Abstract and Applied Analysis

Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order

Djumaklych Amanov and Allaberen Ashyralyev

Full-text: Open access

Abstract

The initial-boundary value problem for partial differential equations of higher-order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 973102, 16 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174056

Digital Object Identifier
doi:10.1155/2012/973102

Mathematical Reviews number (MathSciNet)
MR2947751

Zentralblatt MATH identifier
1246.35201

Citation

Amanov, Djumaklych; Ashyralyev, Allaberen. Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 973102, 16 pages. doi:10.1155/2012/973102. https://projecteuclid.org/euclid.aaa/1365174056


Export citation

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.\setlengthemsep1.5pt
  • G. Sorrentinos, “Fractional derivative linear models for describing the viscoelastic dynamic behavior of polymeric beams,” in Proceedings of IMAS, Saint Louis, Mo, USA, 2006.
  • G. Sorrentinos, “Analytic modeling and experimental identi?cation of viscoelastic mechanical systems,” in Advances in Fractional Calculus, J. Sabatier, O. P. Agrawal, and J. A Tenreiro Machado, Eds.,pp. 403–416, Springer, 2007.
  • Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, Springer, New York, NY, USA, 1997.
  • R. L. Magin, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering, vol. 32, no. 1, pp. 1–104, 2004.
  • M. D. Ortigueira and J. A. Tenreiro Machado, “Special issue on Fractional signal processing and applications,” Signal Processing, vol. 83, no. 11, pp. 2285–2286, 2003.
  • B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 231–248, 2000.
  • K. B. Oldham, “Fractional differential equations in electrochemistry,” Advances in Engineering Software, vol. 41, no. 1, pp. 9–12, 2010.
  • R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000.
  • M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011.\setlengthemsep1.55pt
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, London, UK, 1993.
  • 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 B.V., Amsterdam, The Nrtherlands, 2006.
  • J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,” SIAM Review, vol. 18, no. 2, pp. 240–268, 1976.
  • A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.
  • C. Yuan, “Two positive solutions for $(n-1,1)$-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 930–942, 2012.
  • M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,” Advances in Difference Equations, vol. 2011, Article ID 748608, 32 pages, 2011.
  • O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.
  • R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1–10, 2007.
  • V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, no. 8, pp. 2677–2682, 2008.
  • R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401–411, 2009.
  • A. Ashyralyev and B. Hicdurmaz, “A note on the fractional Schrödinger differential equations,” Kybernetes, vol. 40, no. 5-6, pp. 736–750, 2011.
  • A. Ashyralyev, F. Dal, and Z. P\inar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011.
  • A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differential equations,” AIP Conference Proceeding, vol. 1389, pp. 617–620, 2011.
  • A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011.
  • A. A. Kilbas and O. A. Repin, “Analogue of Tricomi's problem for partial differential equations containing diffussion equation of fractional order,” in Proceedings of the International Russian-Bulgarian Symposium Mixed type equations and related problems of analysis and informatics, pp. 123–127, Nalchik-Haber, 2010.
  • A. A. Nahushev, Elements of Fractional Calculus and Their Applications, Nalchik, Russia, 2010.
  • A. V. Pshu, Boundary Value Problems for Partial Differential Equations of Fractional and Continual Order, Nalchik, Russia, 2005.
  • N. A. Virchenko and V. Y. Ribak, Foundations of Fractional Integro-Differentiations, Kiev, Ukraine, 2007.
  • M. M. Džrbašjan and A. B. Nersesjan, “Fractional derivatives and the Cauchy problem for differential equations of fractional order,” Izvestija Akademii Nauk Armjanskoĭ SSR, vol. 3, no. 1, pp. 3–28, 1968.
  • R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
  • 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.
  • 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.
  • R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532–3554, 2010.
  • D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis, vol. 69, no. 11, pp. 3692–3705, 2008.
  • G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis, vol. 70, no. 5, pp. 1873–1876, 2009.
  • G. M. Mophou and G. M. N'Guérékata, “Mild solutions for semilinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2009, no. 21, 9 pages, 2009.
  • G. M. Mophou and G. M. N'Guérékata, “Existence of the mild solution for some fractional differential equations with nonlocal conditions,” Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009.
  • V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis, vol. 69, no. 10, pp. 3337–3343, 2008.
  • V. Lakshmikantham and J. V. Devi, “Theory of fractional differential equations in a Banach space,” European Journal of Pure and Applied Mathematics, vol. 1, no. 1, pp. 38–45, 2008.
  • V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 395–402, 2007.
  • A. S. Berdyshev, A. Cabada, and E. T. Karimov, “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,” Nonlinear Analysis, vol. 75, no. 6, pp. 3268–3273, 2011.
  • R. Gorenflo, Y. F. Luchko, and S. R. Umarov, “On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order,” Fractional Calculus & Applied Analysis, vol. 3, no. 3, pp. 249–275, 2000.
  • B. Kadirkulov and K. H. Turmetov, “About one generalization of heat conductivity equation,” Uzbek Mathematical Journal, no. 3, pp. 40–45, 2006 (Russian).
  • D. Amanov, “Solvability of boundary value problems for equation of higher order with fractional derivatives,” in Boundary Value Problems for Differential Equations, The Collection of Proceedings no. 17, pp. 204–209, Chernovtsi, Russia, 2008.
  • D. Amanov, “Solvability of boundary value problems for higher order differential equation with fractional derivatives,” in Problems of Camputations and Applied Mathemaitics, no. 121, pp. 55–62, Tashkent, Uzbekistan, 2009.
  • M. M. Djrbashyan, Integral Transformations and Representatation of Functions in Complex Domain, Moscow, Russia, 1966.