Symmetry analysis, conservation laws and invariant solutions of the time-fractional equal width wave equation

Document Type : research paper

Author

Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran

Abstract

Lie symmetry analysis provides an efficient method to get the analytical and exact solutions of the fractional differential equations. In this paper, we discuss Lie symmetry analysis for the time-fractional equal width wave equation with Riemann–Liouville derivative. This equation is used to describe the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. By employing classical and nonclassical Lie symmetry analysis and some technical calculations, new infinitesimal generators are obtained. Then we reduce the fractional equal width wave equation to the ordinary fractional differential equation by changing the coordinates and find invariant solutions to this equation. By means of Ibragimov’s new conservation theorem and the generalization of the Noether operators, we construct the conservation laws for the equation. Also, we derive the adjoint equation and infinitesimal generator associated with Lie symmetries of the underlying equation and we reduce this equation to the ordinary fractional differential equation. In the reduced equations the derivative is in Erdelyi–Kober sense.

Keywords


Article Title [فارسی]

آنالیز تقارن، قوانین بقا و جواب‌های ناوردا از معادله زمان-کسری موج همسان

Author [فارسی]

  • رامین نجفی
گروه ریاضی، واحد ماکو، دانشگاه آزاد اسلامی، ماکو، ایران
Abstract [فارسی]

آنالیز تقارن لی روشی کارآمد برای بدست آوردن جواب‌های تحلیلی و دقیق از معادلات دیفرانسیل ارائه می‌دهد. در این مقاله آنالیز تقارن لی برای معادله دیفرانسیل زمان-کسری موج همسان با مشتق کسری ریمن-لیوویل را مورد بحث قرار می‌دهیم. این معادله برای توصیف شبیه‌سازی انتشار موج تک بعدی در محیط‌های غیرخطی همراه با فرآیندهای پراکندگی مورد استفاده قرارمی‌گیرد. با به کار بردن آنالیز تقارن لی کلاسیک و غیرکلاسیک و بعضی تکنیک‌های محاسباتی، مولد‌های بی‌نهایت کوچک جدید را بدست می‌آوریم. سپس با تغییر مختصات، معادله موج همسان کسری را به معادله دیفرانسیل معمولی کسری تقلیل داده و جواب‌های ناوردایی برای این معادله پیدا می‌کنیم. با استفاده از قضیه بقا جدید ایبراگیموف و تعمیم عملگرهای نوتر، قوانین بقا را برای معادله می سازیم. همچنین معادله الحاقی و مولد بی‌نهایت کوچک آن، که با تقارن های لی معادله اساسی در ارتباط است را بدست می آوریم و این معادله را به معادله دیفرانسیل معمولی کسری تقلیل می‌دهیم. در معادلات کاهش یافته ، مشتق در مفهوم اردلی-کوبر است.

Keywords [فارسی]

  • معادله زمان-کسری موج همسان
  • آنالیز تقارن لی
  • قوانین بقا
  • معادله الحاقی
  • جواب ناوردا
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