Document Type : Article

**Authors**

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

**Abstract**

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.

**Keywords**

- Analytic solution
- Distributed order differential equation
- Reimann-Liouville fractional derivative
- Mittag-Leffler function
- Relaxation process

**Main Subjects**

References:

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2. Bellouquid, A., Nieto, J., and Urrutia, L. "About the kinetic description of fractional diffusion 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 Physical Mathematics, 6(2) (2015). DOI: 10.4172/2090-0902.1000146. https://projecteuclid.org/euclid.jpm/1504144903.

4. Saxena, R.K. and Pagnini, G. "Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion 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., Goren o, R., and Stojanovic, 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 diffusion equations via the Fox H functions", Science Asia, 39, pp. 57-66 (2013).

8. Naber, M. "Distributed order fractional sub-diffusion", Fractals, 12(01), pp. 23-32 (2004).

9. Mainardi, F. and Pagnini, G. "The role of the Fox- Wright functions in fractional sub-diffusion of distributed order", Journal of Computational and Applied Mathematics, 207(2), pp. 245-257 (2007).

10. Stojanovic, 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 modified fractional diffusion equation", Physica A: Statistical Mechanics and Its Applications, 367, pp. 136-144 (2006).

12. Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential 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/differentiator", 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-Leffler functions and their applications", Journal of Applied Mathematics, 2011, Article ID 298628 (2011). DOI: 10.1155/2011/298628.

16. Goren o, R., Kilbas, A.A., Mainardi, F., and Sergei, V. Rogosin, Mittag-Leffler 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 diffusion 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 differential 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 unified framework for numerically inverting Laplace transforms", INFORMS Journal on Computing, 18(4), pp. 408-421 (2006).

Transactions on Computer Science & Engineering and Electrical Engineering (D)

June 2020Pages 1384-1397