Stability analysis of distributed order Hilfer-Prabhakar differential equations

Year 2018, Volume: 47 Issue: 2, 299 - 315, 01.04.2018

M. Mashoof A.h. Refahi Sheikhani H. Saberi Naja

https://izlik.org/JA69SP93DF

Abstract

In the current study we presented a distributed order form of Hilfer-Prabhakar (DHP) derivative, which in special cases reduces to the existent definitions of fractional or distributed order derivatives. Moreover, we analyzed the stability of DHP differential equations, which are the generalized form of all previous distributed or fractional differential equations. The obtained results showed that sufficient conditions on asymptotic stability of these systems have been obtained through the generalized properties of Mittag-Leer functions and the Laplace transform. Moreover, a number of conditions on stability analysis of such systems have been introduced by using a new definition of inertia of a matrix.

Keywords

Distributed order fractional derivative , Hilfer-Prabhakar derivative , Asymptotic stability

References

• Aminikhah, H., Refahi Sheikhani, A. and Rezazadeh, H. Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Bol. Soc. Paran. Mat., 34 (2), 213-229, 2016.
• Aminikhah, H., Refahi Sheikhani, A. H. and Rezazadeh, H. Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, Scientia Iranica, 23 (3), 1048-1054, 2016.
• Aminikhah, H., Refahi Sheikhani, A. H. and Rezazadeh, H. Exact solutions of some nonlinear systems of partial dierential equations by using the functional variable method, MATHEMATICA, 56 (2), 103-116, 2014.
• Aminikhah, H., Refahi Sheikhani, A. H. and Rezazadeh, H. Approximate analytical solutions of distributed order fractional Riccati dierential equation, Ain Shams Engineering Journal, Article in press.
• Aminikhah, H., Refahi Sheikhani, A. and Rezazadeh, H. Stability analysis of linear distributed order system with multiple time delays, U.P.B. Sci. Bull., 77 (2), 207-218, 2015.
• Ansari, A. and Refahi Sheikhani, A. New identities for the Wright and the Mittag-Leer functions using the Laplace transform, Asian-European Journal of Mathematics, 7 (3), 1-8, 2014.
• Ansari, A., Refahi Sheikhani, A. and Kordrostami, S. On the generating function ext+yφ(t) and its fractional calculus, Cent. Eur. J. Phys., 11 (10), 1457-1462, 2013.
• Aminikhah. H, Refahi Sheikhani, A. and Rezazadeh, H. Stability Analysis of Distributed Order Fractional Chen System, The Scientic World Journal, 2013 (2013), 1-13, 2013.
• Ansari, A., Refahi Sheikhani, A. and Saberi Naja, H. Solution to system of partial fractional dierential equations using the fractional exponential operators, Math. Meth. Appl. Sci., 35 (1), 119-123, 2012.
• Atanackovic,T.M. A generalized model for the uniaxial isothermal deformation of a vis- coelastic body, Acta Mech., 159 (1), 77-86, 2002.
• Atanackovic, T.M., Budincevic, M. and Pilipovic, S. On a fractional distributed-order os- cillator, J. Phys. A, Math. Gen., 38 (30), 6703-6713, 2005.
• Atanackovic, T.M., Pilipovic, S. and Zorica, D. Time distributed order diusion-wave equa- tion, II. Applications of the Laplace and Fourier transformations, Proc. R. Soc. A, 465 (2106), 1893-1917, 2009.
• Atanackovic, T.M., Pilipovic, S. and Zorica, D. Distributed-order fractional wave equation on a nite domain. Stress relaxation in a rod, Int. J. Eng. Sci., 49 (2), 175-190, 2011.
• Bagley, R. L. and Torvik, P.J. On the existence of the order domain and the solution of distributed order equations, International Journal of Applied Mathematics, 2 (8), 965-987, 2000.
• Blasius. B., Huppert, A. and Stone, L. Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399 (6734), 354-359, 1999.
• Caputo, M. Mean fractional-order-derivatives dierential equations and lters, Annali dell'Universita di Ferrara. Nuova Serie. Sezione VII. Scienze Matematiche, 41 (1), 73-84, 1995.
• Caputo, M. Distributed order dierential equations modelling dielectric induction and dif- fusion, Fractional Calculus Applied Analysis, 4 (4), 421-442, 2001.
• Datta, B. N. Stability and inertia, Linear Algebra and its Application, 302, 563-600, 1999.
• Denisov, V.I., Shvilkin, B.N., Sokolov, V.A. and Vasili'Ev, M.I. Pulsar radiation in post- Maxwellian vacuum nonlinear electrodynamics, Physical Review D - Particles, Fields, Grav- itation and Cosmology, 94 (4), 1-6, 2016.
• Diethelm, K. The Analysis of Fractional Dierential Equations: An Application-Oriented Exposition Using Dierential Operators of Caputo Type, vol. 2004 of Lecture Notes in Math- ematics, Springer, Berlin, Germany, 2010.
• Duy, D. G. Transform Methods for Solving Partial Dierential Equations, CRC Press, 2nd edition, 2004.
• Garra, R., Goreno, R., Polito, F. and Tomovski, Z. Hilfer-Prabhakar derivatives and some applications, Applied Mathematics and Computation, 242, 576-589, 2014.
• Garrappa, R. GrnwaldLetnikov operators for fractional relaxation in HavriliakNegami mod- els, Communications in Nonlinear Science and Numerical Simulation, 38, 178-191, 2016.
• Goreno, R., Kilbas, A. A., Mainardi, F. and Rogosin, S. V. Mittag-Leer functions, related topics and applications, Berlin: Springer, 2014.
• Han, S. K., Kurrer, C. and Kuramoto, Y. Dephasing and bursting in coupled neural oscil- lators, Phys. Rev. Lett., 75 (17), 3190-3193, 1995.
• Hilfer, R. Applications of fractional calculus in physics, World Scientic, Singapore, 2000.
• Hilfer, R. Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284, 399408, 2002.
• Humbert, P. Quelques resultats d'le fonction de Mittag-Leer, C. R. Acad. Sci. Paris, 236 (1953), 1467-1468, 1953.
• Kilbas, A. A., Saigo, M. and Saxena, R.K. Generalized MittagLeer function and general- ized fractional calculus operators, Integr. Transf. Spec. Funct., 15 (1), 31-49, 2010.
• Kilbas, A. A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Dierential Equations, in: North-Holland Mathematical Studies, vol. 204, Elsevier (North- Holland) Science Publishers, Amsterdam, 2006.
• Lakshmanan, M. and Murali, k. Chaos in Nonlinear Oscillators: Controlling and Synchro- nization,World Scientic Series on Nonlinear Science Series A: 13, 1996.
• Matignon, D. Stability results for fractional dierential equations with applications to con- trol processing, Computational Engineering in Systems and Application Multiconference, IMACS, IEEE-SMC, Lille, France, 2 (1996), 963-968, 1996.
• Mittag-Leer, G. Sur la nouvelle fonction E(x), Comptes Rendus de lAcademie des Sci- ences Paris, 137, 554-558, 1903.
• Mittag-Leer, G. Sur la representation analytique d'une branche uniforme d'une fonction monogene (cinquieme note), Acta Math., 29, 101-181, 1905.
• Polito, F. and Scalas, E. A generalization of the space-fractional Poisson process and its connection to some Levy processes, Electronic Communications in Probability, 21, 1-14, 2016.
• Polito, F.and Tomovski, Z. Some properties of Prabhakar-type fractional calculus operators, Fractional Dier. Calc, 6 (1), 73-94, 2016.
• Prabhakar, T. R. A singular integral equation with a generalized Mittag-Leer function in the kernel, Yokohama Math. J., 19 (1971), 7-15, 1971.
• Qian, D., Li, C., Agarwal, R. P. andWong, P. J. Y. Stability analysis of fractional dierential system with Riemann-Liouville derivative, Mathematical and Computer Modelling, 52 (5- 6), 862-874, 2010.
• Refahi Sheikhani, A., Saberi Naja, H., Ansari, A,. and Mehrdoust, F. Analytic study on linear system of distributed order fractional dierential equation,Le Matematiche, 67 (2), 3-13, 2012.
• Rezazadeh, H., Aminikhah, H. and Refahi Sheikhani, A. Stability analysis of Hilfer frac- tional dierential systems, Math. Commun, 21 (1), 45-64, 2016.
• Saberi Naja, H., Refahi Sheikhani, A. and Ansari, A. Stability Analysis of Distributed Order Fractional Dierential Equations, Abstract and Applied Analysis., 2011 (2011), 1- 12, 2011.
• Sparrow, C. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, NY, USA, 1982.
• Srivastava, H. M. and Zhukovsky, K. V. Solutions of Some Types of Dierential Equations and of Their Associated Physical Problems by Means of Inverse Dierential Operators, Th. M. Rassias and V. Gupta (Editors), In book: Mathematical Analysis, Approximation Theory and Their Applications, Springer optimization and its applications series, 573-629, 2016.
• Wiman, A. ber de Fundamental Satz in der Theorie der Funktionen E(x), Acta Math., 29 (1), 191-201. 1905.
• Yu, Y., Li, H., Sha Wang, S. and Yu, J. Dynamic analysis of a fractional-order Lorenz chaotic system, Chaos, Solitons and Fractals, 42 (2), 1181-1189, 2009.
• Zhukovsky, K. V. Exact solution of Guyer-Krumhansl type heat equation by operational method, International Journal of Heat and Mass Transfer, 96, 132-144, 2016.
• Zhukovsky, K. V. Violation of the maximum principle and negative solutions with pulse propagation in Guyer- Krumhansl model, International Journal of Heat and Mass Transfer, 98, 523-529, 2016.
• Zhukovsky, K. V. A method of inverse dierential operators using ortogonal polynomials and special functions for solving some types of dierential equations and physical problems, Moscow Univ. Phys. Bull., 70 (2), 93-100, 2015. Zhukovsky, K. V. The Operational Solution of Fractional-Order Dierential Equations, as well as BlackScholes and Heat-Conduction Equations, Moscow Univ. Phys. Bull., 71 (3), 237-244, 2016. Zhukovsky, K.V. Operational method of solution of linear non-integer ordinary and partial dierential equations, SpringerPlus, 5 (1), 1-25, 2016.
• Zhukovsky, K. V. and Srivastava, H. M. Analytical solutions for heat diusion beyond Fourier law, Applied Mathematics and Computation, 293, 423-437, 2017.
• Zhukovsky, K.V. Operational solution for some types of second order dierential equations and for relevant physical problems, Journal of Mathematical Analysis and Applications, 446 (1), 628-647, 2017.