fa روش کم‌ترین مانده تعمیم یافته پیش شرط سازشده برای حل معادله انتشار‌-‌‌‌هم رفت کسری جبر alg علمی پژوهشی کاربردی S <p dir="RTL"><span style="font-family:b nazanin;"><span style="font-size:10.0pt;">&nbsp;&nbsp;&nbsp;در این مقاله معادله انتشار‌-‌‌هم رفت با مشتق مرتبه کسری را در نظر می‌گیریم. برای به‌دست آوردن یک روش عددی، مشتقات کسری موجود در معادله با استفاده از تعاریف گرانوالد‌‌-‌لتنیکوف انتقال‌یافته جای‌گزین می‌شوند.&nbsp; برای بهبود روش عددی ارائه شده، مشتقات جزئی معادله را با روش تفاضل متناهی کرانک‌-‌‌نیکلسون کسری گسسته‌سازی می‌کنیم. هم‌چنین نشان می‌دهیم این روش به‌صورت غیرمشروط پایدار است. سپس با معرفی یک ماتریس پیش‌شرط‌ ساز، روش کم‌ترین مانده تعمیم یافته (</span></span><span dir="LTR"><span style="font-family:times new roman,serif;"><span style="font-size:10.0pt;">GMRES</span></span></span><span style="font-family:b nazanin;"><span style="font-size:10.0pt;">) پیش‌شرط ساز شده را برای حل دستگاه معادلات جبری‌ به‌دست آمده معرفی خواهیم کرد. در پایان با هدف تأیید نتایج نظری، با ارائه یک مثال روش </span></span><span dir="LTR"><span style="font-family:times new roman,serif;"><span style="font-size:10.0pt;">GMRES</span></span></span><span style="font-family:b nazanin;"><span style="font-size:10.0pt;"> پیش‌شرط سازشده‌ ‌را آزمایش می‌کنیم.</span></span><br> <br> &nbsp;</p> <strong>Introduction</strong><br> Fractional&nbsp;differential equations (FDEs)&nbsp; have &nbsp;attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or&nbsp;numerical scheme &nbsp;may be a good approach, particularly, the schemes in numerical linear algebra for solving a system, , emerging&nbsp; by&nbsp; discretizing &nbsp;the partial derivatives, with&nbsp; large and sparse dimensions. In the procedure of solving a specified FDE,&nbsp; if the dimension of&nbsp; the corresponding system of linear equations&nbsp; is small,&nbsp;&nbsp;&nbsp; one can use the direct methods or the classical&nbsp; iterative methods for the analysis of these systems.&nbsp; However,&nbsp; if the dimension is large, then the proposed methods are not effective. In this case, we use variants of&nbsp; the Krylov subspace methods that are more robust&nbsp; with respect to the computer memory and&nbsp; time. The GMRES (Generalized Minimal Residual) is a well-known method&nbsp; based on Krylov subspace that is used to solve a system of sparse linear equations with an non-symmetric matrix. &nbsp;A main drawback of iterative methods is the slowness of convergence rate which depends on the condition number of the corresponding coefficient &nbsp;matrix. If the condition number of the coefficient matrix is small, then the rate of convergence will be faster. So, we try to convert the original system to another equivalent system, in which the condition number of its coefficient matrix becomes small. A preconditioner matrix is a matrix that performs this transformation.<br> In this paper, we propose the iterative GMRES method, preconditioned GMRES method&nbsp;&nbsp; and examine capability of these methods by solving the space fractional advection-diffusion equation.<br> <strong>Material and methods</strong><br> We first introduce a space fractional advection-diffusion equation in the sense of the&nbsp;&nbsp; shifted Gr&uuml;nwald-Letnikov fractional derivative. To improve the introduced numerical scheme, we discretize the partial derivatives of equation using the fractional Crank-Nicholson finite difference method. Then we use a preconditioner matrix and present preconditioned GMRES method for solving the derived linear system of algebraic equations.<br> <strong>Results and discussion</strong><br> In this paper, we use the GMRES and preconditioned GMRES to solve a linear system of equations emerging by discretizing partial derivatives appearing in&nbsp; a Advection-Diffusion equation and&nbsp; then&nbsp; asset the accuracy of these methods. Numerical results indicate that we derive a smaller condition number of the equivalent coefficient matrix for different values of M and N, as dimensions of the corresponding linear equations. Hence the convergence rate increases and consequently the number of iterations and the calculation time decreases.<span dir="RTL"></span><br> <strong>Conclusion</strong><br> The following conclusions were drawn from this research. <ul> <li>The GMRES method is a&nbsp; Kyrlov subspace methods to solve large-dimensions non-symmetric system of linear equations, which will be more effective when is applied with preconditioning&nbsp; techniques.</li> <li>One of the common ways to increase the rate of convergence of iterative methods based on the Kyrlov subspace is the applying&nbsp; the &nbsp;preconditioned techniques.</li> <li>An appropriate preconditioner matrix increases the rate of convergence of&nbsp; the iterative method.<a href="./files/site1/files/51/%DA%86%D8%B1%D8%A7%D8%BA%DB%8C.pdf">./files/site1/files/51/%DA%86%D8%B1%D8%A7%D8%BA%DB%8C.pdf</a></li> </ul> معادلۀ انتشار‌-‌هم رفت کسری, کرانک‌-‌نیکلسون کسری, گرانوالد‌-‌لتنیکوف انتقال‌یافته, GMRES پیش‌شرط‌ ساز شده. Fractional advection-diffusion equation, Fractional Crank-Nicholson method, Shifted Grünwald-Letnikov fractional derivative, Preconditioned GMRES method 39 52 http://mmr.khu.ac.ir/browse.php?a_code=A-10-169-1&slc_lang=fa&sid=1 طیبه چراغ زاده tcheraghzadeh@yahoo.com 10031947532846002695 10031947532846002695 No دانشگاه شهرکرد، گروه ریاضی کاربردی، شهرکرد، ایران مهدی قاسمی meh_ghasemi@yahoo.com 10031947532846002696 10031947532846002696 No دانشگاه شهرکرد، گروه ریاضی کاربردی، شهرکرد، ایران رضا خوش سیر reza_khosh@yahoo.com 10031947532846002697 10031947532846002697 No دانشگاه شهرکرد، گروه ریاضی کاربردی، شهرکرد، ایران علیرضا انصاری alireza_1038@yahoo.com 10031947532846002698 10031947532846002698 Yes دانشگاه شهرکرد، گروه ریاضی کاربردی، شهرکرد، ایران