|تعداد مشاهده مقاله||7,639,240|
|تعداد دریافت فایل اصل مقاله||5,858,416|
Bernoulli Wavelet Method for Numerical Solution of Fokker-Planck-Kolmogorov Time Fractional Differential Equations
|Computational Sciences and Engineering|
|دوره 2، شماره 1، تیر 2022، صفحه 143-163 اصل مقاله (488.24 K)|
|نوع مقاله: Original Article|
|شناسه دیجیتال (DOI): 10.22124/cse.2021.21106.1021|
|Shaban Mohammadi* 1؛ S .Reza Hejazi2؛ Hossein Seifi3|
|1Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran|
|2Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.|
|3Bachelor of Mathematics, Sarvelayat Education Organization, Chakaneh, Iran|
|The purpose of this paper is to present a wavelet method for numerical solutions Fokker-Planck-Kolmogorov time-fractional differential equations with initial and boundary conditions. The authors was employed the Bernoulli wavelets for the solution of Fokker-Planck-Kolmogorov time-fractional differential equation. We calculated the Bernoulli wavelet fractional integral operation matrix of the fractional order and the upper error boundary for the Riemann‐Levilleville fractional integral operation matrix and the Bernoulli wavelet fractional integral operation matrix. The Fokker-Planck-Kolmogorov time-fractional differential equation is converted to the linear equation using the Bernoulli wavelet operation matrix in this technique. This method has the advantage of being simple to solve. The simulation was carried out using MATLAB software. Finally, the proposed strategy was used to solve certain problems. the Bernoulli wavelet and Bernoulli fraction of the fractional order, the Bernoulli polynomial, and the Bernoulli fractional functions were introduced. Explaining how functions are approximated by fractional-order Bernoulli wavelets as well as fractional-order Bernoulli functions. The Bernoulli wavelet fractional integral operational matrix was used to solve the Fokker-Planck-Kolmogorov fractional differential equations. The results for some numerical examples are documented in table and graph form to elaborate on the efficiency and precision of the suggested method. The results revealed that the suggested numerical method is highly accurate and effective when used to Fokker-Planck-Kolmogorov time fraction differential equations|
|Fokker-Planck-Kolmogorov differential equations؛ Bernolii wavelet؛ fractional integration|
 K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
 K.S. Miller, B. Ross, An introduction to the fractional calculus and Fractional differential equations , Wiley, New York, (1993).
 R.L. Bagley, P.J. Torvik, Fractional calculus in the transient analysis of vis- coelastically damped structures, AIAA Journal 23 (1985) 918–925.
 E. Keshavarz, Y. Ordokhani, M. Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model. 38 (2014) 6038-6051.
 G. Arfken, Mathematical methods for physicists, Third eddition, Academic Press, San Dieqo, (1985)
 J.E. Kreyszig, Introductory Fractional Analysis with Applications, John Wiley and Sons Press, New York, (1978).
 S. Yuzbasi, Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials, Comput. Appl. Math. 219 (2013) 6328-6343.
 H. Jafari, S.A. Yousefi, M.A. Firoozjaee, S. Momani, C.M. Khalique, Application of Leg- endre wavelets for solving fractional differential equations, Comput. Math. Appl. 62 (2011) 1038-1045
 S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model. 37 (7) (2013) 5498-5510.
 R.T. Baillie, Fractional integration in econometrics, Journal of Econometrics, 73 (1996) 5-59.
 F. Mainardi, Fractional calculus”some basic problems in continuum and statistical mechanics”. in: Carpinteri Aand Mainardi F (eds) Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, (1997).
 Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews, 50 (1997) 15-67.
 T.S. Chew, Fractional dynamic of interfaces between soft-nanoparticales and rough substrates, Physics Letters A, 342 (50) (2005) 148-155.
 L. Gaul, P. Klein, S. Kemple, Damping description involving fractional operators, Mech. Syst. Signal. Pr, 5 (1991) 81-88.
 L. Suarez, A. Shokooh, An eigenvector erpansion method for the solution of motion containing fractional derivatives , J. Appl. Mech, 64 (1999) 629-735.
 I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Application, Academic press. New York (1998).
 S. Momani, K. AlKhaled, Numerical solutions for systems of Fractional Differential Equations by the decomposition method, Appl. Math. Comput, 162 (3) (2005) 1351-1365.
 M. Meerschaert, C. Tadjeran, Finite difference approximations for two- sided spacefractional partial differential equations, Appl. Numer. Math, 56 (1) (2006) 80-90.
 Z. Odibat, N. Shawaghfeh, Generalized Taylor’s formula, Appl. Math. Comput, 186 (1) (2007) 286-293.
 A. Arikoglu, I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fract, 40(2) (2009) 521-529.
 I. Hashim,O .Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun, Nonlinear Sci. Numer. Simul, 14 (3) (2009) 674-684.
 M. Razzaghi, G. Elnagar, Linear quadratic optional control problems via shifted Legendre state parametrization, Nonlinear Int. J. Sci, 25 (1994) 393-399.
 H. Marzban, M. Razzaghi, Hybrid Fractions Approach for Linearly constrained Quadratic Optimal Control problems, Appl. Math. Model, 27 (2003) 393-399.
 M. Razzaghi, M. Razzaghi, Instabilities in the solution of heat condition problem usind Taylor series and alternative approaches, J. Frankl. Inst, 326 (1989) 683-690.
 Hejazi, S.R., Habibi, N., Dastranj,E., Lashkarian, E. (2020), “Numerical approximations for time-fractional Fokker-Planck-Kolmogorov equation of geometric Brownian motion”, Journal of Interdisciplinary Mathematics, Vol(23), pp.1387-1403.
 B. F. Spencer Jr. and L. A. Bergman, On the numerical solution of the Fokker-Planck equation for nonlinear stochastic system, Nonlinear Dynamics, 4, 357–372, (1993).
 C. Floris, "Numeric Solution of the Fokker-Planck-Kolmogorov Equation," Engineering, Vol. 5 No. 12, 2013, pp. 975-988.
 M. Zorzano, H. Mais, L. Vazquez, Numerical solution of two dimensional Fokker--Planck equations, Applied mathematics and computation, Vol. 98, No. 2-3, pp. 109-117, 1999.
 J. Bect, H. Baili, G. Fleury, Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets, in Proceeding of.
 P. Rahimkhani, Y. Ordokhani, E. Babolian, Fractional-Order Bernoulli Wavelets and their applications,Appl. Math. Model. 40 (2016) 8087-8107.
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