On the fractional diffusion-advection-reaction equation in ℝ
-
Victor Ginting
Abstract
We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses.
Appendix A. Several pertinent theorems
Theorem A.1
(Plancherel Theorem (eg. [18] p. 187)). Givenw ∈ L2(ℝ), there is a uniqueŵ ∈ L2(ℝ) so that the following properties hold: 1) ifw ∈ L1(ℝ) ∩ L2(ℝ), thenŵ = 𝓕(w), 2) for everyw ∈ L2(ℝ), ∥w∥2 = ∥ŵ∥2, 3) the mappingw → ŵis a Hilbert space isomorphism ofL2(ℝ) ontoL2(ℝ).
Theorem A.2
([8], p. 189). Givenv, w ∈ L2(ℝ), then (v, w) = (v̂, ŵ). andw = (ŵ)∨.
Theorem A.3
([9], p. 204). Ifv ∈ L2(ℝ), w ∈ L1(ℝ), then
Theorem A.4
([23], p. 191). Let (X, (⋅, ⋅)) be a Hilbert space and letYbe a subspace ofX. Y = Xif and only if 0 ∈ Xis the only one satisfying (x, y) = 0 for ally ∈ Y.
Theorem A.5
([4], p. 107). Letv ∈
Acknowledgements
Yulong Li was partially supported by the UW Science Initiative Scholarship.
References
[1] R. A. Adams and J. F. Fournier, Sobolev Spaces. Academic Press (2003).Search in Google Scholar
[2] B. Baeumer, M. Kovács, M. Meerschaert, and H. Sankaranarayanan, Boundary conditions for fractional diffusion. J. Comput. Appl. Math. 336 (2018), 408–424.10.1016/j.cam.2017.12.053Search in Google Scholar
[3] D. A. Benson, M. Meerschaert, and J. Revielle, Fractional calculus in hydrologic modeling: A numerical perspective. Adv. Water Resour. 51 (2013), 479–497.10.1016/j.advwatres.2012.04.005Search in Google Scholar PubMed PubMed Central
[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer (2011).10.1007/978-0-387-70914-7Search in Google Scholar
[5] O. Defterli, M. D’Elia, Q. Du, M. Gunzburger, R. Lehoucq, and M. Meerschaert, Fractional diffusion on bounded domains. Fract. Calc. Appl. Anal. 18, No 2 (2015), 342–360; 10.1515/fca-2015-0023; https://www.degruyter.com/view/j/fca.2015.18.issue-2/issue-files/fca.2015.18.issue-2.xml.Search in Google Scholar
[6] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.10.1016/j.bulsci.2011.12.004Search in Google Scholar
[7] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22 (2006), 558–576.10.1002/num.20112Search in Google Scholar
[8] L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics # 19, Amer. Math. Soc. (2010).10.1090/gsm/019Search in Google Scholar
[9] C. Gasquet and P. Witomski, Fourier Analysis and Applications. Texts in Applied Mathematics # 30, Springer-Verlag (1999). Filtering, Numerical Computation, Wavelets, Translated by R. Ryan.Search in Google Scholar
[10] A. Goulart, M. Lazo, J. Suarez, and D. Moreira, Fractional derivative models for atmospheric dispersion of pollutants. Physica A: Statistical Mechanics and its Applications477 (2017), 9–19.10.1016/j.physa.2017.02.022Search in Google Scholar
[11] F. Izsák and B. J. Szekeres, Models of space-fractional diffusion: A critical review. Applied Mathematics Letters71 (2017), 38–43.10.1016/j.aml.2017.03.006Search in Google Scholar
[12] M. Japundžić and D. Rajter-Ćirić, Reaction-advection-diffusion equations with space fractional derivatives and variable coefficients on infinite domain. Fract. Calc. Appl. Anal. 18, No 4 (2015), 911–950; 10.1515/fca-2015-0055; https://www.degruyter.com/view/j/fca.2015.18.issue-4/issue-files/fca.2015.18.issue-4.xml.Search in Google Scholar
[13] B. Jin, R. D. Lazarov, J. E. Pasciak, and W. Rundell, Variational formulation of problems involving fractional order differential operators. Math. Comput. 84 (2015), 2665–2700.10.1090/mcom/2960Search in Google Scholar
[14] Y. Li, On Fractional Differential Equations and Related Questions. Ph.D. Dissertation, University of Wyoming (2019).Search in Google Scholar
[15] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies # 204, Elsevier Science B.V., Amsterdam (2006).Search in Google Scholar
[16] F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. Springer-Vienna (1997), 291–348.10.1007/978-3-7091-2664-6_7Search in Google Scholar
[17] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar
[18] W. Rudin, Real and Complex Analysis. McGraw-Hill Book Co. (1987).Search in Google Scholar
[19] W. Rudin, Functional Analysis. International Ser. in Pure and Applied Mathematics, McGraw-Hill, Inc. (1991).Search in Google Scholar
[20] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science Publishers (1993).Search in Google Scholar
[21] E. Scalas, R. Gorenflo, and F. Mainardi, Fractional calculus and continuous-time finance. Physica A: Statistical Mechanics and its Applications284 (2000), 376–384.10.1016/S0378-4371(00)00255-7Search in Google Scholar
[22] R. Stern, F. Effenberger, H. Fichtner, and T. Schäfer, The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions. Fract. Calc. Appl. Anal. 17, No 1 (2014), 171–190; 10.2478/s13540-014-0161-9; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Search in Google Scholar
[23] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana # 3, Springer (2007).Search in Google Scholar
[24] H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51 (2013), 1088–1107.10.1137/120892295Search in Google Scholar
[25] H. Wang, D. Yang, and S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 52 (2014), 1292–1310.10.1137/130932776Search in Google Scholar
[26] H. Wang, D. Yang, and S. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations. J. Sci. Comput. 70 (2017), 429–449.10.1007/s10915-016-0196-7Search in Google Scholar
[27] D. Werner, Funktionalanalysis. Springer-Verlag (2011).10.1007/978-3-642-21017-4Search in Google Scholar
[28] S. W. Wheatcraft and M. Meerschaert, Fractional conservation of mass. Adv. Water Resour. 31 (2008), 1377–1381.10.1016/j.advwatres.2008.07.004Search in Google Scholar
[29] Y. Zhou, J. Wang, and L. Zhang, Basic Theory of Fractional Differential Equations. World Scientific Publishing Co. Pte. Ltd. (2017).10.1142/10238Search in Google Scholar
© 2019 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 22–4–2019)
- Research Paper
- Fractional equations via convergence of forms
- About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
- Survey Paper
- The vertical slice transform on the unit sphere
- Research Paper
- Well-posedness of time-fractional advection-diffusion-reaction equations
- Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
- Exact asymptotic formulas for the heat kernels of space and time-fractional equations
- Estimates of damped fractional wave equations
- A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis
- On the fractional diffusion-advection-reaction equation in ℝ
- Controllability of nonlinear stochastic fractional higher order dynamical systems
- Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2
- Analysis of fractional order error models in adaptive systems: Mixed order cases
- Fractional calculus of variations: a novel way to look at it
- Pricing of perpetual American put option with sub-mixed fractional Brownian motion
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 22–4–2019)
- Research Paper
- Fractional equations via convergence of forms
- About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
- Survey Paper
- The vertical slice transform on the unit sphere
- Research Paper
- Well-posedness of time-fractional advection-diffusion-reaction equations
- Existence of solutions for nonlinear fractional order p-Laplacian differential equations via critical point theory
- Exact asymptotic formulas for the heat kernels of space and time-fractional equations
- Estimates of damped fractional wave equations
- A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis
- On the fractional diffusion-advection-reaction equation in ℝ
- Controllability of nonlinear stochastic fractional higher order dynamical systems
- Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2
- Analysis of fractional order error models in adaptive systems: Mixed order cases
- Fractional calculus of variations: a novel way to look at it
- Pricing of perpetual American put option with sub-mixed fractional Brownian motion
