American Institute of Mathematical Sciences

Advanced
2005, 2005(Special): 233-240. doi: 10.3934/proc.2005.2005.233

On some fractional differential equations in the Hilbert space

Mahmoud M. El-Borai 1,

1. 

Alexandria University, Faculty of Science, Egypt

Received  September 2004 Revised  May 2005 Published  September 2005

Let $A$ be a closed linear operator defined on a dense set in the Hilbert space $H$. Fractional evolution equations of the form $\frac{d^\alpha u(t)}{dt^\alpha} = Au(t), 0 < \alpha \leq 1$, are studied in $H$, for a wide class of the operators $A$. Some properties of the solutions of the Cauchy problem for the considered equation are studied under suitable conditions . It is proved also that there exists a dense set $S$ in $H$, such that if the initial condition $u(0)$ is an element of $S$, then there exists a solution $u(t)$ of the considered Cauchy problem. Applications to general partial differential equations of the form $$ \frac{\partial^\alpha u(x,t)}{\partial t^\alpha} = \sum_{|q| \leq m} a_q(x) D^q u(x,t) $$ are given without any restrictions on the characteristic form $\sum_{|q|=m} a_\alpha(x) \xi^q$, where $D^q = D_1^{q_1} ... D_n^{q_n}, x = (x_1, ..., x_n), D_j= \frac{\partial}{\partial x_j}, \xi^q = \xi_1^{q_1}, ..., \xi_n^{q_n}, |q| = q_1 + ... + q_n$, and $q = (q_1, ..., q_n)$ is a multi index.\par}
Keywords: General partial differential equations., Cauchy problem, Closed operators, Fractional derivatives and integrals, Hilbert space.
Mathematics Subject Classification: 47 D 09, 34 G 10, 34 G 99, 35 K 9.
Citation: Mahmoud M. El-Borai. On some fractional differential equations in the Hilbert space. Conference Publications, 2005, 2005 (Special) : 233-240. doi: 10.3934/proc.2005.2005.233
[1]

Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753
[2]

Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure and Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837
[3]

Anna Karczewska, Carlos Lizama. On stochastic fractional Volterra equations in Hilbert space. Conference Publications, 2007, 2007 (Special) : 541-550. doi: 10.3934/proc.2007.2007.541
[4]

Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006
[5]

Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541
[6]

Abbas Ja'afaru Badakaya, Aminu Sulaiman Halliru, Jamilu Adamu. Game value for a pursuit-evasion differential game problem in a Hilbert space. Journal of Dynamics and Games, 2022, 9 (1) : 1-12. doi: 10.3934/jdg.2021019
[7]

Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022028
[8]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
[9]

P. Chiranjeevi, V. Kannan, Sharan Gopal. Periodic points and periods for operators on hilbert space. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4233-4237. doi: 10.3934/dcds.2013.33.4233
[10]

Salim A. Messaoudi, Ilyes Lacheheb. A general decay result for the Cauchy problem of plate equations with memory. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022026
[11]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817
[12]

Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117
[13]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[14]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428
[15]

Lok Ming Lui, Yalin Wang, Tony F. Chan, Paul M. Thompson. Brain anatomical feature detection by solving partial differential equations on general manifolds. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 605-618. doi: 10.3934/dcdsb.2007.7.605
[16]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215
[17]

Arzu Ahmadova, Nazim I. Mahmudov, Juan J. Nieto. Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022008
[18]

Liping Luo, Zhenguo Luo, Yunhui Zeng. New results for oscillation of fractional partial differential equations with damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3223-3231. doi: 10.3934/dcdss.2020336
[19]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
[20]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435

 Impact Factor: 

Tools

Metrics

  • PDF downloads (148)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy