- Open Access
Modified fractional Cauchy problem in a complex domain
© Ibrahim; licensee Springer 2013
- Received: 8 December 2012
- Accepted: 6 May 2013
- Published: 23 May 2013
In this paper, we make an extension to the Srivastava-Owa fractional operators in the space , where U is the open unit disk and H is a complex Hilbert space. Some recurrent relations are imposed on these extended operators. Moreover, by employing the theory of sums of accretive operators, the existence and uniqueness of the solution of the fractional Cauchy problem (in the sense of extended Srivastava-Owa fractional operators) is studied in a complex Hilbert space. Applications are illustrated.
- fractional calculus
- fractional differential equations
- Srivastava-Owa fractional operators
- unit disk
- analytic function
- Cauchy problem
Fractional derivatives can express the properties of memory and heredity of materials, which is the chief benefit of fractional derivatives compared with integer-order derivatives. Practical problems require definitions of fractional derivatives allowing the use of physically interpretable initial conditions. Fractional time derivatives are linked with irregular sub-diffusion, where a darken of particles spreads more slowly than a classical diffusion. While the fractional space derivatives are used to model irregular diffusion or dispersion, where a particle follow spreads at a rate not in agreement with the classical Brownian motion model, and the follow can be asymmetric [1–3].
Fractional differential and integro-differential equations occur from different real processes, and phenomena arise in physics such as signal processing and image processing, optics, engineering, control system, computer science (such as real neural network, complex neural network, information technology), statistics and probability, astronomy, geophysics, hydrology, chemical technology, materials, robots, earthquake analysis, electric fractal network, statistical mechanics, biotechnology, medicine, and economics [4–10].
Fractional Cauchy problems restore the integer time derivative by its fractional complement. Nigmatullin  posed a physical derivation of the fractional Cauchy problem; Kochubei  introduced the mathematical study of fractional Cauchy problems; Meerschaert et al.  constructed the stochastic solutions for fractional Cauchy problems in a bounded domain; Zaslavsky , falsified fractional Cauchy problems as a model for Hamiltonian chaos; Kexue and Jigen , concerned with fractional abstract Cauchy problems with order , proposed the sufficient conditions for the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem; Li et al.  established an existence theorem for mild solutions to the nonlocal Cauchy problem by virtue of measure of noncompactness and the fixed point theorem for condensing maps; Zhong et al.  studied the Cauchy problem for some local fractional abstract differential equation with fractal conditions; Yang  considered the problem for local fractional derivatives from local fractional functional analysis theory; finally, local fractional Cauchy formula within fractal complex domain was investigated in [19–21]. Recently, the author studied the fractional Cauchy problem in a complex domain [22–26].
In this article, we shall make an extension to the Srivastava-Owa fractional operators in the space , where U is the open unit disk and H is a complex Hilbert space. Some properties are discussed such as the recurrent relations. Moreover, by applying the theory of sums of accretive operators, the existence and uniqueness of the solution of the fractional Cauchy problem (in the sense of extended Srivastava-Owa fractional operators) is established in a complex Hilbert space. Applications are introduced.
In , Srivastava and Owa provided the definitions for fractional operators (derivative and integral) in the complex z-plane ℂ as follows.
where the function is analytic in a simply-connected region of the complex z-plane ℂ containing the origin, and the multiplicity of is removed by requiring to be real when .
where the function is analytic in a simply-connected region of the complex z-plane (ℂ) containing the origin, and the multiplicity of is removed by requiring to be real when .
Throughout the paper, we consider , and .
By employing the concept of sums of accretive operators, we shall prove the maximal regularity of problem (1).
where . We have the following property.
Lemma 2.1 .
Lemma 2.3 .
From the last assertion, we conclude that . □
Lemma 2.4 Let , then .
Combining the last two assertions, we end the proof. □
By virtue of the last discussion, we have the following result.
Lemma 2.5 Let , then , is an accretive operator.
but u is in the domain of , so, consequently, is an accretive operator. □
Lemma 2.6 Let , then is an m-accretive operator.
where . Thus is well defined for all , and bounded in . This implies that is an m-accretive operator. □
under the following assumptions.
(H1) A is a linear m-accretive operator in H, , where H is a complex Hilbert space.
and , .
We provide the main result of this section. We need the following result.
Lemma 3.1 
Let P be m-accretive and let Q be accretive Lipschitz continuous in a Hilbert space H. Then is m-accretive with .
According to Lemma 2.5 and Lemma 2.6, we obtain that is m-accretive. Moreover, by (H2) it follows that the operator is a bounded accretive operator. By virtue of Lemma 3.1, it follows that , , is m-accretive. Together with (H3), we obtain that Eq. (9) has a unique solution . □
Hence, from the last two inequalities, we conclude the inequality (9). This completes the proof.
Corollary 3.1 Let A be accretive and self-adjoint in H, and let the hypotheses (H2) and (H3) hold. Then there exists a unique function satisfying (1) and the inequality (7).
Proof It is well known that an accretive and self-adjoint operator implies a symmetric and m-accretive one . Hence, by virtue of Theorem 3.1, there exists a unique function satisfying (1) and the inequality (7). □
Corollary 3.2 Let C be accretive and Lipschitz continuous in H and let the hypotheses (H1) and (H3) hold. Then there exists a unique function satisfying (1) and the inequality (7).
where and are positive; thus in view of Theorem 3.1, there exists a unique function satisfying (1) and the inequality (7). □
It follows that A and C satisfy the conditions of Corollary 3.1 for some λ, and therefore (10) has a unique function .
Note that self-adjoint operators on a Hilbert space are applied in quantum mechanics to describe a physical observation such as the position, momentum, angular momentum and spin. The differential operators corresponding to the Legendre differential equation and the harmonic motion equation are self-adjoint, while those corresponding to the Laguerre differential equation and Hermite differential equation are not. A nonself-adjoint second-order linear differential operator can be viewed as a self-adjoint by using Sturm-Liouville theory.
It is clear that A is a self-adjoint operator and C is bounded; thus it follows that the problem (11) has a unique solution.
The author is thankful to the referees for helpful suggestions for the improvement of this article. This research has been funded by the university of Malaya, under Grant No. RG208-11AFR.
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