W. McLean, and K. Mustapha, A semidiscrete finite element approximation of a time-fractional Fokker–Planck equation with non-smooth initial data. SIAM J. Sci. Comput. 40, No 6, (2018), A3831–3852; 10.1137/17M1125261.Search in Google Scholar
[21] Kim Ngan Le, W. McLean, and M. Stynes, Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Commun. Pure Appl. Anal. (to appear); 10.13140/RG.2.2.30053.24801.Search in Google Scholar
[22] Y. Li and Q. Zhang, Blow-up and global existence of solutions for a time fractional diffusion equation. Fract. Calc. Appl. Anal. 21, No 6 (2018), 1619–1640; 10.1515/fca-2018-0085; https://www.degruyter.com/view/j/fca.2018.21.issue-6/issue-files/fca.2018.21.issue-6.xml.Search in Google Scholar
[23] H.-L. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform l1 formula for linear reaction-subdiffusion equations. SIAM J. Numer. Anal. 66, No 2 (2018), 1112–1133; 10.1137/17M1131829.Search in Google Scholar
[24] P. Linz, Analytical and Numerical Methods for Volterra Equations. Ser. Studies in Applied and Numerical Mathematics, SIAM, Philadelphia (1985); 10.1137/1.9781611970852.Search in Google Scholar
[25] F. Liu, V. V. Anh, I. Turner, and P. Zhuang, Time fractional advection-dispersion equation. J. Appl. Math. Computing13 (2003), 233–245; 10.1007/BF02936089.Search in Google Scholar
[26] Ch. Lubich, Runge–Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comp. 41 (1983), 87–102; 10.1090/S0025-5718-1983-0701626-6.Search in Google Scholar
[27] Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems. Fract. Calc. Appl. Anal. 19, No 3 (2016), 676–695; 10.1515/fca-2016-0036; https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml.Search in Google Scholar
[28] W. McLean, Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52 (2010), 123–138; 10.1017/S1446181111000617.Search in Google Scholar
[29] W. McLean, K. Mustapha, R. Ali, and O. M. Knio, Regularity theory for time–fractional advection–diffusion–reaction equations. Comput. Math. Appl. (2019), (In press); 10.1016/j.camwa.2019.08.008.Search in Google Scholar
[30] R. Metzler, E. Barkai, and J. Klafter, Deriving fractional Fokker–Planck equations from a generalised master equation. Europhys. Lett. 46 (1999), 431–436; 10.1209/epl/i1999-00279-7.Search in Google Scholar
[31] K. Mustapha, Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130, No 3 (2015), 497–516; 10.1007/s00211-014-0669-2.Search in Google Scholar
[32] J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations. Adv. Math. 22 (1976), 278–304; 10.1016/0001-8708(76)90096-7.Search in Google Scholar
[33] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
[34] 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, No 1 (2011), 426–447; 10.1016/j.jmaa.2011.04.058.Search in Google Scholar
[35] C.-S. Sin and L. Zheng, Existence and uniqueness of global solutions of Caputo-type fractional differential equations. Fract. Calc. Appl. Anal. 19, No 3 (2016), 765–774. 10.1515/fca-2016-0040; https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml.Search in Google Scholar
[36] M. Stynes, E. O’Riordan, and J. Luis Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, No 2 (2017), 1057–1079; 10.1137/16M1082329.Search in Google Scholar
[37] G. Vainikko, Weakly Singular Integral Equations. Lecture Notes, University of Tartu, Helsinki University of Technology (2006–2007).Search in Google Scholar
[38] S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann stability analysis for fractional diffusion equations. SIAM. J. Numer. Anal. 42, No 5 (2005), 1862–1874; 10.1137/030602666.Search in Google Scholar
[39] R. Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcial. Ekvac. 52 (2009), 1–18; 10.1619/fesi.52.1.Search in Google Scholar
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