Mathematical Researches

fa روشی کارا در بررسی رفتار معادلات دیفرانسیل کسری Numerical solution of multi order fractional differential equations using Lucas polynomials جبر alg مقاله مستقل Original Manuscript در این مقاله یک روش جدید برای حل معادلۀ دیفرانسیل کسری و چندمرتبه‌ای کسری مورد مطالعه قرار می‌گیرد. مشتق کسری از نوع کاپوتو می‌باشد. در این روش ابتدا به کمک چند جمله‌ای‌های لوکاس به عنوان پایه، یک تقریب برای جواب معادله در نظر می‌گیریم. به کمک این تقریب، این معادله را به یک مسأله کمترین مربعات تبدیل می‌کنیم. برای حل مسأله کمترین مربعات از روش انتگرال گیری گاوس استفاده می‌کنیم. سپس با استفاده از قضیۀ ضرائب لاگرانژ یک مسأله بهینه‌سازی مقید را حل می‌کنیم. با حل این مسأله&rlm;،&lrm; جواب تقریبی برای معادلۀ دیفرانسیل حاصل می‌شود. همگرایی و آنالیز خطای این روش مورد بررسی قرار می‌گیرد و مثال‌های عددی نشان می‌دهد که این روش، مؤثر و کارا است. Introduction This paper presents a reliable numerical technique based on Lucas polynomials for a family of fractional differential equations and multi order fractional differential equations by means of the least square method. The fractional derivative is in the Caputo sense. A relevant feature of this approach is the analyzing of the suggested technique by Gauss quadrature method and using the theory of Lagrange multipliers to solve a constrained optimization problem.  An upper error bound, the convergence, and error analysis of the scheme are investigated and the CPU time used, the values of maximum errors, the numerical convergence analysis based on the proposed technique for different values of parameters are discussed. Furthermore the results of present technique are compared with the, operational matrix of hybrid basis functions, the Jacobi orthogonal functions and pseudo-spectral scheme. In order to introduce the numerical behavior of the proposed technique in case of non-smooth solutions, this issue is discussed. In this case, the obtained results imply an elegant superiority of our proposed technique. The numerical examples illustrate the accuracy and performance of the technique. Finally extending the proposed technique to high dimensions and system of fractional differential equations can be examined as a further works. Material and methods In this study, the least square method, the Gauss quadrature method and the theory of Lagrange multipliers are used to solve a constrained optimization problem. Results and discussion Several numerical examples are examined using the proposed technique. The numerical examples illustrate the accuracy and performance of the technique. Also, the numerical results reported in the tables indicate that the accuracy improve by increasing the degree of the Lucas polynomials. Conclusion In this paper, Lucas polynomials have been successfully applied to compute the approximate solution of the fractional differential equations and multi order fractional differential equations. The results show that: • The proposed technique provides the solutions in terms of convergent series with easily computable components in a direct way, without using linearization, perturbation or restrictive assumption. • The proposed technique is very straightforward and the solution procedure can be done easily. • The numerical behavior of the proposed technique in case of non-smooth solutions, demonstrated that the obtained results imply an elegant superiority of our proposed technique. مشتق و انتگرال کسری, معادله دیفرانسیل چند-مرتبه ای کسری, چندجمله ای لوکاس, انتگرال گیری گاوس, تحلیل خطا. Fractional derivative and integral, Multi-Order fractional differential equations, Lucas polynomial, Gauss integration method, Error analysis. 184 204 http://mmr.khu.ac.ir/browse.php?a_code=A-10-992-1&slc_lang=fa&sid=1 Mohammadreza Rostami محمدرضا رستمی rostami.mohamadreza@gmail.com 10031947532846005771 10031947532846005771 Yes Mahallat Institute of Higher Education مرکز آموزش عالی محلات Khosro Sayevand خسرو سایوند Ksayehvand@Malayeru.ac.ir 10031947532846005772 10031947532846005772 No Malayer university دانشگاه ملایر