en عمومى General پژوهشي Research paper <div><br> <br> &nbsp;</div> <div style="text-align: justify;"><span class="fontstyle0">In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and embeded in the equation to achieve a linear system of equations which give the expansion coefficients of the solution. We prove that the rate of the convergence is </span><span class="fontstyle2">O</span><span class="fontstyle0">(</span><span class="fontstyle2">h</span>^{<span class="fontstyle3">2</span>}<span class="fontstyle0">) and </span><span class="fontstyle2">O</span><span class="fontstyle0">(</span><span class="fontstyle2">h</span>^{<span class="fontstyle3">4</span>}<span class="fontstyle0">) for these two different bases under some conditions. Two examples are solved and the results are compared with those of block pulse functions method (BPFs) to show the accuracy and reliability of the methods.</span></div> Generalized hat functions, Improved hat functions, Stochastic operational matrix, Stochastic Volterra-Fredholm integral equation, Brownian motion. 145 164 http://ijmsi.ir/browse.php?a_code=A-10-1636-2&slc_lang=en&sid=1 N. Momenzade neda_momenzadeh@yahoo.com 100319475328460010025 100319475328460010025 No Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran A. R. Vahidi alrevahidi@yahoo.com 100319475328460010026 100319475328460010026 Yes Department of Mathematics, College of Science, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran E. Babolian babolian@khu.ac.ir 100319475328460010027 100319475328460010027 No Department of Mathematics, College of Science, Yadegar-e-Emam Khomeyni (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran