روش محاسباتی برای حل معادلات دیفرانسیل تصادفی تأخیری از مرتبه کسری
محورهای موضوعی : آماربهروز پارسا مقدم 1 , زینب سلامت مستقیم 2 , الهام السادات هاشمی زاده 3
1 - گروه ریاضی، دانشکده علوم ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
2 - گروه ریاضی، دانشکده علوم ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
3 - گروه ریاضی، واحد کرج، دانشگاه آزاد اسلامی، کرج، ایران
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.
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.
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