Abstract
Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright’s function for the case of equations of rational order, is presented. An example is treated in detail.
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T. Atanackovic, Lj. Oparnica, S. Pilipovic, Distributional framework for solving fractional differential equations. Integral Transforms Spec. Funct. 20 (2009), 215–222.
A. Al-Rabath, V.S. Erturk, S. Momani, Solutions of a fractional oscillator by using differential transform method. Computers and Mathematics with Applications 59 (2010), 833–842.
W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 2001.
B.S. Baclic, T.M. Atanackovic, Stability and creep of a fractional order viscoelastic rod. Bull. de l’Académie Serbe des Sciences et des Arts, Classe des Sciences mathématiques et naturelles 25 (2000), 115–131.
E. Bazhlekova, Properties of the fundamental and the impulse-response solutions of multi-term fractional differential equations. Complex Analysis and Applications’ 13, Proc. of Intern. Conference, Sofia, 31 Oct.–2 Nov. 2013 (2013), 55–64; http://www.math.bas.bg/complan/caa13/.
B. Bonilla, M. Rivero, J.J. Truillo, On a system of linear fractional differential equations with constant coefficients. Applied Mathematics and Computation 187 (2007), 68–78.
K. Diethelm, N.J. Ford, Multi-order fractional differential equations and their numerical solution. Applied Mathematics and Computation 154 (2004), 621–640.
K. Diethelm, Smoothness properties of solutions of Caputo-type fractional differential equations. Fract. Calc. Appl. Anal. 10 (2007), 151–160; at http://www.math.bas.bg/~fcaa.
K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics # 2004, Springer, Berlin, 2010.
G. Doetsch, Handbuch der Lalace-Transformation, I. Birkhäuser-Verlag, Basel, 1950.
A. Ghorbani, Toward a new analytical method for solving nonlinear fractional differential equations. Comput. Methods Appl. Mech. Engrg. 97 (2008), 4173–4179.
S.B. Hadid, Y.F. Luchko, An opperational method for solving fractional differential equations of an arbitrary real order. PanAmer. Math. J. 6 (1996), 57–73.
A.A. Kilbas, H.M. Srivastava, J.T. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
V. Kiryakova, Transmutation method for solving hyper-Bessel differential equations based on the Poisson-Dimovski transformation. Frac. Calc Appl. Anal. 11 (2008), 299–316; http://www.math.bas.bg/~fcaa.
H. Komatsu, Laplace transforms of hyperfunctions — A new fondation of the Heaviside calculus. J. Fac. Eci. Univ. Tokyo Sec. IA 34 (1987), 805–820.
Z. Li, M. Yamamoto, Initial-boundary value problems for linear diffusion equation with multiple time-fractional derivatives. http://arxiv.org/abs/1306.2778.
Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24 (1999), 207–233.
Y. Luchko, H. M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus. Computers Math. Applic. 29 (1995), 73–85.
J. Mikusiński, Operational Calculus, Vol. I and Vol. II (with T. K. Boehme). Pergamon Press, Oxford, 1987.
A. Pálfalvi, Efficient solution of a vibration equation involiving fractional derivatives. Intern. Journal Non-Linear Mechanics 45 (2010), 169–175.
A. Pedas, E. Tamme, Spline collocation methods for linear multi-term fractional differential equations. J. of Computational and Applied Mathematics 236 (2011), 167–176.
I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999.
A.V. Pshu, Partial Differential Equations with Fractional Derivatives. Nauka, Moscow, 2005 (in Russian).
M. Rivero, L. Rodríguez-Germá, J.J. Trujillo, Linear fractional differential equations with variable coefficients. Applied Mathematics Letters 21 (2008), 892–897.
B. Stanković, Sur une classe d’équations intégrales singuliéres. Recueil des travaux de l’Académie Serbe des Sciences XLIII, No 4 (1955), 81–130.
V.S. Vladimirov, Generalized Functions in Mathematical Physics. Mir Publishers, Moscow, 1979.
Z.H. Wang, X. Wang, General solution of the Bagley-Torvik equation with fractional-order derivatives. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 1279–1285.
E.M. Wright, On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 8 (1953), 71–79.
E.M. Wright, The generalized Bessel function of order qreater then one. Quart. J. Math. Oxford Ser. 2 (1940), 36–48.
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Atanackovic, T., Dolicanin, D., Pilipovic, S. et al. Cauchy problems for some classes of linear fractional differential equations. Fract Calc Appl Anal 17, 1039–1059 (2014). https://doi.org/10.2478/s13540-014-0213-1
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DOI: https://doi.org/10.2478/s13540-014-0213-1