On the fractional diffusion-advection-reaction equation in ℝ
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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.
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© 2019 Diogenes Co., Sofia
Artikel in diesem Heft
- 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
Artikel in diesem Heft
- 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
