اسپلاین نمایی برای حل معادله ریکاتی کسری

نویسندگان
رضا جلیلیان 1
هومن عمادیفر 2
1 گروه ریاضی، دانشکده علوم پایه، دانشگاه رازی، کرمانشاه، ایران.
2 گروه ریاضی، دانشکده علوم پایه، دانشگاه آزاد اسلامی، همدان، ایران.
چکیده
در این مقاله با استفاده از دو روش پیشنهادی، تقریبی به‌مراتب کاراتر نسبت به­ روش‌های عددی موجود برای دسترسی به جواب‌های عددی معادله ریکاتی کسری ارائه می­گردد. این تقریب برای جواب معادله ریکاتی کسری بر پایه استفاده از توابع اسپلاین نمائی پیشنهاد می­شود. سپس روابط سازگار اسپلاین نمائی به‌دست‌آمده و معادله دیفرانسیل ریکاتی کسری گسسته سازی می­گردد و براثر اجرای این عملیات ریاضی یک دستگاه جبری از معادلات به دست می­آید. به جهت بیان مزایای روش­های پیشنهادی در این مقاله تحلیل خطا و همگرایی نیز بر اساس این اسپلاین نمائی مورد بحث قرار می­گیرد. و نهایتاً هر دو روش پیشنهادی از مرتبه همگرایی دو برخوردار می­باشند. از اهم مزایای این روش­های پیشنهادی این است که این روش­ها نه‌تنها برای حل معادلات ریکاتی کسری، بلکه برای انواع معادلات کسری می­تواند مورد استفاده قرار بگیرد. برای نشان دادن کارایی این روش­ها، مثال­های عددی از نوع معادلات ریکاتی کسری به واسطه این روش­ها حل و نتایج به دست آمده را با نتایج سایر روش­های عددی موجود مقایسه و این ادعا ثابت می­گردد، که روش­های موصوف تقریب خوبی برای معادله دیفرانسیل ریکاتی کسری می­باشند.
کلیدواژه‌ها
معادله دیفرانسیل ریکاتی کسری
مشتق کاپوتو
اسپلاین نمایی
همگرایی

عنوان مقاله English

An exponential spline for solving the fractional riccati differential equation

نویسندگان English

reza jalilian 1
hooman emadifar 2
1 Department of Mathematics, Faculty of Basic Sciences, Razi University, Kermanshah, Iran.
2 Department of Mathematics, Faculty of Basic Sciences, Islamic Azad University, Hamedan, Iran.
چکیده English

In this Article, proposes an approximation for the solution of the Riccati equation based on the use of exponential spline functions. Then the exponential spline equations are obtained and the differential equation of the fractional Riccati is discretized. The effect of performing this mathematical operation is obtained from an algebraic system of equations. To illustrate the benefits of the method proposed here the error analysis and convergence article are also discussed based on this exponential spline. Finally, a second-order method is obtained. One of the benefits of this proposed method is that it is not only for solving fractional Riccati equations, but also for a variety of Fractional equations that can be used. To illustrate the effectiveness of this method by solving numerical examples and a comparison of the results obtained from the implementation of this proposed method with the results of other existing numerical methods proves the claim that the proposed method is a good approximation for the fractional Riccati equations.

کلیدواژه‌ها English

Fractional riccati differential equation
Caputo derivative
Exponential Spline
Convergence
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