ﺣﻞ ﻋﺪدي ﻣﻌﺎدﻟﻪ اﻧﺘﮕﺮال ﺗﺼﺎدﻓﯽ ﻏﯿﺮﺧﻄﯽ ﻧﻮع ﺳﻮم ﺑﻪ ﮐﻤﮏ ﻣﺎﺗﺮﯾﺲ ﻋﻤﻠﯿﺎﺗﯽ ﺑﺎ اﺳﺘﻔﺎده از ﭼﻨﺪ ﺟﻤﻠﻪ اي ﻫﺎي ﺑﺮﻧﺸﺘﺎﯾﻦ

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

در اين مقاله به حل عددي معادلات انتگرال تصادفي نوع سوم با استفاده از ماتريس هاي عملياتي چندجمله اي هاي برنشتاين مي پردازيم. براي اين منظور ابتدا ماتريس عملياتي و ماتريس عملياتي تصادفي چندجمله اي هاي برنشتاين را به دست مي آوريم. تمامي توابع موجود درمعادلۀ انتگرال تصادفي نوع سوم را بااستفاده از سري چندجمله اي هاي برنشتاين تقريب مي زنيم و سپس از ماتريس هاي عملياتي چندجمله اي هاي برنشتاين استفاده مي كنيم. با اين كار حل معادلۀ انتگرال تصادفي نوع سوم به حل يك دستگاه معادلات جبري تبديل مي شود،كه با روش نيوتن مي توان آن را حل كرد. تجزيه وتحليل همگرايي روش مطرح مي شود و براي بررسي دقت و كارايي روش دو مثال عددي ارائه مي نماييم.

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.