SOLUTIONS OF LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS OF ORDER 𝒏−𝟏<𝒏𝒒<𝒏

Year 2020, Issue: 045, 81 - 89, 31.12.2020

Sertaç Erman

Abstract

In this study, the linear Caputo fractional differential equation of order 𝑛−1<𝑛𝑞<𝑛 is investigated. First, the solution of the equation of order 0<𝑞<1, with variable coefficients, is obtained by using the solution of differential equation of integer order which is the least integer greater than fractional order. Moreover, the solution of linear fractional differential equations of order 𝑛−1<𝑛𝑞<𝑛 is considered. The solutions of the equation are presented in terms of Mittag-Leffler function with three parameters. The main goal of this study is to present a closed-series form of the solutions. To demonstrate the accuracy and the effectiveness of the proposed approach, some numerical solutions are given.

Keywords

Fractional Differential Equation, Mittag-Leffler function, Three parameters

References

• [1] Debnath, L., (2003), Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54, 3413–3442.
• [2] Ross, B., (1975), A Brief History and Exposition of the Fundamental Theory of Fractional Calculus. Fractional calculus and its Applications, Lecture notes in Mathematics, Springer: Berlin, Germany, 457, 1–36.
• [3] Oldham, K.B., Spanier, J., (1974), The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press: Newyork.
• [4] Podlubny, I., (1998), Fractional Differential Equations, Volume 198: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their solutions, Mathematics in Science and Engineering, Academic Press: San Diego.
• [5] Podlubny, I., (2002), Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis 5, 4, 367–386.
• [6] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., (2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier: Amsterdam, The Netherlands.
• [7] Deithelm, K., (2004), The Analysis of Fractional Differential Equations, Volume 2004 of Lecture Notes in Mathematics, Springer: Berlin, Germany.
• [8] Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin S.V., (2014), Mittag-Leffler Functions, Related Topics and Applications, Springer: Berlin, Germany.
• [9] Pooseh, S., Rodrigues, H.S., Torres, D.F.M., (2011), Fractional Derivatives in Dengue Epidemics, AIP Conf. Proc., 1389, 739–742.
• [10] Debnath, L., (2003), Recents Applications of Fractional Calculus to Science and Engineering, Int. J. Math. Appl. Sci, 54, 3413–3442.
• [11] Singh, J., Kumar, D., Kılıçman, A., (2014), Numerical Solutions of Nonlinear Fractional Partial Differential Equations Arising in Spatial Diffusion of Biological Populations, Abstr. Appl. Anal., 2014, 535793.
• [12] Koeller, R.C., (1984), Applications of Fractional Calculus to the Theory of Viscoelasticity, J. Appl. Mech., 51, 299–307.
• [13] Saha, R.S., Bera, R.K., (2000), The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339, 2000, 1-77
• [14] Waggas, G.A., (2010), Application of Fractional Calculus Operators for a New Class of Univalent Functions with Negative Coefficients Defined by Hohlov Operator, Math. Slovaca, 1, 75–82.
• [15] Aliev, F.A., Aliev, N.A., Safarova, N.A., Gasimova K.G.,, Velieva N.I., (2018), Solution of Linear Fractional-Derivative Ordinary Differential Equations with Constant Matrix Coefficients, Applied and Computational Mathematics, 17, 3, 317-322
• [16] Demir, A., Erman, S., Özgür, B., and Korkmaz, E., (2013), Analysis of fractional partial differential equations by Taylor series expansion, Boundary Value Problems, 68