Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-25T23:14:38.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Cosine Radial Basis Function Neural Networks for Solving Fractional Differential Equations

Part of: Real functions Stochastic processes

Published online by Cambridge University Press:  17 January 2017

Haidong Qu
Haidong Qu*
Affiliation:
Department of Mathematics, Hanshan Normal University, Chaozhou, Guangdong 521041, China
*
*Corresponding author. Email:qhaidong@163.com (H. D. Qu)
Get access

Abstract

In this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

Keywords

Fractional Riccati equationfractional Langevin equationnumerical solutionradial basis function neural network

MSC classification

Secondary: 26A33: Fractional derivatives and integrals 60G22: Fractional processes, including fractional Brownian motion
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 3 , June 2017 , pp. 667 - 679
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Mary, M., Randolph-Gips, , Karayiannis, Nicolaos B., Cosine radial basis function neural networks, Proceedings of the International Joint Conference on Neural Networks, 1 (2003), pp. 96–101.Google Scholar
[2] Mitter, Anderson and Reid, W. T., Riccati Differential Equations, Mathematics in Science and Engineering, Vol. 86. Academic Press, New York, 1972.Google Scholar
[3] Anderson, B. D. and Moore, J. B., Optimal Control-linear Quadratic Methods, Prentice-Hall, New Jersey, 1999.Google Scholar
[4] Mitter, S. K., Filtering and stochastic control: a historical perspective, IEEE Control Syst. Mag. 16(3) (1996), pp. 67–76.Google Scholar
[5] Diethelm, K., Ford, J. M., Ford, N. J. and Weilbeer, W., Pitfalls in fast numerical solvers for fractional differential equations, J. Comput. Appl. Math., 186 (2006), pp. 482–503.Google Scholar
[6] Cang, J., Tan, Y., Xu, H. and Liao, S., Series solutions of non-linear Riccati differential equations with fractional order, Chaos, Solitons and Fractals, 40 (2009), pp. 1–9.Google Scholar
[7] Odibat, Zaid and Momani, S., Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos, Solitons and Fractals, 36 (2008), pp. 167–174.Google Scholar
[8] Momani, S. and Shawagfeh, N., Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput., 182 (2006), pp. 1083–1092.Google Scholar
[9] Kosinski, R. A. and Grabowski, A., Theory and Langevin equations for modeling evacuation processes, Acta Phys. Pol., 2 (2006), pp. 365–377.Google Scholar
[10] Wodkiewicz, K. and Zubairy, M. S., Exact solution of a nonlinear Langevin equation with applications to photoelectron counting and noise-induced instability, J. Math. Phys., 6 (1983), pp. 1401–1404.Google Scholar
[11] Bouchaud, J. P. and Cont, R., A Langevin approach to stock market fluctuations and crashes, Euro. Phys. J., 4 (1998), pp. 543–550.CrossRefGoogle Scholar
[12] Guo, P., Zeng, C., Li, C. P. and Chen, Y. Q., Numerics for the fractional langevin equation driven by the fractional brownian motion, Fract. Calc. Appl. Anal., 16 (2013), pp. 123–141.Google Scholar
[13] Kobelev, V. and Romanov, E., Fractional Langevin equation to describe anomalous diffusion, Prog. Theory Phys., 139 (2000), pp. 470–479.Google Scholar
[14] Coffey, W. T., Kalmykov, Y. P. and Waldron, J. T., The Langevin Equation: with Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering. World Scientific Press, Singapore, 2004.Google Scholar
[15] Eab, C. H. and Lim, S. C., Fractional generalized Langevin equation approach to Single-File diffusion, Phys. A, 389(13) (2010), pp. 2510–2521.CrossRefGoogle Scholar
[16] Lim, S. C., Li, M. and Teo, L. P., Langevin equation with two fractional orders, Phys. Lett. A, 372(42) (2008), pp. 6309–6320.Google Scholar
[17] Ahmad, B. N., Alsaedi, J. J. and El-Shahed, A. M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl., 13 (2012), pp. 599–606.Google Scholar
[18] Kilbsa, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Google Scholar
[19] Podlubny, I., Mittag-Leffler function, File ID 8738, Matlab Central, http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function, 2009.Google Scholar