Volume 10, Issue 1, pp 60--70
Publication Date: 2017-01-26
- King Saud University, College of Science, Mathematics Department, P. O. Box 2455, Riyadh 11451, Saudi Arabia.
Imed Bachar - King Saud University, College of Science, Mathematics Department, P. O. Box 2455, Riyadh 11451, Saudi Arabia.
This paper deals with the study of the well-posedness of a mixed fractional problem for the wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. We prove the existence and uniqueness of solution as well as its dependence on the given data. Our results develop and show the efficiency and effectiveness of the functional analysis method when we deal with fractional partial differential equations instead of the nonfractional equations which have been extensively studied by many authors during the last three decades.
Caputo derivative, solvability of the problem, fractional differential equation, initial boundary value problem.
 G. Adomian,/ A review of the decomposition method in applied mathematics,/ J. Math. Anal. Appl.,/ 135 (1988), 501–544.
 G. Adomian,/ Solving frontier problems of physics: the decomposition method, With a preface by Yves Cherruault,/ Fundamental Theories of Physics, Kluwer Academic Publishers Group, Dordrecht,/ (1994).
 A. A. Alikhanov,/ A priori estimates for solutions of boundary value problems for fractional-order equations,/ Dier. Equ.,/ 46 (2010), 660–666.
 L. Debnath, D. D. Bhatta,/ Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics,/ Fract. Calc. Appl. Anal.,/ 7 (2004), 21–36.
 A. M. A. El-Sayed, M. Gaber,/ The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,/ Phys. Lett. A,/ 359 (2006), 175–182.
 N. Engheta,/ On fractional calculus and fractional multipoles in electromagnetism,/ IEEE Trans. Antennas and Propagation,/ 44 (1996), 554–566.
 A. Freed, K. Diethelm, Y. Luchko,/ Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus: First annual Report,/ NASA Technical Reports Server (NTRS), United States,/ (2002).
 R. Gorenfo,/ Abel integral equations with special emphasis on applications,/ Lectures in Mathematical Sciences, The University of Tokyo, Graduate School of Mathematical Sciences,/ (1996).
 J. H. He,/ Some applications of nonlinear fractional differential equations and their approximations,/ Bull. Sci. Technol.,/ 15 (1999), 86–90.
 R. Hilfer (Ed.),/ Applications of fractional calculus in physics,/ World Scientific Publishing Co., Inc., River Edge, NJ,/ (2000).
 H. Jafari, V. Daftardar-Gejji,/ Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,/ Appl. Math. Comput.,/ 180 (2006), 488–497.
 T. Kaplan, L. J. Gray, S. H. Liu,/ Self-affine fractal model for a metal-electrolyte interface,/ Phys. Rev. B,/ 35 (1987), 5379– 5381.
 A. M. Keighttey, J. C. Myland, K. B. Oldham, P. G. Symons,/ Reversible cyclic voltammetry in the presence of product,/ J. Electroanal. Chem.,/ 322 (1992), 25–54.
 A. A. Kilbas, H. M. Srivastava, J. J. Trujillo,/ Theory and applications of fractional differential equations,/ North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam,/ (2006).
 O. A. Ladyzhenskaya,/ The boundary value problems of mathematical physics,/ Translated from the Russian by Jack Lohwater [Arthur J. Lohwater], Applied Mathematical Sciences, Springer-Verlag, New York,/ (1985).
 A. Le Mehaute, G. Crepy,/ Introduction to transfer and motion in fractal media: the geometry of kinetics,/ Solid State Ion.,/ 9 (1983), 17–30.
 F. Mainardi,/ Fractional relaxation-oscillation and fractional diffusion-wave phenomena,/ Chaos Solitons Fractals,/ 7 (1996), 1461–1477.
 F. Mainardi,/ Fractional calculus: some basic problems in continuum and statistical mechanics,/ Fractals and fractional calculus in continuum mechanics, Udine, (1996), 291–348, CISM Courses and Lectures, Springer, Vienna,/ (1997).
 F. Mainardi,/ Fractional calculus and waves in linear viscoelasticity,/ An introduction to mathematical models. Imperial College Press, London,/ (2010).
 F. Mainardi, Y. Luchko, G. Pagnini,/ The fundamental solution of the space-time fractional diffusion equation,/ Fract. Calc. Appl. Anal.,/ 4 (2001), 153–192.
 F. Mainardi, P. Paradisi,/ Fractional diffusive waves,/ J. Comput. Acoust.,/ 9 (2001), 1417–1436.
 S. Mesloub, A. Bouziani,/ On a class of singular hyperbolic equation with a weighted integral condition,/ Int. J. Math. Math. Sci.,/ 22 (1999), 511–519.
 S. Mesloub, R. Mezhoudi, M. Medjeden,/ A mixed problem for a parabolic equation of higher order with integral conditions,/ Bull. Polish Acad. Sci. Math.,/ 50 (2012), 313–322.
 K. S. Miller, B. Ross,/ An introduction to the fractional calculus and fractional differential equations,/ A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York,/ (1993).
 S. Momani,/ An explicit and numerical solutions of the fractional KdV equation,/ Math. Comput. Simulation,/ 70 (2005), 110–118.
 S. Momani,/ Non-perturbative analytical solutions of the space- and time-fractional Burgers equations,/ Chaos Solitons Fractals,/ 28 (2006), 930–937.
 S. Momani, R. Qaralleh,/ Numerical approximations and Padé approximants for a fractional population growth model,/ Appl. Math. Model.,/ 31 (2007), 1907–1914.
 S. Momani, N. Shawagfeh,/ Decomposition method for solving fractional Riccati differential equations,/ Appl. Math. Comput., /182 (2006), 1083–1092.
 R. Nigmatullin,/ The realization of the generalized transfer equation in a medium with fractal geometry,/ Phys. Status Solidi B, /133 (1986), 425–430.
 I. Podlubny,/ Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications,/ Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA,/ (1999).
 S. G. Samko, A. A. Kilbas, O. I. Marichev,/ Fractional integrals and derivatives,/ Theory and applications, Edited and with a foreword by S. M. Nikolski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon,/ (1993).