روشی کارا در بررسی رفتار معادلات دیفرانسیل کسری

نویسندگان
محمدرضا رستمی 1
خسرو سایوند 2
1 مرکز آموزش عالی محلات
2 دانشگاه ملایر
چکیده
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،‎ 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.
کلیدواژه‌ها
مشتق و انتگرال کسری
معادله دیفرانسیل چند-مرتبه ای کسری
چندجمله ای لوکاس
انتگرال گیری گاوس
تحلیل خطا

عنوان مقاله English

Numerical solution of multi order fractional differential equations using Lucas polynomials

نویسندگان English

Mohammadreza Rostami 1
Khosro Sayevand 2
1 Mahallat Institute of Higher Education
2 Malayer university
چکیده English

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.

کلیدواژه‌ها English

Fractional derivative and integral
Multi-Order fractional differential equations
Lucas polynomial
Gauss integration method
Error analysis
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