The use of radial basis functions by variable shape parameter for solving partial differential equations

Document Type: research paper

Authors

1 Department of Applied Mathematics, Faculty of Basic Science, Islamic Azad University, Science and Research Branch, Tehran, Iran

2 Department of Applied Mathematics, Faculty of Basic Science , Imam Khomeini International University, Qazvin, Iran

Abstract

In this paper, some meshless methods based on the local Newton basis functions are used to solve some time dependent partial differential equations. For stability reasons, used variably scaled radial kernels for constructing Newton basis functions. In continuation, with considering presented basis functions as trial functions, approximated solution functions in the event of spatial variable with collocation method.  Then, with aid of method of lines obtained a system of ordinary differential equations according to solution function in the event of time. Methods applied for solving the nonlinear Burgers’ equation and couple Burgers’ equation. The numerical results show that the proposed method is efficient, accurate and stable.

Keywords


Article Title [Persian]

استفاده از توابع پایه شعاعی با پارامتر شکلی متغیر برای حل معادلات دیفرانسیل جزیی

Authors [Persian]

  • حنانه نوجوان 1
  • سعید عباسبندی 2
  • توفیق الهویرنلو 1
1 گروه ریاضی، دانشکده علوم پایه، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران
2 گروه ریاضی کاربردی، دانشکده علوم پایه، دانشگاه بین المللی امام خمینی (ره)، قزوین، ایران
Abstract [Persian]

در این مقاله از روش­های بدون شبکه مبتنی بر توابع پایه­ای نیوتن موضعی برای حل معادلات دیفرانسیل با مشتقات جزیی وابسته به زمان استفاده شده است. به منظور پایداری بیشتر از هسته­های شعاعی به طور متغیر مقیاس شده برای ساخت توابع پایه­ای نیوتن استفاده شده است. در ادامه با در نظر گرفتن توابع پایه­ای معرفی شده به عنوان توابع آزمون، تابع جواب در راستای متغیر مکان با استفاده از توابع آزمون به روش هم مکانی تقریب زده می­شود. سپس با استفاده از روش خطوط، به دستگاهی از معادلات با مشتقات معمولی بر حسب تابع جواب در راستای متغیر زمان دست یافتیم. روش­های معرفی شده را برای حل معادله غیرخطی برگرز به کار گرفته و با مشاهده نتایج عددی دقت و کارآیی روش مشخص خواهد شد.
 

Keywords [Persian]

  • روش بدون شبکه موضعی
  • روش خطوط
  • توابع پایه‌ای نیوتن
  • هسته شعاعی به طور متغیر مقیاس شده
  • معادله غیرخطی برگرز
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