Numerical Solution of Fokker-Planck-Kolmogorov Time Fractional Differential Equations Using Haar Wavelet Method and convergence and error analysis
محورهای موضوعی : Numerical Analysis
شعبان محمدی
1
(Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.)
S. Reza Hejazi
2
(Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.)
کلید واژه: Haar wavelet, Time fractional differential equations, Fokker-Planck-Kolmogorov differential equations,
چکیده مقاله :
The purpose of this paper is to present an efficient numerical method for finding numerical solutions Fokker-Planck-Kolmogorov time-fractional differential equations. The Haar Wave was the first to be introduced. The Fokker-Planck-Kolmogorov time-fractional differential equation is converted to the linear equation using the Haar 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 results revealed that the suggested numerical method is highly accurate and effective when used to Fokker-Planck-Kolmogorov time fraction 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. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis.
هدف از این مقاله ارائه یک روش عددی کارآمد برای یافتن جواب های عددی معادلات دیفرانسیل کسری-زمانی فوکر-پلانک-کلموگروف است. موج هار اولین بار بود که معرفی شد. معادله دیفرانسیل کسری زمان فوکر-پلانک-کلموگروف با استفاده از ماتریس عملیات موجک هار در این تکنیک به معادله خطی تبدیل می شود. این روش این مزیت را دارد که حل آن ساده است. شبیه سازی با استفاده از نرم افزار MATLAB انجام شده است. در نهایت، استراتژی پیشنهادی برای حل مشکلات خاص مورد استفاده قرار گرفت. نتایج نشان داد که روش عددی پیشنهادی هنگام استفاده از معادلات دیفرانسیل کسر زمانی فوکر-پلانک-کلموگروف بسیار دقیق و موثر است. نتایج برخی از مثال‌های عددی به صورت جدول و نمودار مستند شده است تا کارایی و دقت روش پیشنهادی را توضیح دهد. علاوه بر این، برای همگرایی تکنیک پیشنهادی، نابرابری در زمینه تحلیل خطا به دست می‌آید
Bect, J., Baili, H. Fleury, G.(2006), “Generalized Fokker-Planck equation for piecewise-diffusion processes with boundary hitting resets”, MTNS, pp.1360-1367.
Chen,C., Tsay, Y., Wu, T.(1977),“Walsh operational matrices for fractional calculus and their application to distributed parameter systems”, Journal of Franklin Institute, vol503,pp.267–284.
Chen, C., Hsiao, C.(1997), “Haar wavelet method for solving lumped and distributed parameter systems”, IEE Proceedings–Control Theory and Applications,vol.144, pp. 87–94.
Dal, F.(2009) “Application of variational iteration method to fractional hyperbolic partial differential equations”, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Article ID 824385 , pp.1-10.
Floris, C. (2013), “Numeric Solution of the Fokker-Planck-Kolmogorov Equation”, Engineering, Vol. 5, pp. 975-988.
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.
Ismail, M., Rehman, M.u. and Saeed, U. (2020), “Green-Haar method for fractional partial differential equations”, Engineering Computations, Vol.37, pp. 1473-1490.
Jafari, H., daftardar-Gejji, V. (2006), “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition”, Applied Mathematics and Computation, vol.180, pp. 488–497.
Jumarie, G.(2006), “Modified Riemann - Liouville derivative and fractional Taylor series of nondifferentiable fractional further results”, Camputers and Mathematics with Applications,vol.51, pp.1367-1376.
Jumarie, G,(2007), “Fractional partial differential equations and modified Riemann – Liouville derivative new methods for solution”, J. Appl. Math. Computing, vol.24,pp.31 - 48.
Kurulay, M., Ibrahimoglu, B., Bayram,M.(2010), “Solving a system of nonlinear fractional partial differential equations using three dimensional differential transform method”, International Journal of the Physical Sciences, 5 (2010) 906-912.
Lepik, U.(2009), “Solving fractional integral equations by the Haar wavelet method”, Applied Mathematics and Computation,vol. 214, pp. 468–478.
Li, Y., Zhao, W.(2010), “Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations”, Applied Mathematics and Computation, vol.216,pp. 2276-2285.
Meerschaert, M., Tadjeran, C.(2006), “Finite difference approximations for two-sided space fractional partial differential equations”, Applied Numerical Mathematics, vol56, pp.80–90.
Momani, S., Odibat, Z. (2006), “Analytical approach to linear fractional partial differential equations arising in fluid mechanics”, Physics Letters A, vol.355, pp. 271–279.
Odibat, Z., Momani, S.(2008), “A generalized differential transform method for linear partial differential equations of fractional order”, Applied Mathematics Letters, vol.21,pp.194–199.
Podlubny,I.(1999) “Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractiorlal Differential Eqnations, to Methods of their Solution and Some of Their Applications”, Mathematics in science and engineering, Academic Press, New York, 198.
Paraskevopoulos, P., Sparis,P., Mouroutsos, S.(1985), “The Fourier series operational matrix of integration”, International Journal of System Science,vol.16, pp.171–176.
Razzaghi, M., Yousefi, S, (2000), “Legendre wavelets direct method for
variational problems”, Math. Comp. Simulat, vol.53, pp.185-192.
Spencer, B,. Bergman, L.(1993), “On the numerical solution of the Fokker-Planck equation for nonlinear stochastic system”, Nonlinear Dynamics, vol.4, pp.357–372.
Shih, D., Kung, F., Chao, C.(1986), “Laguerre series approach to the analysis of a linear control system incorporation observers”, International Journal of Control, vol.43,pp. 123–128.
Saeed, U. and Umair, M. (2019), “A modified method for solving non-linear time and space fractional partial differential equations”, Engineering Computations, Vol. 36, pp. 2162-2178.
Wu, J.(2009), “wavelet operational method for solving fractional partial differential equations numerically”, Applied Mathematics and Computation,vol.214, pp,31–40.
Wang,L.(2014).Haar wavelet method for solving fractional partial differential equations numerically,Appl. Math. Comput. 227 66–76(2014)
Yulita Molliq R,. Noorani, M., Hashim,I.(2009), “Variational iteration method for fractional heat- and wave-like equations”, Nonlinear Analysis: Real World Applications, vol.10, pp.1854–1869.
Zorzano, M., Mais, H., Vazquez, L.(1999), “Numerical solution of two dimensional Fokker--Planck equations”, Applied mathematics and computation, Vol. 98, pp. 109-117.
