حل عددی معادله انتگرال تصادفی غیر خطی نوع سوم به کمک ماتریس عملیاتی با استفاده از چند جمله ای های برنشتاین

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضیات دانشگاه آزاد اسلامی واحد کرج، کرج، ایران

چکیده

در این مقاله به حل عددی معادلات انتگرال تصادفی نوع سوم با استفاده از ماتریس های عملیاتی چندجمله ای های برنشتاین می پردازیم. برای این منظور ابتدا ماتریس عملیاتی و ماتریس عملیاتی تصادفی چندجمله ای های برنشتاین را به دست می آوریم. تمامی توابع موجود درمعادلۀ انتگرال تصادفی نوع سوم را بااستفاده از سری چندجمله ای های برنشتاین تقریب می زنیم و سپس از ماتریس های عملیاتی چندجمله ای های برنشتاین استفاده می کنیم. با این کار حل معادلۀ انتگرال تصادفی نوع سوم به حل یک دستگاه معادلات جبری تبدیل می شود،که با روش نیوتن می توان آن را حل کرد. تجزیه وتحلیل همگرایی روش مطرح می شود و برای بررسی دقت و کارایی روش دو مثال عددی ارائه می نماییم.

کلیدواژه‌ها


عنوان مقاله [English]

Numerical Solution of Nonlinear Stochastic Integral Equation of the Third Kind by Stochastic Operational Matrix Based on Bernstein Polynomials

نویسندگان [English]

  • M Khodabin
  • parvaneh jami
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
چکیده [English]

In the present research, we were numerically solved nonlinear stochastic integral equation of the third kind by stochastic operational matrix based on Bernstein polynomials. For this aim, we were obtaining the Bernstein polynomials operation matrix and the stochastic operation matrix. Also we approximated all the functions in the Volterra integral equation of the third kind using the Bernstein polynomials series and then use the Bernstein polynomials operation matrix. By doing this, solving the third kind of stochastic Volterra integral equation turns into solving a system of algebraic equations, which could be a more suitable solution. Then we were analysed the convergence of the proposed method and provide several numerical examples to evaluate the accuracy and efficiency of this method. The current results were obtained by running a program written in Mathematica software.

کلیدواژه‌ها [English]

  • Bernstein polynomials
  • Stochastic integral equation of the third kind
  • Stochastic integration operational matrixes
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