حل عددی معادله برگرز کسری از مرتبه متغیر با ترکیب روشهای المان طیفی دارای نرخ جریمه تطبیقی و کرانک-نیکلسون لیپ فراگ

نویسندگان
پیام مختاری اقدمی
نگیسا ایازی
چکیده
هدف از این مقاله ارائه یک رویکرد عددی ترکیبی به منظور تقریب جواب های معادله برگرز کسری یک ‌بعدی مبتنی بر مشتق ریمان-لیوویل از مرتبه متغیر می باشد. الگوریتم عددی مد نظر طوری طراحی گردیده است که برای گسسته سازی مکانی از روش المان طیفی بر پایه رویکرد هم مکانی و برای گسسته سازی زمانی از رویکرد کرانک- نیکلسون لیپ فراگ استفاده می شود. بعلاوه، به منظور پایداری ماتریس مشتق بدست آمده از پیاده سازی رویکرد المان طیفی تکنیک اعمال نرخ جریمه تطبیقی پیشنهاد می شود که در آن ضریب جریمه بصورت پویا و بر اساس تغییرات موضعی مرتبه مشتق کسری تنظیم می شود. کارآیی رویکرد اتخاذ شده توسط مثالهای متنوع بررسی و در تمامی موارد دقت بالا، پایداری و کارآیی محاسباتی روش پیشنهادی تایید شده اند.
کلیدواژه‌ها
معادله برگرز کسری از مرتبه متغیر
روش المان طیفی
جریمه تطبیقی
روش کرانک- نیکلسون لیپ فراگ

عنوان مقاله English

Numerical Solution of the Variable-Order Fractional Burger’s Equation Using a Hybrid Spectral Element Method with Adaptive Penalty Rate and Crank-Nicolson Leap-Frog Scheme

نویسنده English

Negisa Ayazi
چکیده English

The primary objective of this study is to develop a hybrid numerical scheme for approximating the solutions of the one-dimensional fractional Burgers’ equation governed by a variable-order Riemann–Liouville derivative. In the proposed framework, spatial discretization is accomplished via a spectral element method based on the collocation approach, whereas temporal discretization is handled using a Crank–Nicolson Leap-Frog scheme. To enhance the stability of the derivative matrix resulting from the spectral element formulation, an adaptive penalty enforcement strategy is employed, wherein the penalty coefficient is dynamically adjusted in response to the local variations in the fractional derivative order. The accuracy, stability, and computational efficiency of the proposed method are validated through a series of numerical experiments, demonstrating its robustness and effectiveness across diverse test cases.

کلیدواژه‌ها English

Variable-Order Fractional Burger’s equation
Spectral Element Method
Adaptive Penalty
Crank-Nicolson Leap-Frog Scheme
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