A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus
Summary (2 min read)
1 Introduction
- Properties and applications of ∗The final version is published in Mathematical Methods in the Applied Science, in press.
- Additionally, in [4] the authors also investigated some interesting connections between the Teodorescu operator and Hermitian regular functions.
- Actually, in classical three-dimensional vector analysis it is nothing else than the decomposition of an arbitrary sufficiently regular vector field into the sum of a divergence free field (having a vector potential) and a curl free vector field (having a scalar potential).
- In Section 3, the authors present the fundamental solutions of the fractional Laplace and Dirac operators in Rn, defined by left Riemann-Liouville and Caputo fractional derivatives.
3 Fundamental solutions revisited
- In [10] and [12] the authors considered the so-called three-parameter fractional Laplace and Dirac operators defined in terms of the left Riemann-Liouville and Caputo fractional derivatives, and obtained families of eigenfunctions and fundamental solutions for both operators.
- For the fractional Dirac operator CDαa+ the authors can obtain a family of fundamental solutions by applying the operator CDαa+ to the family of fundamental solutions of the operator C∆αa+ .
- Now the authors present the corresponding results for the Riemann-Liouville case.
4 Fractional Teodorescu and Cauchy-Bitsadze operators
- In this section the authors introduce and study the main properties of the fractional analogues of the classical Teodorescu and Cauchy-Bitsadze operators described in [17].
- Suppose that f and g satisfy the above mentioned conditions, also known as Proof.
- Before the authors deduce two properties of the fractional integral operators (59) and (60), they need to understand the behaviour of their fractional derivatives when the argument of the function over which they apply the derivatives is only translated.
- Concerning (63) the deduction is even more direct because the operator C a+k ∂ 1+αk 2 xk only acts on the functions g0 and g1.
5 Hodge-type decomposition
- The aim of this section is to obtain a Hodge-type decomposition and to present an immediate application of this decomposition for the resolution of boundary value problems involving the fractional Laplace operator.
- To realize this the authors need first the following lemma.
- Hence, the intersection of these subspaces only contains the zero function, which implies that the sum is direct.
- Since f ∈ Lq(Ω) was arbitrarily chosen their decomposition is a direct decomposition of the space Lq(Ω).
- The proof is based on applying the properties of the operator CTα and of the projector CQα.
6 Conclusion
- In this work the authors presented a generalization of several results of the classical continuous Clifford function theory developed in [17] in the context of fractional Clifford analysis.
- Due to the “double duality” indicated previously, some of the previous results admit alternative versions, for instance, for the operator RLDαa+ .
- Moreover, it is desirable to obtain an explicit expression for the fundamental solutions finding appropriate functions g0 and g1.
Did you find this useful? Give us your feedback
Citations
3,828 citations
10 citations
1 citations
1 citations
References
3 citations
Additional excerpts
...Therefore, the derivation rule (12) must be replaced in these cases by the following derivation rule: Da+ ( (x − a)Eμ,α−p (k(x − a)) ) = (x − a)kEμ,μ−p (k(x − a)) , p = 0, ....
[...]
...From the power series (10) and the operators (1), (3), and (5), we can obtain by straightforward calculations the following fractional integral and differential formulae involving Eμ,ν(z) (see Gorenflo et al25, pp87-88): I a+ ( (x − a)Eμ,ν (k(x − a)) ) = (x − a)Eμ,ν+α (k(x − a)) (11) for all α > 0, k ∈ C, x > a, μ > 0, ν > 0, Da+ ( (x − a)Eμ,ν (k(x − a)) ) = (x − a)Eμ,ν−α (k(x − a)) (12) for all α > 0, k ∈ C, x > a, μ > 0, ν > 0, ν ≠ α − p, where p = 0, ....
[...]
1 citations
"A higher dimensional fractional Bor..." refers methods in this paper
...Therefore, the derivation rule (13) must be replaced in these cases by the following derivation rule: Da+ ( (x − a)Eμ,p (k(x − a)) ) = (x − a)kEμ,μ+p−α (k(x − a)) , p = 1, ....
[...]
...Applying the operator Ca+1 ∂ 1+α1 2 x1 to Cα+ and relying on (13), we obtain the expression (62)....
[...]
...,m − 1 with m = [α] + 1, and Da+ ( (x − a)Eμ,ν (k(x − a)) ) = (x − a)Eμ,ν−α (k(x − a)) (13) for all α > 0, k ∈ C, x > a, μ > 0, ν > 0, ν ≠ p, where p = 1, ....
[...]