روش محاسباتی برای حل معادلات دیفرانسیل تصادفی تأخیری از مرتبه کسری

محورهای موضوعی : آمار

بهروز پارسا مقدم ¹ (گروه ریاضی، دانشکده علوم ریاضی، دانشگاه آزاد اسلامی واحد لاهیجان، لاهیجان، ایران)
زینب سلامت مستقیم ² (گروه ریاضی، دانشکده علوم ریاضی، دانشگاه آزاد اسلامی واحد لاهیجان، لاهیجان، ایران)
الهام السادات هاشمی زاده ³ (عضو هیات علمی گروه ریاضی دانشگاه آزاد کرج و معاون آموزشی دانشکده)

تاریخ دریافت : 1397/10/14 تاریخ پذیرش : 1398/08/12 تاریخ انتشار : 1399/08/01

کلید واژه: Fractional calculus, Stochastic calculus, Bilinear spline interpolation, Stochastic delay differential equations,

چکیده مقاله :

سیستم های دینامیکی در بسیاری از شاخه های علوم و صنعت غالبا با انواع مختلف از نویزهای محیطی آشفته می شوند. آنالیز این سیستم ها از اهمیت ویژه ای مابین پژوهشگران برخودار می باشند. در این مقاله، ما روشی برای محاسبه جواب تقریبی معادلات دیفرانسیل غیر خطی تصادفی تاخیری از مرتبه کسری حاصل از حرکت برآونی را ارائه می‌دهیم. مشتقات از مرتبه کسری از نوع کاپوتو در نظر گرفته شده است. اساس روش محاسباتی بر پایه درون‌یابی اسپلاین دو خطی و تقریب تفاضلات متناهی می باشد. مرتبه همگرایی روش پیشنهادی با استفاده از نرم میانگین مجذور اثبات شده است و دقت روش از منظر میانگین خطای مطلق و مرتبه همگرایی تجربی آنالیز شده است. روش ارایه شده برای تعیین شاخص‌های آماری در مدل‌های گومبرتزیان و نیکولسون بکار گرفته شده است. معادله دیفرانسیل تاخیری و تصادفی گومبرتزیان از مرتبه کسری مدلسازی شده است برای توصیف رشد فرایند سرطان و معادله دیفرانسیل تاخیری و تصادفی نیکولسون از مرتبه کسری برای بیان دینامیک جمعیت آشفتگی های نیکولسون در محیط زیست، فرموله شده است.

چکیده انگلیسی:

Dynamic systems in many branches of science and industry are often perturbed by various types of environmental noise. Analysis of this class of models are very popular among researchers. In this paper, we present a method for approximating solution of fractional-order stochastic delay differential equations driven by Brownian motion. The fractional derivatives are considered in the Caputo sense. The computational method is based on bilinear spline interpolation and finite difference approximation. The convergence order of the proposed method investigated in the mean square norm and the accuracy of proposed scheme is analyzed in the perspective of the mean absolute error and experimental convergence order. The proposed method is considered in determining statistical indicators of Gompertzian and Nicholson models. The fractional stochastic delay Gompertzian equation is modeled for describing the growth process of a cancer and the fractional stochastic delay Nicholson equation is formulated for explaining a population dynamics of the well-known Nicholson blowflies in ecology.

منابع و مأخذ:

 [1] D. Henderson, P. Plaschko, Stochastic Differential Equations in Science and Engineering, World Scientific, 2006.

[2] S. Singh, S. S. Ray, Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion, International Journal of Biomathematics 10 (07) (2017) 1750103.

[3] K. Hattaf, M. Mahrouf, J. Adnani, N. Yousfi, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Physica A: Statistical Mechanics and its Applications 490 (2018) 591-600.

[4] G. I. Zmievskaya, A. L. Bondareva, T. V. Levchenko, G. Maino, Computational stochastic model of ions implantation, AIP Publishing LLC, 2015.

[5] N. Gillard, E. Belin, F. Chapeau-Blondeau, Stochastic antiresonance in qubit phase estimation with quantum thermal noise, Physics Letters A 381 (32) (2017) 2621-2628.

[6] A. Farhadi, G. H. Erjaee, M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Computers & Mathematics with Applications 73 (9) (2017) 2066-2075.

[7] X. Chen, P. Hu, S. Shum, Y. Zhang, Dynamic stochastic inventory management with reference price effects, Operations Research 64 (6) (2016) 1529-1536.

[8] A. Einstein, On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat, Annalen der Physik 17 (1905) 549-560.

[9] M. von Smoluchowski, Zur kinetischen theorie der Brownschen molekularbewegung und der suspensionen, Annalen der Physik 326 (14) (1906) 756-780.

[10] P. Langevin, Sur la theorie du mouvement Brownien, CR Acad. Sci. Paris 146 (530-533) (1908) 530.

[11] G. E. Uhlenbeck, L. S. Ornstein, On the theory of the Brownian motion, Physical Review 36 (5) (1930) 823.

[12] Z. S. Mostaghim, B. P. Moghaddam, and H. S. Haghgozar, Computational technique for simulating variable-order fractional Heston model with application in US stock market, Mathematical Sciences, vol. 12, no. 4, pp. 277–283, Oct. 2018.

[13] Z. S. Mostaghim, B. P. Moghaddam, and H. S. Haghgozar, Numerical simulation of fractional-order dynamical systems in noisy environments, Computational and Applied Mathematics, vol. 37, no. 5, pp. 6433–6447, Aug. 2018.

[14] G. S. Ladde, L. Wu, Development of modified geometric Brownian motion models by using stock price data and basic statistics, Nonlinear Analysis: Theory, Methods & Applications 71 (12) (2009) e1203{e1208. doi:10.1016/j.na.2009.01.151.

[15] D. N. Tien, Fractional stochastic differential equations with applications to finance, Journal of Mathematical Analysis and Applications 397 (1) (2013) 334-348.

[16] B. B. Mandelbrot, J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review 10 (4) (1968) 422-437. doi:10.1137/1010093.

[17] A. Farhadi, G. H. Erjaee, M. Salehi, Derivation of a new Merton's optimal problem presented by fractional stochastic stock price and its applications, Computers & Math- ematics with Applications 73 (9) (2017) 2066-2075.

[18] Z.-G. Yu, V. Anh, Y. Wang, D. Mao, J. Wanliss, Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition, Journal of Geophysical Research: Space Physics 115 (A10) (2010) n/a-n/a.

[19] T. H.Thao, On some classes of fractional stochastic dynamical systems. East-West Journal of Mathematics, 15 (1) (2013).

[20] F. Mirzaee, N. Samadyar, Application of hat basis functions for solving two-dimensional stochastic fractional integral equations, Computational and Applied Mathematics (2018) 1-18.

[21] L. Yan, X. Yin, Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 22 (11) (2017). 1–21.

[22] P. Tamilalagan, P. Balasubramaniam, Moment stability via resolvent operators of fractional stochastic differential inclusions driven by fractional Brownian motion, Applied Mathematics and Computation 305 (2017) 299-307.

[23] S. A. Asogwa, E. Nane, Intermittency fronts for space-time fractional stochastic partial differential equations in (d+1) dimensions, Stochastic Processes and their Applications 127 (4) (2017) 1354-1374.

[24] X. L. Ding, J. Nieto, Analytical solutions for multi-time scale fractional stochastic differential equations driven by fractional brownian motion and their applications, Entropy 20 (1) (2018) 63.

[25] D. Chowdhury, Statistical physics of vehicular traffic and some related systems, Physics Reports 329 (4-6) (2000) 199-329.

[26] X. Mao, Stochastic differential equations, in: Stochastic Differential Equations and Applications, Elsevier, 2011, pp. 47-90.

[27] X. X. Liao, X. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Analysis and Applications 14 (2) (1996) 165-185.

[28] Z. Yan, F. Lu, Existence of an optimal control for fractional stochastic partial neutral integro-differential equations with infinite delay, Journal of Nonlinear Sciences and Applications 08 (05) (2015) 557-577.

[29] C. Rajivganthi, P. Muthukumar, B. G. Priya, Successive approximation and optimal controls on fractional neutral stochastic differential equations with Poisson jumps, Optimal Control Applications and Methods 37 (4) (2015) 627-640.

[30] C. Chatfield, Statistics for technology: a course in applied statistics, Routledge, 2018.

[31] C. Li, A. Chen, J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, Journal of Computational Physics 230 (9) (2011) 3352-3368.

[32] E. I. Dureman, C. Boden, Fatigue in simulated car driving, Ergonomics 15 (3) (1972) 299-308.

[33] F. M. Atay (Ed.), Complex Time-Delay Systems, Springer Berlin Heidelberg, 2010.

[34] B. P. Moghaddam, L. Zhang, A. M. Lopes, J. A. Tenreiro Machado, and Z. S. Mostaghim, Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, Stochastics, (2019) 1–18.

 [35] S. Banerjee, Mathematical modeling: models, analysis and applications, Chapman and Hall/CRC, 2014.

[36] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach Sci. Publishers, 1993.

[37] Quarteroni, Alfio, and Alberto Valli. Numerical approximation of partial differential equations. Vol. 23. Springer Science & Business Media, 2008.

 [38] Byrne, H. M. (1997). The effect of time delays on the dynamics of avascular tumor growth. Mathematical Biosciences, 144(2), 83–117. 

[39] Mazlan, M. S. A., Rosli, N., Azmi, N. S., & Bahar, A. (2015). Modelling the Cervical Cancer Growth Process by Stochastic Delay Differential Equations. Sains Malaysiana, 44(8), 1153–1157. 

[40] Shaikhet, L. (2013). Lyapunov Functionals and Stability of Stochastic Functional Differential Equations.