آﻧﺎﻟﯿﺰ ﺗﻘﺎرن، ﻗﻮاﻧﯿﻦ ﺑﻘﺎ و ﺟﻮاب ﻫﺎي ﻧﺎوردا از ﻣﻌﺎدﻟﻪ زﻣﺎن -ﮐﺴﺮي ﻣﻮج ﻫﻤﺴﺎن

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

آﻧﺎﻟﯿﺰ ﺗﻘﺎرن ﻟﯽ روﺷﯽ ﮐﺎرآﻣﺪ ﺑﺮاي ﺑﺪﺳﺖ آوردن ﺟﻮابﻫﺎي ﺗﺤﻠﯿﻠﯽ و دﻗﯿﻖ از ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ اراﺋﻪ ﻣﯽدﻫﺪ. در اﯾﻦ ﻣﻘﺎﻟﻪ آﻧﺎﻟﯿﺰ ﺗﻘﺎرن ﻟﯽ ﺑﺮاي ﻣﻌﺎدﻟﻪ دﯾﻔﺮاﻧﺴﯿﻞ زﻣﺎن-ﮐﺴﺮي ﻣﻮج ﻫﻤﺴﺎن ﺑﺎ ﻣﺸﺘﻖ ﮐﺴﺮي رﯾﻤﻦ-ﻟﯿﻮوﯾﻞ را ﻣﻮرد ﺑﺤﺚ ﻗﺮار ﻣﯽدﻫﯿﻢ. اﯾﻦ ﻣﻌﺎدﻟﻪ ﺑﺮاي ﺗﻮﺻﯿﻒ ﺷﺒﯿﻪﺳﺎزي اﻧﺘﺸﺎر ﻣﻮج ﺗﮏ ﺑﻌﺪي در ﻣﺤﯿﻂﻫﺎي ﻏﯿﺮﺧﻄﯽ ﻫﻤﺮاه ﺑﺎ ﻓﺮآﯾﻨﺪﻫﺎي ﭘﺮاﮐﻨﺪﮔﯽ ﻣﻮرد اﺳﺘﻔﺎده ﻗﺮارﻣﯽﮔﯿﺮد. ﺑﺎ ﺑﻪ ﮐﺎر ﺑﺮدن آﻧﺎﻟﯿﺰ ﺗﻘﺎرن ﻟﯽ ﮐﻼﺳﯿﮏ و ﻏﯿﺮﮐﻼﺳﯿﮏ و ﺑﻌﻀﯽ ﺗﮑﻨﯿﮏﻫﺎي ﻣﺤﺎﺳﺒﺎﺗﯽ، ﻣﻮﻟﺪﻫﺎي ﺑﯽﻧﻬﺎﯾﺖ ﮐﻮﭼﮏ ﺟﺪﯾﺪ را ﺑﺪﺳﺖ ﻣﯽآورﯾﻢ. ﺳﭙﺲ ﺑﺎ ﺗﻐﯿﯿﺮ ﻣﺨﺘﺼﺎت، ﻣﻌﺎدﻟﻪ ﻣﻮج ﻫﻤﺴﺎن ﮐﺴﺮي را ﺑﻪ ﻣﻌﺎدﻟﻪ دﯾﻔﺮاﻧﺴﯿﻞ ﻣﻌﻤﻮﻟﯽ ﮐﺴﺮي ﺗﻘﻠﯿﻞ داده و ﺟﻮابﻫﺎي ﻧﺎورداﯾﯽ ﺑﺮاي اﯾﻦ ﻣﻌﺎدﻟﻪ ﭘﯿﺪا ﻣﯽﮐﻨﯿﻢ. ﺑﺎ اﺳﺘﻔﺎده از ﻗﻀﯿﻪ ﺑﻘﺎ ﺟﺪﯾﺪ اﯾﺒﺮاﮔﯿﻤﻮف و ﺗﻌﻤﯿﻢ ﻋﻤﻠﮕﺮﻫﺎي ﻧﻮﺗﺮ، ﻗﻮاﻧﯿﻦ ﺑﻘﺎ را ﺑﺮاي ﻣﻌﺎدﻟﻪ ﻣﯽﺳﺎزﯾﻢ. ﻫﻤﭽﻨﯿﻦ ﻣﻌﺎدﻟﻪ اﻟﺤﺎﻗﯽ و ﻣﻮﻟﺪ ﺑﯽﻧﻬﺎﯾﺖ ﮐﻮﭼﮏ آن، ﮐﻪ ﺑﺎ ﺗﻘﺎرنﻫﺎي ﻟﯽ ﻣﻌﺎدﻟﻪ اﺳﺎﺳﯽ در ارﺗﺒﺎط اﺳﺖ را ﺑﺪﺳﺖ ﻣﯽآورﯾﻢ و اﯾﻦ ﻣﻌﺎدﻟﻪ را ﺑﻪ ﻣﻌﺎدﻟﻪ دﯾﻔﺮاﻧﺴﯿﻞ ﻣﻌﻤﻮﻟﯽ ﮐﺴﺮي ﺗﻘﻠﯿﻞ ﻣﯽدﻫﯿﻢ. در ﻣﻌﺎدﻻت ﮐﺎﻫﺶ ﯾﺎﻓﺘﻪ، ﻣﺸﺘﻖ در ﻣﻔﻬﻮم اردﻟﯽ-ﮐﻮﺑﺮ اﺳﺖ.

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.