Presentation of two models for the numerical analysis of fractional integro-differential equations and their comparison

Document Type: research paper


Professor, Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Mazandaran, Iran


In this paper, we exhibit two methods to numerically solve the fractional integro differential equations and then proceed to compare the results of their applications on different problems. For this purpose, at first shifted Jacobi polynomials are introduced and then operational matrices of the shifted Jacobi polynomials are stated. Then these equations are solved by two methods: Caputo fractional
derivative method and the Riemann-Liouville fractional integral method. In the both method, a set of linear or nonlinear algebraic equations are achieved using collocation technique. Tow presented methods are implemented on some test problems. Numerical results explain the high performance of tow methods. Note that all calculations have been done by Mathematica software. Numerical results show that it should be used the first method when the exact solution of differential equation is a polynomial and the second method should be used when the exact solution of differential equation is a transcendental function.


Article Title [Persian]

ارائه دو مدل برای تحلیل عددی جواب معادلات دیفرانسیل-انتگرال کسری و مقایسه آنها

Author [Persian]

  • محمود بهروزی فر
استاد، دانشکده علوم پایه ، دانشگاه صنعتی نوشیروانی بابل، مازندران، ایران.
Abstract [Persian]

در این مقاله، دو مدل برای تحلیل عددی جواب معادلات دیفرانسیل-انتگرال کسری ارائه می دهیم و سپس به مقایسه نتایج بکارگیری آنها بر روی مسائل متنوع می پردازیم. برای این منظور ابتدا ماتریس‌های عملیاتی چند جمله‌ای‌های ژاکوبی
را بیان کرده و سپس هر مسئله را با دو روش حل می‌نماییم: روش ماتریس عملیاتی مشتق مرتبه کسری کاپوتو و روش ماتریس عملیاتی انتگرال مرتبه کسری ریمان-لیوویل. در هر دو روش با استفاده از تکنیک نقطه گذاری به یک دستگاه معادلات جبری خطی یا غیرخطی خواهیم رسید با کمک روش تکراری نیوتن حل می شوند. روش‌های ارائه شده روی چند مثال پیاده سازی شده است و نتایج عددی حاصل بیانگر کارایی بالای هر دو روش است.
لازم به ذکر است که تمامی محاسبات با کمک نرم افزار متمتیکا
انجام شده است. نتایج عددی نشان می‌دهد که برای معادله دیفرانسیلی که جواب آن بصورت چند جمله ای می باشد بهتر است از روش اول و در معادله غیر خطی که جواب آن بصورت تابع متعالی است بهتر است که روش دوم استفاده شود.

Keywords [Persian]

  • معادله دیفرانسیل-انتگرال کسری
  • ماتریس عملیاتی
  • ماتریس عملیاتی مشتق مرتبه کسری کاپوتو
  • ماتریس عملیاتی انتگرال مرتبه کسری ریمان-لیوویل
  • تکنیک نقطه گذاری
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