Journal Menu
- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391062, 17 pages
doi:10.1155/2012/391062
Research Article
On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Received 27 December 2011; Revised 10 April 2012; Accepted 14 April 2012
Academic Editor: Bashir Ahmad
Copyright © 2012 M. D. Qassim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at Publisher · View at Google Scholar
- R. L. Bagley and P. J. Torvik, “A different approach to the analysis of viscoelastically damped structures,” AIAA Journal, vol. 21, pp. 741–748, 1983.
- R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of real material,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294–298, 1984.
- R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750, 25 pages, 2010. View at Zentralblatt MATH
- R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009. View at Zentralblatt MATH
- B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, article 36, 9 pages, 2011. View at Publisher · View at Google Scholar
- B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,” Computers and Mathematics with Applications. In press. View at Publisher · View at Google Scholar
- V. V. Anh and R. Mcvinish, “Fractional differential equations driven by Lévy noise,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 97–119, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- B. de Andrade, C. Cuevas, and J. P. C. dos Santos, “Existence results for a fractional equation with state-dependent delay,” Advances in Difference Equations, vol. 2011, Article ID 642013, 15 pages, 2011. View at Zentralblatt MATH
- M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order,” Surveys in Mathematics and its Applications, vol. 3, pp. 1–12, 2008. View at Zentralblatt MATH
- S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. M. A. El-Sayed, “Fractional differential equations,” Kyungpook Mathematical Journal, vol. 28, no. 2, pp. 119–122, 1988. View at Zentralblatt MATH
- A. M. A. El-Sayed, “On the fractional differential equations,” Applied Mathematics and Computation, vol. 49, no. 2-3, pp. 205–213, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. M. A. El-Sayed, “Fractional order evolution equations,” Journal of Fractional Calculus, vol. 7, pp. 89–100, 1995. View at Zentralblatt MATH
- A. M. A. El-Sayed and S. A. Abd El-Salam, “Weighted Cauchy-type problem of a functional differ-integral equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 30, pp. 1–9, 2007. View at Zentralblatt MATH
- A. M. A. El-Sayed and S. A. Abd El-Salam, “-solution of weighted Cauchy-type problem of a diffre-integral functional equation,” International Journal of Nonlinear Science, vol. 5, no. 3, pp. 281–288, 2008.
- K. M. Furati and N.-E. Tatar, “Power-type estimates for a nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 62, no. 6, pp. 1025–1036, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. M. Furati and N.-E. Tatar, “An existence result for a nonlocal fractional differential problem,” Journal of Fractional Calculus, vol. 26, pp. 43–51, 2004. View at Zentralblatt MATH
- K. M. Furati and N.-E. Tatar, “Behavior of solutions for a weighted Cauchy-type fractional differential problem,” Journal of Fractional Calculus, vol. 28, pp. 23–42, 2005. View at Zentralblatt MATH
- K. M. Furati and N.-E. Tatar, “Long time behavior for a nonlinear fractional model,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 441–454, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. A. Kilbas, B. Bonilla, and J. J. Trujillo, “Existence and uniqueness theorems for nonlinear fractional differential equations,” Demonstratio Mathematica, vol. 33, no. 3, pp. 583–602, 2000. View at Zentralblatt MATH
- A. A. Kilbas, B. Bonilla, J. J. Trujillo, et al., “Fractional integrals and derivatives and differential equations of fractional order in weighted spaces of continuous functions,” Doklady Natsionalnoy Akademii Nauk Belarusi, vol. 44, no. 6, 2000 (Russian).
- A. A. Kilbas and S. A. Marzan, “Cauchy problem for differential equation with Caputo derivative,” Fractional Calculus & Applied Analysis, vol. 7, no. 3, pp. 297–321, 2004. View at Zentralblatt MATH
- A. A. Kilbas and S. A. Marzan, “Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions,” Differential Equations, vol. 41, no. 1, pp. 84–89, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006, Edited by Jan van Mill.
- A. A. Kilbas and A. A. Titioura, “Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions,” Mathematical Modelling and Analysis, vol. 12, no. 3, pp. 343–356, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. Kosmatov, “Integral equations and initial value problems for nonlinear differential equations of fractional order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp. 2521–2529, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- C. Kou, J. Liu, and Y. Ye, “Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 142175, 15 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, 1987, (Translation from Russian 1993).
- C. Yu and G. Gao, “Existence of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 26–29, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Zhou, “Existence and uniqueness of solutions for a system of fractional differential equations,” Fractional Calculus & Applied Analysis, vol. 12, no. 2, pp. 195–204, 2009.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012.
- L. Gaul, P. Klein, and S. Kempe, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, pp. 81–88, 1991.
- W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach of selfsimilar protein dynamics,” Biophysical Journal, vol. 68, pp. 46–53, 1995.
- T. T. Hartley, C. F. Lorenzo, H. K. Qammar, et al., “Chaos in a fractional order Chua system,” IEEE Transactions on Circuits & Systems I, vol. 42, no. 8, pp. 485–490, 1995.
- R. Hilfer, Fractional Time Evolution, Applications of Fractional Calculus in Physics, World Scientific, New-Jersey, NJ, USA, 2000. View at Zentralblatt MATH
- R. Hilfer, “Experimental evidence for fractional time evolution in glass materials,” Chemical Physics, vol. 284, pp. 399–408, 2002.
- R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” American Society of Mechanical Engineers, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Laskri and N.-E. Tatar, “The critical exponent for an ordinary fractional differential problem,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1266–1270, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanis,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, Vienna, Austria, 1997.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Pres, World Scientific, London, UK, 2010. View at Publisher · View at Google Scholar
- B. Mandelbrot, “Some noises with 1/f spectrum, a bridge between direct current and white noise,” IEEE Transactions on Information Theory, vol. 13, no. 2, pp. 289–298, 1967.
- F. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, pp. 7180–7186, 1995.
- I. Podlubny, I. Petráš, B. M. Vinagre, P. O'Leary, and L'. Dorčák, “Analogue realizations of fractional-order controllers,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 281–296, 2002. View at Publisher · View at Google Scholar
- K. M. Furati, M. D. Kassim, and N.-E. Tatar, “Existence and stability for a differential problem with Hilfer-Hadamard fractional derivative,” submitted.
- P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 1–15, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- È. Mitidieri and S. I. Pokhozhaev, “A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,” Proceedings of the Steklov Institute of Mathematics, vol. 234, pp. 1–383, 2001.