رویکرد عددی متمایز در جواب نوعی از مسئله مقدار اولیه شامل معادلات دیفرانسیل q-کسری غیرخطی

نویسندگان
محمد اسماعیل سامعی
اعظم فتحی پور
دانشگاه بوعلی سینا
چکیده
معادله دیفرانسیل کسری و q-کسری تعمیمی بر انتگرال و مشتق معمولی هستند که در آن مشتق و انتگرال از هر مرتبه­ای می­تواند باشد. معمولا روند فیزیکی تحمیل شده بر اشیاء در مقیاس زمانی را توصیف می­کنند. در این پژوهش ابتدا یک رابطه تفاضلی برای q-مشتق کسری از نوع کاپوتو با مرتبه برای ارائه می­دهیم و ثابت می­کنیم که این رابطه تفاضلی به طور مطلق پایدار است. سپس روش تفاضلی را برای حل مسئله مقدار اولیه معادله دیفرانسیل q-کسری در نظر می­گیریم. همچنین یکتایی وجود جواب، پایداری و همگرایی جواب حاصل را ثابت می­کنیم. در پایان چند مثال ارائه می­دهیم که نتایج عددی آنها نشان دهنده دقت بالای روش تفاضلی مذکور خواهد بود.

کلیدواژه‌ها
: q-مشتق کسری
روش تفاضلی
خطای برش
پایداری بدون قید
معادله دیفرانسیل q-کسری
تحلیل همگرایی

عنوان مقاله English

A distinct numerical approach for the solution of some kind of initial value problem involving nonlinear q-fractional differential equations

نویسندگان English

اعظم فتحی پور
azam fathipour
چکیده English

The fractional calculus deals with the generalization of integration and differentiation of integer order to those ones of any order. The q-fractional differential equation usually describe the physical process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative of Caputo type with order and scale index . We establish a rigorous truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial problem of q-fractional differential equation: . We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.


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