Método de Monte Carlo aplicado ao Cálculo Fracionário

Autores

  • L. N. Ferreira Federal University of Rio Grande
  • M. J. Lazo

DOI:

https://doi.org/10.5540/tcam.2022.023.02.00243

Palavras-chave:

Cálculo Fracionário, Monte Carlo, Python.

Resumo

O presente trabalho analisa e desenvolve um método para resolver equações diferenciais fracionárias utilizando o Método de Monte Carlo. Uma simulação numérica é realizada para algumas equações diferenciais, comparando os resultados com o que existe na literatura matemática. A linguagem Python é usada para criar modelos computacionais.

Referências

M.N. Akhtar, M.H. Durad & A. Ahmed. Solving initial value ordinary differential equations by Monte Carlo method. Proc. IAM, 4 (2015), 149–174.

W.F. Ames. “Numerical methods for partial differential equations”. Academic press (2014).

W. Bauer. The monte carlo method. Journal of the Society for Industrial and Applied Mathematics, 6(4) (1958), 438–451.

K. Diethelm. “The Analysis of Fractional Differential Equations”. Lectures Notes in Mathematics. Springer Verlag, London (2004).

K. Diethelm, N.J. Ford, A.D. Freed & Y. Luchko. Algorithms for the fractional calculus: a selection of numerical methods. Computer methods in applied mechanics and engineering, 194(6-8) (2005), 743–773.

W.L. Dunn & J.K. Shultis. “Exploring monte carlo methods”. Elsevier (2011).

G. Evans, J. Blackledge & P. Yardley. “Numerical methods for partial differential equations”. Springer Science & Business Media (2012).

R. Hilfer. “Applications of fractional calculus in physics”. World scientific (2000).

M.H. Kalos & P.A. Whitlock. “Monte carlo methods”. John Wiley & Sons (2009).

A. Keller, S. Heinrich & H. Niederreiter. “Monte carlo and quasi-monte carlo methods 2006”. Springer (2007).

R.L. Magin. “Fractional calculus in bioengineering”. Begell House Redding (2006).

F. Mainardi. “Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models”. World Scientific (2010).

R. Metzler & J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 339(1) (2000), 1–77.

G.M. Mittag-Leffler. Sur la nouvelle fonction Ea (x). CR Acad. Sci. Paris, 137(2) (1903), 554–558.

S. Mordechai. “Applications of Monte Carlo method in science and engineering”. London (2011).

K. Oldham & J. Spanier. “The fractional calculus theory and applications of differentiation and integration to arbitrary order”. Elsevier (1974).

I. Podlubny. Geometric and physical interpretation of fractional integration and fractional differentiation. arXiv preprint math/0110241, (2001).

M. Rivero, J.J. Trujillo, L. V´azquez & M.P. Velasco. Fractional dynamics of populations. Applied Mathematics and Computation, 218(3) (2011), 1089–1095.

J. Sabatier, O.P. Agrawal & J.T. Machado. “Advances in fractional calculus”. Springer (2007).

J.W. Thomas. “Numerical partial differential equations: finite difference methods”, volume 22. Springer Science & Business Media (2013).

M. Weilbeer. “Efficient numerical methods for fractional differential equations and their analytical background”. Papierflieger Clausthal-Zellerfeld, Germany (2006).

Downloads

Publicado

2022-06-27

Como Citar

Ferreira, L. N., & Lazo, M. J. (2022). Método de Monte Carlo aplicado ao Cálculo Fracionário. Trends in Computational and Applied Mathematics, 23(2), 243–255. https://doi.org/10.5540/tcam.2022.023.02.00243

Edição

Seção

Artigo Original