یک الگوریتم سه مرحله‌ای با دقت فوق‌بهینه برای حل معادلات برگرز-هاکسلی وبرگرز-فیشر در حالت کلی

نوع مقاله: مقاله پژوهشی

نویسنده

گروه ریاضی، دانشگاه کردستان

چکیده

در این مقاله یک روش جدید سه مرحله‌ای برای حل عددی دسته‌ای از معادلات دیفرانسیل با مشتقات جزئی موسوم به برگرز-هاکسلی و برگرز-فیشر در حالت کلی ایجاد خواهد شد. همان‌گونه که می‌دانیم حداکثر دقت روش اسپلاین مکعبی برای درونیابی برابر است، اما این دقت هنگام حل معادلات دیفرانسیل به روش کلاسیک افت می‌کند. در اینجا با تعریف شرایط انتهایی مناسب برای اسپلاین مکعبی و با ساختن یک الگوریتم سه مرحله‌ای تصحیح- تصحیح، تقریب‌هایی با مرتب دقت برای جواب مسائل از نوع برگرز-هاکسلیوبرگرز-فیشر ایجاد خواهیم نمود. همگرایی و کران خطای روش را با استفاده از مفهوم تابع گرین به ‌تفصیل مورد بررسی قرار خواهیم داد. همچنین برای تایید کران‌های خطای به دست آمده، چند مثال عددی نیز ارائه خواهیم نمود. در نهایت سعی می‌کنیم با مقایسه نتایج عددی به‌دست آمده با نتایج ارائه شده در مراجع دیگر برتری و کارایی روش را به صورت عملی نمایش دهیم.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

A three step superconvergent algorithm for the solution of generalized Burgers’-Huxley and Burgers’-Fisher equations

نویسنده [English]

  • Mohammad Ghasemi
Department of Mathematics, University of Kurdistan, Sanandaj, Iran
چکیده [English]

In this paper, a new three-step method based on cubic spline will be construct to the numerical solution of a class of partial differential equations well-known as Burgers’-Huxley and Burgers’-Fisher. As we know, the maximum order achieved using cubic spline for interpolating is , but this order is reduced when it is used for the solution of differential equations. Here we will find an  superconvergent approximation for the solution of Burgers’-Huxley and Burgers’-Fisher equations by defining some proper end conditions and constructing a three step deferred-correction algorithm. We will discuss the convergence and error bounds of the method using Green’s function definition in details. In addition, to verify the obtained error bounds, some numerical examples will be presented. Finally, we will try to show the applicability and efficiency of the method by comparing the results with other existing methods.

کلیدواژه‌ها [English]

  • Spline
  • Superconvergence approximations
  • Burgers’-Huxley equation
  • Burgers’-Fisher equation
  • Green’s function
  • Convergence
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