یک روش عددی دقیق برای حل مسأله نفوذ کسری مرتبه متغیر

نویسندگان
فرناز خیرخواه
مجتبی حاجی پور
دانشگاه صنعتی سهند
چکیده
در این مقاله برای حل یک مسأله مقدار مرزی برگرفته از معادله نفوذ کسری مرتبه-متغیر یک روش عددی با مرتبه دقت بالا طراحی و پیاده‌سازی شده است. این معادله حاوی یک مشتق کسری با مرتبه متغیر نسبت به زمان و یک مشتق صحیح مرتبه دوم نسبت به مکان است. برای ساختن این روش جدید، از یک فرمول تفاضلات متناهی فشرده برای گسسته­سازی مکانی و از یک فرمول گرانوالد-لتنیکوف انتقال‌یافته وزن‌دار برای گسسته‌سازی زمانی استفاده شده است. نشان داده شده است که این روش نسبت به مکان و زمان به ترتیب دارای نرخ همگرایی چهار و دو است. همچنین حل‌پذیری، پایداری و همگرایی روش ساخته شده بررسی شده است. به منظور نشان دادن کارایی و نرخ همگرایی بالای این روش، چند مثال عددی و برخی نتایج مقایسه‌ای ارائه شده است.
کلیدواژه‌ها
: مشتق کسری مرتبه-متغیر
معادله نفوذ کسری
روش عددی

عنوان مقاله English

An accurate numerical method for solving the variable-order fractional diffusion problem

نویسندگان English

Farnaz Kheirkhah
Mojtaba Hajipour
Sahand University of Technology
چکیده English

In this paper, a high-order numerical method is designed and implemented to solve a boundary value problem governed by the variable-order fractional diffusion equation. This equation contains a variable-order fractional time-derivative and a second-order spatial-derivative. To develop this novel method, a compact finite difference formula and a weighted shifted Grunwald-Letnikov operator are used for spatial and temporal discretization, respectively. It is shown that this method is of fourth- and second-order of convergence accuracy in spatial and time directions, respectively. Also, the solvability, stability and convergence of the peresent method are investigated. To verify the efficiency and high accuracy of this method, some numerical examples and comparative results are presented.

کلیدواژه‌ها English

Variable-order fractional derivative
fractional diffusion equation
Numerical method
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