حل عددی مسأله ریلی- استوکس کسری با استفاده از توابع پایه شعاعی مکان- زمان

محورهای موضوعی : آمار

نفیسه نقره ای ¹ , اصغر کرایه چیان ² , علیرضا سهیلی ³

1 - گروه ریاضیات کاربردی، دانشکده علوم ریاضی، دانشگاه فردوسی مشهد، مشهد، ایران
2 - گروه ریاضیات کاربردی، دانشکده علوم ریاضی، دانشگاه فردوسی مشهد، مشهد، ایران
3 - گروه ریاضیات کاربردی، دانشکده علوم ریاضی، دانشگاه فردوسی مشهد، مشهد، ایران

تاریخ دریافت : 1398/11/04 تاریخ پذیرش : 1399/07/23 تاریخ انتشار : 1400/07/01

کلید واژه: ‎Gaussian radial basis function, ‎Single exponential ‎transformation, ‎Space-time formulation, ‎Fractional calculus, ‎Sinc quadrature rule,

چکیده مقاله :

در این مقاله، جواب مسأله دو بعدی ریلی- استوکس برای یک جریان گرمایی درجه دوم تعمیم یافته با مشتق کسری ‏‎را‎‏ تقریب می‌زنیم. ‏این تقریب بر پایه استفاده از توابع ‏پایه شعاعی (‏RBFs‏) مکان- زمان و روش انتگرال گیری عددی سینک می‌باشد. در این روش، از تابع پایه ‏شعاعی گاوسین استفاده شده و بین متغیرهای زمان و مکان تمایز قائل نمی‌شویم و نقاط هم‌محلی‏، هم شامل مختصات ‏زمان و هم شامل مختصات ‏مکان هستند.‏ از روش انتگرال گیری عددی سینک با تبدیل نمایی یگانه برای تقریب قسمت انتگرالی مشتق کسری استفاده ‏‏می‌کنیم. ‏مشتق کسری، ریمان- لیوویل انتخاب شده است.روش ارائه شده روی دو مثال با مقادیر مختلف برای مرتبه مشتق کسری، پیاده سازی شده که نتایج حاصل، اثر بخشی روش را تأیید می‌کند ‏و نشان می‌دهد که با استفاده از تعداد کمی از نقاط هم‌محلی برای تابع پایه شعاعی می‌توان نتایج دقیقی بدست آورد‏.‏ لازم به ذکر است که تمامی ‏محاسبات با کمک نرم‎ ‎افزار متمتیکا انجام شده است.‏

چکیده انگلیسی:

In this paper, we approximate the solution of two-dimensional Rayleigh-Stokes problem ‎for a heated generalized second grade fluid with fractional derivatives. This approximation is ‎based on the space-time radial basis functions (RBFs) and the Sinc quadrature rule. In this ‎method, we use Gaussian radial basis function and don't distinguish between time and place ‎variables and the collocation points have both the coordinates of time and space. We use the ‎Sinc quadrature rule with single exponential transformation to approximate the integral part of ‎fractional derivatives. The chosen fractional derivatives is Riemann – Liouville.‎This method is implemented on two examples with different values of the fractional ‎derivative order. Obtained results illustrate the effectiveness of our method and sh ow that ‎one can obtain accurate results with a small number of the collocation points for the radial ‎basis function. It should be noted that all calculations in this paper have been done using ‎Mathematica software.‎

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