# A fractional Malthusian growth model with variable order using an optimization approach

### Abstract

The objective is to study the population's growth with a fractional differential equation. The order of the fractional derivative is a function depending on time and the goal is to determine the fractional order function that better fits the given data. The model is than tested to describe the world population growth and of some countries. All the numerical experiments were done in MATLAB, using the routines lsqcurvefit, fminunc and spline.### References

R. Almeida, N.R.O. Bastos and M.T.T. Monteiro, Modelling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, vol. 39, pp. 4846-4855, 2016.

R.L. Bagley and P.J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, vol. 30, pp. 133–155,

Y. Cao, Y. Li,W. Ren and Y. Chen, Distributed coordination of networked fractional-order systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 40, pp. 362–370, 2010.

M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529–539, 1967.

Y. Chen, L. Liu, B. Li and Y. Sun, Numerical solution for the variable order linear cable equation with Bernstein polynomials, Applied Mathematics and Computation, vol. 238, pp. 329–341, 2014.

Gapminder World, http://www.gapminder.org/data/

J.F. Gomez Aguilar, M.G. Lopez L´opez, V.M. Alvarado Martinez, J. Reyes–Reyes and M. Adam–Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Physica A. Statistical Mechanics and its Applications, vol. 447, pp. 467–481, 2016.

R. Herrmann, Folded potentials in cluster physics - a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals, Journal of Physics A: Mathematical and Theoretical, vol. 46, pp. 405203, 2013.

H.A. Jalab, R.W. Ibrahim and A. Ahmed, Image denoising algorithm based on the convolution of fractional Tsallis entropy with the Riesz fractional derivative, Neural Computing and Applications, vol. 28, Supplement 1, pp. 217–223, 2017.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

Y.-l Li, H.-q Tang and H.-x. Chen, Fractional-order derivative spectroscopy for resolving simulated overlapped Lorenztian peaks, Chemometrics and Intelligent Laboratory Systems, vol 107, pp. 83–89, 2011.

J. Lu, J. Shen, J. Cao and J. Kurths, Consensus of Networked Multi-agent Systems with Delays and Fractional-Order Dynamics, In Consensus and Synchronization in Complex Networks, (Ed. L. Kocarev) 69–110 Springer, 2013.

F. Meral, T. Royston and R. Magin, Fractional calculus in viscoelasticity: An experimental study, Communications in Nonlinear Science and Numerical Simulation, vol. 15, pp. 939–945, 2010.

C. Moler, J. Little, and S. Bangert, Matlab User’s Guide - The Language of Technical Computing, The MathWorks, Sherborn, Mass. (2001).

X. Pan, Y. Ye and J. Wang, Fractional directional derivative and identification of blur parameters of motion-blurred image, Signal, Image and Video Processing, vol. 8, pp. 565–576, 2014

I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms and Special Functions, vol. 1, pp. 277–300, 1993.

M. F. Silva, J. A. T. Machado and A. M. Lopes, Fractional order control of a hexapod robot, Nonlinear Dynamics, vol. 38, pp. 417–433, 2004.

T. Steihaug, The Conjugate Gradient Method and Trust Regions in Large Scale Optimization, SIAM Journal on Numerical Analysis, vol. 20, pp. 626–637, 1983.

D. Tavares, R. Almeida and D.F.M. Torres, Caputo derivatives of fractional variable order: numerical approximations, Communications in Nonlinear Science and Numerical Simulation, vol. 35, pp. 69–87, 2016.

United Nations, The World at Six Billion Off Site, Table 1, World Population From Year 0 to Stabilization, 5, 1999.

R.A. Waltz, J. L. Morales, J. Nocedal and D. Orban, An interior algorithm for nonlinear optimization that combines line search and trust region steps, Mathematical Programming, vol. 107, pp. 391–408, 2006.

D. Wollscheid and A. Lion, The benefit of fractional derivatives in modelling the dynamics of filler-reinforced rubber under large strains: a comparison with the Maxwell-element approach, Computational Mechanics, vol. 53, pp. 1015–1031, 2014.

J. Xu and J. Li, Stochastic dynamic response and reliability assessment of controlled structures with fractional derivative model of viscoelastic dampers, Mechanical Systems and Signal Processing, vol. 72–73, pp. 865–896, 2016.

M. Zayernouri and G.E. Karniadakis, Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, Journal of Computational Physics, vol. 293, pp. 312–338, 2015.

J. Zhong and L. Li, Fractional-order system identification and proportional-derivative control of a solid-core magnetic bearing, ISA Transactions, vol. 53, pp. 1232–1242, 2014.

P. Zhuang, F. Liu, V. Anh and I. Turner, Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term, SIAM Journal on Numerical Analysis, vol. 47, pp. 1760–1781, 2009.

C. Zopf, S.E. Hoque and M. Kaliske, Comparison of approaches to model viscoelasticity based on fractional time derivatives, Computational Materials Science, vol. 98, pp. 287–296, 2015.

*Statistics, Optimization & Information Computing*,

*6*(1), 4-11. https://doi.org/10.19139/soic.v6i1.465

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