حل عددی و آنالیز خطای معادله‌ی دیفرانسیل تاخیری خطی و غیرخطی
محورهای موضوعی : آمار
ابراهیم امینی
1
(گروه ریاضی دانشگاه پیام نور، صندوق پستی 4697-19395، تهران، ایران)
Ali Ebadian
2
(دانشکده علوم پایه، گروه ریاضی، دانشگاه ارومیه، ارومیه، ایران)
کلید واژه: Reproducing kernel, Error analysis, Convergence analysis, Delay differential equations,
چکیده مقاله :
اﯾﻦ ﻣﻘﺎﻟﻪ ، جواب ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﺗﺎﺧﯿﺮی ﺧﻄﯽ و ﻏﯿﺮ ﺧﻄﯽ را در فضای هسته‌ی بازتولید بدست می‌آوریم. ﺑﺪین ﻣﻨﻈﻮر با توجه به ﻣﻌﺎدله مذکور و ﺷﺮاﯾﻂ ﺣﺎﮐﻢ ﺑﺮ آن، یک ﻋﻤﻠﮕﺮ ﺧﻄﯽ ﺗﻌﺮﯾﻒ می‌کنیم و در ادامه با استفاده از ﻋﻤﻠﮕﺮ اﻟﺤﺎﻗﯽ آن و ﺗابع ﻫﺴته ﺑﺎزﺗﻮﻟید یک دستگاه متعامد یکه کامل برای فضای هسته‌ی بازتولید بدست می‌آوریم. سپس جواب ﻣﻌﺎدﻻت ﻣﺬﮐﻮر را بر حسب یک سری از ﺗﻮاﺑﻊ پایه‌ای بدست می‌آوریم. در واﻗﻊ ﺟﻮاب ﺗﺤﻠﯿﻠﯽ به‌صورت ﯾﮏ ﺳﺮی ﻧﺎﻣﺘﻨﺎﻫﯽ ﻧﻤﺎﯾﺶ داده می‌شود و با اﺳﺘﻔﺎده از ﯾﮏ روش تکراری، ﺟﻮاب ﺗﻘﺮﯾﺒﯽ ﻧﻈﯿﺮ ﺳﺮی ﻣﺬﮐﻮر ﺑﺪﺳﺖ آورده می‌شود. به‌عنوان یکی از اهداف اصلی, آﻧﺎﻟﯿﺰ ﻫﻤﮕﺮاﯾﯽ و ﺧﻄﺎ را برای روش ﻣﻮرد ﻧﻈﺮ در حل معادلات دیفرانسیل تاخیری بررسی می‌کنیم. در ﭘﺎﯾﺎن ﺑﺮﺧﯽ از ﻣﺜﺎﻟﻬﺎی ﻋﺪدی ﺑﺮای ﻧﺸﺎن دادن درﺳﺘﯽ و ﮐﺎرﺑﺮد روش ﭘﯿﺸﻨﻬﺎدی ﻣﻮرد ﺑﺮرﺳﯽ ﻗﺮار ﮔﺮﻓﺘﻪ اﺳﺖ و ﻧﺘﺎﯾﺞ ﺣﺎﺻﻞ از اﯾﻦ روش ﺑﺎ ﺟﻮاب دﻗﯿﻖ ﮐﺎرﻫﺎی ﻗﺒﻠﯽ ﻣﻘﺎﯾﺴﻪ می‌شوند. ﻧﺘﺎﯾﺞ ﺑﺪﺳﺖ آﻣﺪه از ﻣﺜﺎﻟﻬﺎی ﻋﺪدی ﻧﺸﺎن می‌دهد ﮐﻪ روش ﭘﯿﺸﻨﻬﺎدی ﻣﻔﯿﺪ و مناسب است.
In this paper, we obtain the solution of linear and nonlinear delay differential equations in reproducing kernel space. For this purpose, regarding the equation and conditions governing it, a linear operator is defined and subsequently an orthonormal complete system for reproducing kernel space is obtained by using the adjoint operator and reproducing kernel function. Then, the solution of these equations is obtained in the form of a series of the basic functions. Indeed, the analytical solution is represented by infinite series, and the approximate solution is obtained by using an iterative method. As one of the main aims, the convergence analysis and error behavior are discussed for the proposed method. Finally, some numerical examples are studied to demonstrate the validity and applicability of the proposed method. The obtained results of the proposed method are compared with the exact solutions and the earlier works. The outcomes from numerical examples illustrate that the proposed method is very effective and convenient.
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