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.
[1] T. Atanackovic, Lj. Oparnica, S. Pilipovic, Distributional framework for solving fractional differential equations. Integral Transforms Spec. Funct. 20 (2009), 215–222. http://dx.doi.org/10.1080/1065246080256806910.1080/10652460802568069Search in Google Scholar
[2] 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. Search in Google Scholar
[3] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-5075-910.1007/978-3-0348-5075-9Search in Google Scholar
[4] 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. Search in Google Scholar
[5] 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/. Search in Google Scholar
[6] 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. http://dx.doi.org/10.1016/j.amc.2006.08.10410.1016/j.amc.2006.08.104Search in Google Scholar
[7] K. Diethelm, N.J. Ford, Multi-order fractional differential equations and their numerical solution. Applied Mathematics and Computation 154 (2004), 621–640. http://dx.doi.org/10.1016/S0096-3003(03)00739-210.1016/S0096-3003(03)00739-2Search in Google Scholar
[8] 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. Search in Google Scholar
[9] K. Diethelm, The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics # 2004, Springer, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar
[10] G. Doetsch, Handbuch der Lalace-Transformation, I. Birkhäuser-Verlag, Basel, 1950. http://dx.doi.org/10.1007/978-3-0348-6984-310.1007/978-3-0348-6984-3Search in Google Scholar
[11] A. Ghorbani, Toward a new analytical method for solving nonlinear fractional differential equations. Comput. Methods Appl. Mech. Engrg. 97 (2008), 4173–4179. http://dx.doi.org/10.1016/j.cma.2008.04.01510.1016/j.cma.2008.04.015Search in Google Scholar
[12] 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. Search in Google Scholar
[13] A.A. Kilbas, H.M. Srivastava, J.T. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006. Search in Google Scholar
[14] 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. Search in Google Scholar
[15] H. Komatsu, Laplace transforms of hyperfunctions — A new fondation of the Heaviside calculus. J. Fac. Eci. Univ. Tokyo Sec. IA 34 (1987), 805–820. Search in Google Scholar
[16] Z. Li, M. Yamamoto, Initial-boundary value problems for linear diffusion equation with multiple time-fractional derivatives. http://arxiv.org/abs/1306.2778. Search in Google Scholar
[17] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24 (1999), 207–233. Search in Google Scholar
[18] 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. http://dx.doi.org/10.1016/0898-1221(95)00031-S10.1016/0898-1221(95)00031-SSearch in Google Scholar
[19] J. Mikusiński, Operational Calculus, Vol. I and Vol. II (with T. K. Boehme). Pergamon Press, Oxford, 1987. Search in Google Scholar
[20] A. Pálfalvi, Efficient solution of a vibration equation involiving fractional derivatives. Intern. Journal Non-Linear Mechanics 45 (2010), 169–175. http://dx.doi.org/10.1016/j.ijnonlinmec.2009.10.00610.1016/j.ijnonlinmec.2009.10.006Search in Google Scholar
[21] A. Pedas, E. Tamme, Spline collocation methods for linear multi-term fractional differential equations. J. of Computational and Applied Mathematics 236 (2011), 167–176. http://dx.doi.org/10.1016/j.cam.2011.06.01510.1016/j.cam.2011.06.015Search in Google Scholar
[22] I. Podlubny, Fractional Differential Equations. Academic Press, New York, 1999. Search in Google Scholar
[23] A.V. Pshu, Partial Differential Equations with Fractional Derivatives. Nauka, Moscow, 2005 (in Russian). Search in Google Scholar
[24] M. Rivero, L. Rodríguez-Germá, J.J. Trujillo, Linear fractional differential equations with variable coefficients. Applied Mathematics Letters 21 (2008), 892–897. http://dx.doi.org/10.1016/j.aml.2007.09.01010.1016/j.aml.2007.09.010Search in Google Scholar
[25] B. Stanković, Sur une classe d’équations intégrales singuliéres. Recueil des travaux de l’Académie Serbe des Sciences XLIII, No 4 (1955), 81–130. Search in Google Scholar
[26] V.S. Vladimirov, Generalized Functions in Mathematical Physics. Mir Publishers, Moscow, 1979. Search in Google Scholar
[27] 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. http://dx.doi.org/10.1016/j.cnsns.2009.05.06910.1016/j.cnsns.2009.05.069Search in Google Scholar
[28] E.M. Wright, On the coefficients of power series having exponential singularities. J. Lond. Math. Soc. 8 (1953), 71–79. Search in Google Scholar
[29] E.M. Wright, The generalized Bessel function of order qreater then one. Quart. J. Math. Oxford Ser. 2 (1940), 36–48. http://dx.doi.org/10.1093/qmath/os-11.1.3610.1093/qmath/os-11.1.36Search in Google Scholar
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