روشی ماتریسی به همراه موجک‌های لژاندر در حل برخی معادلات انتگرالی غیرخطی از نوع اول با آنالیز خطا و وجود جواب

نوع مقاله : مقاله پژوهشی

نویسنده

گروه ریاضی، دانشگاه سراوان، سراوان، ایران

چکیده

از آنجایی که بیشتر معادلات انتگرال غیرخطی از نوع اول، بد‌ وضع هستند، در این پژوهش سعی شده روشی بر اساس موجک‌های لژاندر برای حل دسته خاصی از آنها ارائه شود. روش پیشنهادی بدین صورت است که در ابتدا تمام اجزای معادله به فرم ماتریس بر اساس توابع موجک لژاندر بیان می‌شوند، سپس این ماتریسها در معادله اصلی جایگزین می‌گردند. این روش به همراه روش هم‌محلی معادله را به یک سیستم از معادلات جبری تعدیل می‌کند که این سیستم جبری با روش‌های عددی شناخته شده و یا تحلیلی قابل حل است. برای نشان دادن همگرایی روش، کران بالای خطا با استفاده از چند لم و قضیه محاسبه شده است. علاوه بر این، آنالیز پایداری، وجود و منحصربه‌فرد بودن جواب نیز مورد بررسی قرار گرفته است. به منظور نشان دادن قابلیت اطمینان روش پیشنهادی، نتایج عددی همراه با مقایسه داده شده‌اند. در عمل به منظور بررسی پایداری، در مثال‌ها عدد شرط را محاسبه کردیم و همچنین نرخ همگرایی محاسبه شده و همگرایی توسط آزمون نسبت در نمودارهایی نمایش داده شده است. نتایج نشان می‌دهند که این روش بسیار مؤثر و دقیق است و در برخی موارد به جواب دقیق مساله همگرا می‌شود.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Matrix method with Legendre wavelets for solving some nonlinear integral equations of the first kind with error analysis and existence of the solution

نویسنده [English]

  • Mohsen Riahi Beni
Department of Mathematics, University of Saravan, Saravan, Iran
چکیده [English]

Since most integral equations of the first kind are ill-posed, in this article we tried to present a method based on Legendre wavelets to solve some of them. The proposed method is that at first all the components of the equation are expressed in matrix form based on Legendre wavelet functions, then these matrices are replaced in the main equation. This method, along with the collocation method, adjusts the equation to a system of algebraic equations, which can be solved by usual numerical or analytical methods. To show the convergence of the method, some computable error bounds are obtained by preparing some theorems and lemmas. Moreover, the stability analysis, the existence, and uniqueness of the solution are investigated. Numerical results with comparisons are given to confirm the reliability of the proposed method. In practice, in order to check the stability, we calculated the condition number in the examples, and also the convergence rate was calculated and the convergence was displayed in graphs by the ratio test. The outcomes reveal that this method is very effective and more accurate than those of the literature. Also, in many cases, one can get the exact solution to the problem.

کلیدواژه‌ها [English]

  • Integral equation of first kind
  • Nonlinear equations
  • Legendre wavelets
  • Error analysis
  • Matrix method

مراجع

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سابقه مقاله

  • تاریخ دریافت: 02 اردیبهشت 1403
  • تاریخ بازنگری: 19 شهریور 1403
  • تاریخ پذیرش: 18 آبان 1403

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