×
Singh, Abhishek Kumar; Mehra, Mani; Gulyani, Samarth

Learning parameters of a system of variable order fractional differential equations. (English) Zbl 07776993

Numer. Methods Partial Differ. Equations 39, No. 3, 1962-1976 (2023).
Summary: We introduce a machine learning framework that uses the differential evolution algorithm in combination with Adam-Bashforth-Moulton method to learn the parameters in a system of variable order fractional differential equations. In this work, we present out developments with regards to taking care of a class of problem: data-driven discovery of system of variable order fractional differential equations. The main advantage of the proposed framework is that it works even if data corresponding to only one of the variables in the system of equations is given. We illustrate the working of our framework on several Examples including modeling the 2014–15 Ebola outbreak in Africa via fractional SEIR (susceptible, exposed, infected, removed) model.
{© 2021 Wiley Periodicals LLC.}

Cited in 5 Documents

MSC:

65-XX Numerical analysis
35-XX Partial differential equations

Keywords:

Adam-Bashforth-Moulton method; differential evolution method; Ebola outbreak; SEIR epidemic model; variable order fractional differential equations
Cite Review PDF
Full Text: DOI

References:

[1] A.Dénes and A. B.Gumel, Modeling the impact of quarantine during an outbreak of ebola virus disease, Infect. Dis. Model.4 (2019), 12-27.
[2] G.Diaz and C. F. M.Coimbra, Nonlinear dynamics and control of a variable order oscillator with application to the van der pol equation, Nonlinear Dyn.56(1-2) (2009), 145-157. · Zbl 1170.70012
[3] K.Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn.71(4) (2013), 613-619.
[4] K.Diethelm, N. J.Ford, and A. D.Freed, A predictor‐corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn.29(1‐4) (2002), 3-22. · Zbl 1009.65049
[5] K.Diethelm and A. D.Freed, On the solution of nonlinear fractional‐order differential equations used in the modeling of viscoplasticity, in Scientific computing in chemical engineering II, Springer, Berlin, Heidelberg, 1999, 217-224.
[6] Y.Ding and H.Ye, A fractional‐order differential equation model of hiv infection of cd4+ t‐cells, Math. Comput. Model.50(3‐4) (2009), 386-392. · Zbl 1185.34005
[7] S.Esmaeili and M.Shamsi, A pseudo‐spectral scheme for the approximate solution of a family of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul.16(9) (2011), 3646-3654. · Zbl 1226.65062
[8] S.Esmaeili, M.Shamsi, and Y.Luchko, Numerical solution of fractional differential equations with a collocation method based on müntz polynomials, Comput. Math. Appl.62(3) (2011), 918-929. · Zbl 1228.65132
[9] R.Gorenflo, F.Mainardi, E.Scalas, and M.Raberto, Fractional calculus and continuous‐time finance iii: The diffusion limit, in Mathematical Finance, Springer, Basel, 2001, 171-180. · Zbl 1138.91444
[10] M. M.Khader and M. M.Babatin, Numerical treatment for solving fractional sirc model and influenza a, Comput. Appl. Math.33(3) (2014), 543-556. · Zbl 1328.92077
[11] A. A. A.Kilbas, H. M.Srivastava, and J. J.Trujillo, Theory and Applications of Fractional Differential Equations, Vol 204, Elsevier B.V., Amsterdam, 2006. · Zbl 1092.45003
[12] N.Kumar and M.Mehra, Legendre wavelet collocation method for fractional optimal control problems with fractional bolza cost, Numer. Methods Partial Differ. Equations37(2) (2021), 1693-1724. https://doi.org/10.1002/num.22604. · Zbl 07776039 · doi:10.1002/num.22604
[13] N.Kumar and M.Mehra, Collocation method for solving nonlinear fractional optimal control problems by using hermite scaling function with error estimates, Optim. Control Appl. Methods42 (2021), 417-444. https://doi.org/10.1002/oca.2681. · Zbl 1468.49032 · doi:10.1002/oca.2681
[14] C.Li and Y.Wang, Numerical algorithm based on adomian decomposition for fractional differential equations, Comput. Math. Appl.57(10) (2009), 1672-1681. · Zbl 1186.65110
[15] C.Li and F.Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, Florida, 2015. · Zbl 1326.65033
[16] S.Ma, Y.Xu, and W.Yue, Numerical solutions of a variable‐order fractional financial system, J. Appl. Math.2012 (2012), 417942. https://doi.org/10.1155/2012/417942. · Zbl 1251.91070 · doi:10.1155/2012/417942
[17] R. L.Magin, Fractional Calculus in Bioengineering, Vol 2, Begell House Redding, Danbury, 2006.
[18] V.Mehandiratta and M.Mehra, A difference scheme for the time‐fractional diffusion equation on a metric star graph, Appl. Numer. Math.158 (2020), 152-163. · Zbl 1452.65171
[19] V.Mehandiratta, M.Mehra, and G.Leugering, Existence and uniqueness results for a nonlinear caputo fractional boundary value problem on a star graph, J. Math. Anal. Appl.477(2) (2019), 1243-1264. · Zbl 1421.34005
[20] V.Mehandiratta, M.Mehra, and G.Leugering, Fractional optimal control problems on a star graph: Optimality system and numerical solution, Math. Control Relat. Fields11(1) (2020), 189. · Zbl 1460.34017
[21] M.Mehra and R. K.Mallik, Solutions of differential-difference equations arising from mathematical models of granulocytopoiesis, Differ. Equations Dyn. Syst.22(1) (2014), 33-49. · Zbl 1325.92016
[22] K. S.Patel and M.Mehra, Fourth order compact scheme for space fractional advection‐diffusion reaction equations with variable coefficients, J. Comput. Appl. Math. 380 (2020), 112963. · Zbl 1440.65097
[23] I.Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, New York, 1999. · Zbl 0924.34008
[24] M.Raissi, P.Perdikaris, and G. E.Karniadakis, Machine learning of linear differential equations using gaussian processes, J. Comput. Phys.348 (2017), 683-693. · Zbl 1380.68339
[25] S. G.Samko, Fractional integration and differentiation of variable order, Anal. Math.21(3) (1995), 213-236. · Zbl 0838.26006
[26] A.Shukla, M.Mehra, and G.Leugering, A fast adaptive spectral graph wavelet method for the viscous burgers’ equation on a star‐shaped connected graph, Math. Methods Appl. Sci.43(13) (2020), 7595-7614. · Zbl 1512.65318
[27] A. K.Singh and M.Mehra, Uncertainty quantification in fractional stochastic integro‐differential equations using legendre wavelet collocation method, in Computational Science - ICCS 2020. Lecture Notes in Computer Science, Vol 12138, V.Krzhizhanovskaya (ed.), G. Zavodszky, M. H. Lees, J. J. Dangarra, P. M. A. Sloot, S. Brissos, J. Teixeira et al., Eds., Springer, Cham, Switzerland, 2020, 58-71.
[28] R.Storn and K.Price, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim.11(4) (1997), 341-359. · Zbl 0888.90135
[29] H. G.Sun, A.Chang, Y.Zhang, and W.Chen, A review on variable‐order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fractional Calculus Appl. Anal.22(1) (2019), 27-59. · Zbl 1428.34001
[30] H. G.Sun, W.Chen, and Y. Q.Chen, Variable‐order fractional differential operators in anomalous diffusion modeling, Physica A: Stat. Mech. Appl.388(21) (2009), 4586-4592.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.