كليدواژه :
معادلات انتگرال-ديفرانسيل ولترا , معادلات كسري تأخيري , چندجمله اي هاي گنوچي , روش گالركين , ماتريس عملياتي , تحليل خطا
چكيده فارسي :
در اين مقاله، روش گالركين[1] با پايه چندجمله‌اي‌هاي گنوچي[2] متعامد انتقال يافته را براي حل يك رده از معادلات انتگرال-ديفرانسيل كسري تأخيري به‌كار مي‌بريم. براي اين منظور ابتدا جواب تقريبي معادله را بر حسب اين چندجمله‌اي‌ها بسط مي‌دهيم و سپس ماتريس عملياتي چندجمله‌اي‌هاي گنوچي را براي مشتقات كسري با مفهوم كاپاتو[3] به‌دست مي‌آوريم. مزيت اين روش اين است كه با جاي‌گذاري جواب تقريبي در معادله اصلي و به كاربردن ماتريس هاي عملياتي، معادله به يك دستگاه معادلات جبري تبديل مي شود كه با روش نيوتن قابل حل است. به علاوه، در تحليل همگرايي روش نشان مي‌دهيم كه تحت شرايطي جواب روش گالركين به كمك توابع متعامد گنوچي به جواب واقعي همگراست. در پايان نتايج عددي ارائه شده است تا كارايي و اعتبار روش و نيز مناسب بودن كران خطا را نشان دهند؛ به‌علاوه با توجه به نتايج عددي مشاهده مي‌شود كه با اين روش در بازه‌هاي بزرگ نيز جواب‌هاي قابل قبولي به دست مي‌آيد.
چكيده لاتين :
Recently, various authors have studied fractional calculus and its applications in earthquake,
solid mechanics, dynamic fluid traffic, statistical mechanics, viscoelastic materials, economics,
transportation, bioengineering, etc. These problems usually do not have analytic solution.
Consequently, different numerical methods have been developed to approximate these equations.
Also, fractional delay differential equations were considered in various papers. In the present
paper, we are going to solve the equation
The main goal is to approximate the solution of equation (1) by Galerkin method with shifted
orthogonal Genocci polynomials and study the error of the method.
Material and methods
In this scheme, we first obtain the operational matrix for fractional derivatives in the Caputo
sense and delayed operational matrices based on shifted orthogonal Genocci polynomials and
then by using these matrices, fractional delay differential equation is transformed to a system of
algebraic equations which can be solved via a suitable numerical method.
Results and discussion
We solve some test examples by using present technique to demonstrate the efficiency, high
accuracy and the simplicity of the present method, then compare the proposed method with other
methods. The reported results demonstrate that there is a good agreement between approximate
solution and exact solution. Also, the numerical results reported in the tables indicate that the
accuracy improve by increasing the number of basis functions. In addition, the results show that
the method is also effective for long intervals.
Conclusion
The following conclusions were drawn from this research.
• shifted orthogonal Genocci polynomials are simple basis functions, so proposed method
is easy to implement and it is a powerful mathematical tool to obtain the numerical
solution of various kind of problems with little additional works.
• The main characteristic of this method is that it reduces the underlying problem to a
system of algebraic equations which can be easily solved by using direct method or
iterative method.
• Convergence analysis shows the convergence of the method, and the method is also
effective for long intervals.
