Analytic solution of a system of linear distributed order differential equations in the Reimann-Liouville sense

Document Type : Article


Department of Electrical Engineering Sharif University of Technology, Tehran, Iran


In this paper, solution of a system of linear differential equations of distributed order in the Riemann-Liouville sense is analytically obtained. The distributed order relaxation equation is a special case of the system investigated in this paper. The solution of the mentioned system is introduced on the basis of a function which can be considered as the distributed order generalization of the matrix Mittag-Leffler functions. It is shown that this generalized function in two special cases of the weight function can be expressed in terms of the multivariate Mittag-Leffler functions and the Wright functions.


Main Subjects

1. Rocca, M.C., Plastino, A.R., Plastino, A., Ferri, G.L., and de Paoli, A. General solution of a fractional di_usion-advection equation for solar cosmic-ray transport", Physica A: Statistical Mechanics and its Applications, 447, pp. 402{410 (2016). 2. Bellouquid, A., Nieto, J., and Urrutia, L. About the kinetic description of fractional di_usion equations modeling chemotaxis", Mathematical Models and Methods in Applied Sciences, 26(02), pp. 249{268 (2016). 3. Rosa, C.F.A.E. and Capelas de Oliveira, E. Relaxation equations: fractional models", Journal of H. Taghavian and M.S. Tavazoei/Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 1384{1397 1397 Physical Mathematics, 6(2) (2015). DOI: 10.4172/2090-0902.1000146 4. Saxena, R.K. and Pagnini, G. Exact solutions of triple-order time-fractional di_erential equations for anomalous relaxation and di_usion I: The accelerating case", Physica A: Statistical Mechanics and Its Applications, 390(4), pp. 602{613 (2011). 5. Bobylev, A.V. and Cercignani, C. The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation", Applied Mathematics Letters, 15(7), pp. 807{813 (2002). 6. Mainardi, F., Mura, A., Goreno, R., and Stojanovi_c, M. The two forms of fractional relaxation of distributed order", Journal of Vibration and Control, 13(9-10), pp. 1249{1268 (2007). 7. Ansari, A. and Moradi, M. Exact solutions to some models of distributed-order time fractional di_usion equations via the Fox H functions", Science Asia, 39, pp. 57{66 (2013). 8. Naber, M. Distributed order fractional sub-di_usion", Fractals, 12(01), pp. 23{32 (2004). 9. Mainardi, F. and Pagnini, G. The role of the Fox- Wright functions in fractional sub-di_usion of distributed order", Journal of Computational and Applied Mathematics, 207(2), pp. 245{257 (2007). 10. Stojanovi_c, M. Fractional relaxation equations of distributed order", Nonlinear Analysis: Real World Applications, 13(2), pp. 939{946 (2012). 11. Langlands, T.A.M. Solution of a modi_ed fractional di_usion equation", Physica A: Statistical Mechanics and Its Applications, 367, pp. 136{144 (2006). 12. Podlubny, I., Fractional di_erential equations: an introduction to fractional derivatives, fractional di_erential equations, to methods of their solution and some of their applications 198, Academic press, San Diego, USA (1998). 13. Li, Y., Sheng, H., and Chen, Y.Q. On distributed order integrator/di_erentiator", Signal Processing, 91(5), pp. 1079{1084 (2011). 14. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F., Tables of Integral Transforms, 1 (1954). 15. Haubold, H.J., Mathai, A.M., and Saxena, R.K. Mittag-Le_er functions and their applications", Journal of Applied Mathematics, 2011, Article ID 298628 (2011). DOI: 10.1155/2011/298628 16. Goreno, R., Kilbas, A.A., Mainardi, F., and Sergei, V. Rogosin, Mittag-Le_er Functions, Related Topics and Applications, Springer, Berlin, Germany (2014). 17. Sandev, T., Chechkin, A.V., Korabel, N., Kantz, H., Sokolov, I.M., and Metzler, R. Distributed-order di_usion equations and multifractality: Models and solutions", Physical Review E, 92(4), 042117 (2015). 18. Jiao, Z., Chen, Y.Q., and Podlubny, I., Distributed- Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer Briefs in Electrical and Computer Engineering (2012). 19. Chrouda, M.B., El Oued, M., and Ouerdiane, H. Convolution calculus and applications to stochastic di_erential equations", Soochow Journal of Mathematics, 28(4), pp. 375{388 (2002). 20. Gossett, E., Discrete Mathematics with Proof, John Wiley & Sons (2009). 21. Abate, J. and Whitt, W. A uni_ed framework for numerically inverting Laplace transforms", INFORMS Journal on Computing, 18(4), pp. 408{421 (2006).