On the Fractional Diffusion-Advection-Reaction Equation in ℝ

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.

Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval

“…Of particular note is that for the fractional diffusion, reaction problem, and the fractional diffusion, advection, reaction problem, the regularity of the solution u is bounded, regardless of the regularity of f . This boundedness in the regularity of u is not the case for the fractional diffusion, advection, reaction equation on R, as was recently established by Ginting and Li in [16].…”

Analysis of one-sided 1-D fractional diffusion operator

“…Now we are ready to establish the weak solution of Equation (3.1) in complexvalued H s 0 (Ω), s = 1+µ 2 . By setting the corresponding sesquilinear form B [•, •] for the operator A in Equation (3.1) and establishing the boundedness and coercivity of B [•, •] with the aid of the second identity of Theorem 4.1, p. 1048, [12], it is not difficult to obtain the following Lemma, of which the proof is omitted.…”

“…In pioneer work [5], via defining intermediate functional spaces in terms of R-L derivatives that are essentially equivalent to the fractional Sobolev spaces, the variational formulation of (1.1) with zero boundary conditions was constructed and the weak solution was successfully established, which suggests the analogy to the classic integer-order elliptic equations. Later on in sequel work [12,18,19,15,8], the regularity of weak solution was thoroughly investigated. In particular, the decomposition of the solution was discovered in [18], consisting of a "smoother" part that is trouble-free and a "less-smooth" part that limits the regularity of the whole solution.…”

Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

“…Another equivalent definition is achieved with the aid of left or right fractionalorder weak derivative, which is a generalization of integer-order weak derivative: Definition 6. ( [19], Section 3) Given 0 ≤ s and assume u(x) ∈ L 2 (R), then u(x) ∈ H s (R) if and only if there exists a unique…”