بررسی عددی یک طرح تفاضلی برای معادلات انتشار کسری زمانی- مکانی کپوتو- ریس چندجمله‌ای
محورهای موضوعی : آنالیز عددیمجتبی فردی 1 , ابراهیم امینی 2 *
1 - گروه ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه شهرکرد، صنوق پستی 88186-34141، شهرکرد، ایران
2 - گروه ریاضی، دانشگاه پیام نور، صندوق پستی ۴۶٩٧- ١٩٣٩۵، تهران، ایران
کلید واژه: Fractional diffusion equations, Caputo-Riesz derivative, Convergence, stable Conditionally, Difference scheme,
چکیده مقاله :
چکیده: در این مقاله، یک طرح تفاضلی برای حل معادلات انتشار کسری زمانی- مکانی چند جمله‌ای ارائه می‌شود. در معادلات انتشار کسری، مشتق زمانی از نوع کپوتو چند جمله‌ای و مشتق مکانی از نوع ریس هستند. معادلات مذکور در بعدهای یک و دو در نظر گرفته شده‌اند. در بعد یک مشتق مکانی ریس از مرتبه‌ی و در بعد دو مشتق مکانی ریس از مرتبه‌های و هستند. هم‌چنین، مشتق کپوتو چند جمله‌ای از مرتبه‌های هستند. آنالیز پایداری و همگرایی طرح تفاضلی ارائه می‌شود و شرایط پایداری طرح تفاضلی ارائه شده را مورد بررسی قرار می‌دهیم. اثبات می‌کنیم که طرح تفاضلی پیشنهادی پایدار مشروط است. علاوه‌بر‌این نشان می‌دهیم که طرح تفاضلی با مرتبه در زمان و با مرتبه‌ی دو در مکان همگراست. در پایان دو مثال عددی به ترتیب در بعدهای یک و دو داده می‌شود تا کارآیی و قابل اجرا بودن طرح تفاضلی پیشنهادی را از نظر دقت و سرعت همگرایی نشان دهیم.
.Abstract: In this paper, we provide a difference scheme for solving multi-term the time-space fractional diffusion equations. In fractional diffusion equations, the time derivative is of the Caputo type and the space derivative is of the Riesz type. The aformentional equations are considered for one and two dimensional. In one dimentional the Riesz space derivative is of the order and in two dimentional the Riesz space derivative is of the orders and . Also, the multi-term Caputo derivative is of orders . We provided the stability and convergence analysis of the proposed difference scheme and investigate the stability conditions of the proposed difference scheme. We prove that the proposed difference scheme is stable conditionally. Furthermore, we show that difference scheme is convergent with order in time and order 2 in space. Finally, we give two numerical examples for one and two dimensional to illustrate the efficiency and applicability of the proposed difference scheme in the sense of accuracy and convergence ratio.
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