روش عددی برای یک کلاس از معادله کسری انتگرال- دیفرانسیل کسری مرتبه متغییر با مشتقات کسری آتانگانا- بالینو-کاپوتو

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم، دانشگاه بیرجند، بیرجند، ایران.

چکیده

هدف اصلی ما در این مقاله، بررسی معادله انتگرال-دیفرانسیل کسری مرتبه متغیر شامل مشتقات کسری آتانگانا-بالینو-کاپوتو به‌صورت
\begin{align*}
&\mathfrak{D}_{\alpha(t)}^{ABC}\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY+\int_{0}^{t}u(x,Y).k(x,Y)dY\\
=&f(x,t),
\end{align*}
است. سعی کردیم با استفاده از یک روش عددی مبتنی بر عملگرهای ماتریسی شامل چند‌جمله‌ای چبیشف به حل عددی این معادله بپردازیم. این عملگرهای ماتریسی باعث تبدیل معادله انتگرال-دیفرانسیلی مرتبه کسری به یک سیستم‌ جبرخطی خواهد شد که با حل کردن این معادلات، جواب عددی معادله انتگرال-دیفرانسیل کسری فوق را به‌دست می‌آوریم. برای نشان دادن دقت و کارایی این روش چند مثال‌ عددی را که توسط نرم افزار متلب محاسبه شده است، بیان می‌کنیم. 

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Numerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivative

نویسندگان [English]

  • Hajimohammad Mohammadinejad
  • Hassan Khosravi
Department of Mathematics, Faculty of Science, University of Birjand, Birjand, Iran.
چکیده [English]

In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows:
\begin{align*}
\mathfrak{D}_{\alpha(t)}^{ABC}&\Big[u(x,t).g(x,t)\Big]+\frac{\partial u(x,t)}{\partial t}+\int_{0}^{t}u(x,Y)dY\nonumber\\
&+\int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t),
\end{align*}
We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.

کلیدواژه‌ها [English]

  • Atangana-Baleanu-Caputo fractional derivative
  • Chebyshev polynomials
  • Operational matrixes
  • Fractional integral-differential equations
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