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 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.
2 Fractional calculus
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. □
3 Existence and uniqueness
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.
- Jeon J-H, Metzler R: Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 2010., 81: Article ID 021103Google Scholar
- Burnecki K, Kepten E, Janczura J, Bronshtein I, Garini Y, Weron A: Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion. Biophys. J. 2012, 103: 1839-1847. 10.1016/j.bpj.2012.09.040View ArticleGoogle Scholar
- Bojdecki T, Gorostiza LG, Talarczyk A: Oscillatory fractional Brownian motion. Acta Appl. Math. 2013. doi:10.1007/s10440-013-9798-3Google Scholar
- Bǎleanu D, Mustafa OG, Agarwal RP: Asymptotically linear solutions for some linear fractional differential equations. Abstr. Appl. Anal. 2010., 2010: Article ID 865139. doi:10.1155/2010/865139Google Scholar
- Bǎleanu D, Mustafa OG, Agarwal RP: On the solution set for a class of sequential fractional differential equations. J. Phys. A 2010., 43: Article ID 385209Google Scholar
- Bǎleanu D, Agarwal RP, Mustafa OG, Coşulschi M: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A 2011., 44: Article ID 055203Google Scholar
- Bǎleanu D, Mustafa OG, Agarwal RP: Asymptotic integration of (1+ α )-order fractional differential equations. Comput. Math. Appl. 2011, 62(3):1492-1500. 10.1016/j.camwa.2011.03.021MathSciNetView ArticleGoogle Scholar
- Jalab HA, Ibrahim RW: Texture enhancement for medical images based on fractional differential masks. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 618536. doi:10.1155/2013/618536Google Scholar
- Jalab HA, Ibrahim RW: On generalized fractional differentiator signals. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 795954. doi:10.1155/2013/795954Google Scholar
- Ibrahim RW, Jalab HA: The fractional complex step method. Discrete Dyn. Nat. Soc. 2013., 2013: Article ID 515973. doi:10.1155/2013/515973Google Scholar
- Nigmatullin RR: The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 1986, 133: 425-430. 10.1002/pssb.2221330150View ArticleGoogle Scholar
- Kochubei AN: The Cauchy problem for evolution equations of fractional order. Differ. Uravn. (Minsk) 1989, 25: 1359-1368.MathSciNetGoogle Scholar
- Meerschaert MM, Nane E, Vellaisamy P: Fractional Cauchy problems in bounded domains. Ann. Probab. 2009, 37: 979-1007. 10.1214/08-AOP426MATHMathSciNetView ArticleGoogle Scholar
- Zaslavsky GM: Fractional kinetic equation for Hamiltonian chaos. Physica D 1994, 76: 110-122. 10.1016/0167-2789(94)90254-2MATHMathSciNetView ArticleGoogle Scholar
- Li K, Jigen P: Fractional abstract Cauchy problems. Integral Equ. Oper. Theory 2011, 70: 333-361. 10.1007/s00020-011-1864-5MATHMathSciNetView ArticleGoogle Scholar
- Li F, Liang J, Lu TT, Zhu H: A nonlocal Cauchy problem for fractional integrodifferential equations. J. Appl. Math. 2012., 2012: Article ID 901942Google Scholar
- Zhong WP, Yang XJ, Gao F: A Cauchy problem for some local fractional abstract differential equation with fractal conditions. J. Appl. Funct. Anal. 2013, 8(1):92-99.MATHMathSciNetGoogle Scholar
- Yang XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York; 2012.Google Scholar
- Yang XJ: Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4: 1-225.Google Scholar
- Yang XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, Hong Kong; 2011.Google Scholar
- Guo Y: Local fractional Z transform in fractal space. Adv. Digit. Multimed. 2012, 1(2):96-102.Google Scholar
- Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations. J. Math. Anal. Appl. 2008, 345: 871-879. 10.1016/j.jmaa.2008.05.017MATHMathSciNetView ArticleGoogle Scholar
- Ibrahim RW: Solutions of fractional diffusion problems. Electron. J. Differ. Equ. 2010, 147: 1-11.Google Scholar
- Ibrahim RW: Existence of fractional parametric Cauchy problem. Appl. Sci. 2010, 12: 30-36.MathSciNetGoogle Scholar
- Ibrahim RW: Existence and uniqueness of holomorphic solutions for fractional Cauchy problem. J. Math. Anal. Appl. 2011, 380: 232-240. 10.1016/j.jmaa.2011.03.001MATHMathSciNetView ArticleGoogle Scholar
- Ibrahim RW: On generalized Srivastava-Owa fractional operators in the unit disk. Adv. Differ. Equ. 2011., 2011: Article ID 55Google Scholar
- Srivastava HM, Owa S: Univalent Functions, Fractional Calculus, and Their Applications. Wiley, New York; 1989.MATHGoogle Scholar
- Duren PL: Univalent Functions. Springer, Berlin; 1983.MATHGoogle Scholar
- Brezis H Math. Studies 5. In Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam; 1973.Google Scholar
- Bajlekova, E: Fractional evolution equations in Banach spaces. PhD thesis, Eindhoven University of Technology. http://alexandria.tue.nl/extra2/200113270.pdf (2001)Google Scholar
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