Regular fractional dissipative boundary value problems
- Dumitru Baleanu^{1}Email author and
- Ekin Uğurlu^{1}
https://doi.org/10.1186/s13662-016-0883-6
© Baleanu and Uğurlu 2016
Received: 1 February 2016
Accepted: 29 May 2016
Published: 1 July 2016
Abstract
In this manuscript we present a regular dissipative fractional operator associated with a fractional boundary value problem. In particular, we present two main dissipative boundary value problems and one of them contains the spectral parameter in the boundary conditions. To construct the associated dissipative operator we present a direct sum Hilbert space.
Keywords
1 Introduction
Fractional calculus is one of the most useful tools to investigate the hidden properties of dynamical systems. During the last few decades a lot of developments were done in both theoretical and the applied view points [4–14], namely, some important results from classical analysis were generalized to the fractional case and fractional calculus was successfully applied to real problems which appear in science and engineering. One of the most challenging hot topics in fractional calculus is to find real world application in physics. We recall that an one-dimensional dissipative Schrödinger-type operators together with their dilations and eigenfunction expansions was discussed in [15] and it was motivated by the problems appearing in semiconductor physics (also see [16, 17]).
2 Boundary value problem
Theorem 2.1
The operator \(\mathbb{L}\) is dissipative in \(L_{w_{\alpha}}^{2}[a,b]\).
Proof
Then we arrive at the following corollary ([18], p.176).
Corollary 2.1
Let λ be an eigenvalue of the operator \(\mathbb{L}\). Then \(\operatorname{Im}\lambda\geq0\).
For the special case of α we have additional results [19, 20, 25–29].
Corollary 2.2
3 Eigenparameter dependent boundary value problem
Theorem 3.1
The operator \(\mathbb{L}_{\lambda}\) is dissipative in H.
Proof
Corollary 3.1
Let λ be an eigenvalue of the operator \(\mathbb{L}_{\lambda}\). Then \(\operatorname{Im}\lambda \geq 0\). For the special case of α we have additional results [21–24].
Corollary 3.2
4 Conclusion
It is well known that the dissipative operators arise in several real world applications and even naturally in mathematics. In this manuscript we considered new operators, namely they are both dissipative and of fractional calculus type. The generalization proposed in this manuscript will extend considerably the possibility to extract new features from the dynamics of complex systems involving non-local effects. Bearing this in mind we discussed first of all the boundary value problem (2.1)-(2.3) and we showed that the corresponding fractional operator \(\mathbb{L}\) is dissipative in \(L_{w}^{2}[a,b]\). After that we investigated the boundary value problem (3.1)-(3.3) and we proved that the corresponding operator \(\mathbb {L}_{\lambda}\) is dissipative in \(L_{w}^{2}[a,b]\oplus \mathbb{C}\).
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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