An optimal analytical method for nonlinear boundary value problems based on method of variation of parameter

Document Type: research paper


1 Department of Mathematics, Neka Branch, Islamic Azad University, Neka, Iran

2 Department of Mathematics Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran


In this paper, the authors present a modified convergent analytic algorithm for the solution of nonlinear boundary value problems by means of a variable parameter method and briefly, the method is called optimal variable parameter method. This method, based on the embedding of a parameter and an auxiliary operator, provides a computational advantage for the convergence of the approximate solutions of nonlinear differential equations. The developed convergence has been shown and its details are discussed. Additionally, a convenient method is considered for selecting an optimal value of the auxiliary parameter, via minimizing the residual error over the domain of problem.
The effectiveness of the method and the accuracy of the proposed algorithm are illustrated by the implementation of physical problems such as Sturm-Liouville problem, Airy equation, and Quantum mechanical harmonic oscillator problem.
The numerical results and obtained demonstrate clearly reflect the accuracy of the method and its convergence.


Article Title [Persian]

یک روش تحلیلی بهینه برای حل مسائل مقدار مرزی غیرخطی بر پایه روش تغییر پارامتر

Authors [Persian]

  • رحمت درزی 1
  • بهرام عاقلی 2
1 گروه ریاضی، دانشگاه آزاد اسلامی، واحد نکا، نکا، ایران
2 گروه ریاضی، دانشگاه آزاد اسلامی، واحد قائمشهر، قائمشهر، ایران
Abstract [Persian]

در این مقاله، نویسندگان یک الگوریتم تحلیلی همگرای اصلاح شده را برای جواب مسائل مقدار مرزی و مقدار اولیه غیرخطی به واسطه روش تغییر پارامتر ارائه می­کنند و بطور خلاصه روش تغییر پارامتر بهینه می­نامند. این روش براساس تعبیه یک پارامتر و یک عملگر کمکی، یک مزیت محاسباتی برای همگرایی جواب­های تقریبی معادلات دیفرانسیل غیرخطی مهیا می­کند. همگرایی توسعه یافته مذ کور نشان داده شده و جزییات آن نیز مورد بحث قرار می­گیرد.
علاوه بر این، یک روش مناسب برای انتخاب مقدار بهینه پارامتر کمکی در نظر گرفته می­شود که تحت مینیمم­سازی خطا روی دامنه مساله می­باشد. موثر بودن روش و دقت الگوریتم پیشنهادی، با اجرا روی مسائل فیزیکی همچون مساله استورم- لیوویل، مساله ایری و مساله نوسانگر هارمونیک کوانتومی نشان داده می­شود. نتایج عددی و شکل­های بدست آمده بوضوح دقت الگوریتم و همگرایی آن را منعکس می­کند.

Keywords [Persian]

  • روش تغییر پارامتر بهینه
  • مسائل مقدار مرزی
  • مساله استورم- لیوویل
  • مساله ایری و مساله نوسانگر هارمونیک مکانیکی کوانتومی


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