یک الگوریتم بهینه‌سازی ماتریسی عملگری جدید برای حل معادله‌ی انتشار و انتقال غیرخطی کسری مرتبه‌ی متغیر زمانی

نوع مقاله: مقاله پژوهشی

نویسندگان

1 گروه ریاضی کاربردی، دانشگاه شهرکرد

2 گروه ریاضی، دانشگاه یاسوج

چکیده

در این مقاله، یک الگوریتم بهینه‌سازی جدید و موثر برای حل معادله‌ی غیرخطی انتشار و انتقال کسری مرتبه‌ی متغیر زمانی با استفاده از مفهوم مشتق کسری مرتبه‌ی متغیر از نوع کاپوتو ارائه می‌دهیم. برای بدست آوردن جواب، ابتدا رده‌ای از چندجمله‌ای‌های تعمیم‌یافته را معرفی، سپس ماتریس‌های عملگری وابسته به آن‌ها ساخته می‌شوند. در تکنیک بهینه‌سازی ارائه شده، جواب مسئله‌ی مورد بررسی بر حسب چندجمله‌ای‌های تعمیم‌یافته با ضریب آزاد و پارامترهای کنترل‌کننده‌ی نامعلوم توسیع داده می‌شود. مهمترین فایده‌ی این روش تبدیل معادله‌ی دیفرانسیل با مشتقات جزئی کسری مرتبه‌ی متغیر زمانی به یک سیستم از معادلات جبری غیرخطی می‌باشد. در ادامه، ضرایب آزاد و پارامترهای کنترل‌کننده به طور بهینه با مینیمم‌سازی خطای جواب تقریبی بدست خواهند آمد. تحلیل همگرایی روش ارائه شده با بدست آوردن قضیه‌ای جدید در خصوص توابع دو متغیره تضمین می‌شود. در پایان، نتایج عددی بدست آمده نشان خواهند داد که الگوریتم ارائه شده برای حل معادله‌ی ذکر شده موثر و از میزان دقت بسیار بالایی برخوردار است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A new optimization operational matrix algorithm for solving nonlinear variable-order time fractional convection-diffusion equation

نویسندگان [English]

  • Hossein Hassani 1
  • Eskandar Naragirad 2
1 Department of َApplied Mathematics, Shahrekord University, Shahrekord , Iran
2 Department of Mathematics, Yasouj University, Yasouj, Iran.
چکیده [English]

In this paper, a new and effective optimization algorithm is proposed for solving the nonlinear time fractional convection-diffusion equation with the concept of variable-order fractional derivative in the Caputo sense. For finding the solution, we first introduce the generalized polynomials (GPs) and construct the variable-order operational matrices. In the proposed optimization technique, the solution of the problem under consideration is expanded in terms of GPs with unknown free coefficients and control parameters. The main advantage of the presented method is to convert the variable-order fractional partial differential equation to a system of nonlinear algebraic equations. Also, we obtain the free coefficients and control parameters optimally by minimizing the error of the approximate solution. Finally, the numerical examples confirm the high accuracy and efficiency of the proposed method in solving the problem under study.

کلیدواژه‌ها [English]

  • Nonlinear variable-order time fractional convection-diffusion equation
  • Operational matrices
  • Optimization algorithm
  • Generalized polynomials (GPs)
  • Control parameters
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