روش هم‌محلی برای حل عددی معادلات انتگرال دوبعدی خطی ولترا و اثبات همگرایی آن

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضیات و کاربردها، دانشکده علوم پایه، دانشگاه شاهد، تهران، ایران

چکیده

در این مقاله روش هم‌محلی را برای حل عددی معادلات انتگرال دوبعدی ولترا تعمیم می‌دهیم. برای این منظور ابتدا وجود و یکتایی جواب این نوع معادلات را ثابت کرده و یک نمایش هسته حلال برای جواب آنها ارائه‌ می‌کنیم. سپس روش هم‌محلی با استفاده از چند‌جمله‌ای‌های قطعه‌ای را برای حل معادلات مذکور تعمیم داده و دستگاه معادلات جبری متناظر را به‌دست‌ آورده و نشان‌ می‌دهیم دستگاه مذکور دارای جواب یکتاست. هم‌چنین همگرایی روش را ثابت کرده و مرتبه‌ی همگرایی روش را با اثبات قضیه‌ای به‌دست‌ می‌آوریم. سرانجام چند مثال عددی برای نشان‌دادن کارایی روش و تأیید نتایج نظری به‌دست‌ آمده، ارائه می‌کنیم.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Collocation method for numerical solution of two-dimensional Linear Volterra integral equations and prove its convergence

نویسندگان [English]

  • Abolfazl Tari Marzabad
  • Somayeh Kazemi
Department of Mathematics, Faculty of Basic Science, Shahed University, Tehran, Iran
چکیده [English]

In this paper, we extend the collocation method for the numerical solution of two-dimensional Volterra integral equations. For this purpose, we first prove the existence and uniqueness of the solution of these types of equations and present a resolvent kernel representation for their solution. Then, we extend the collocation method using piecewise polynomials to solve the mentioned equations and obtain the corresponding algebraic system of equations and show that the system has a unique solution. We also prove the convergence of the method and obtain the order of convergence of the method by proving a theorem. Finally, we present some numerical examples to show the efficiency of the method and confirm the obtained theoretical results.

کلیدواژه‌ها [English]

  • Two-dimensional Volterra integral equations
  • Collocation method
  • Piecewise polynomials
  • convergence
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