On space-time periodic solutions of the one-dimensional heat equation
On space-time periodic solutions of the one-dimensional heat equation
复制标题
一维热方程的时空周期解
DOI：
--
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Chia
中科院分区：
文献类型：
--
作者：
Dong;Chia
We look for solutions egin{document}$ uleft( x,t
ight) $end{document} of the one-dimensional heat equation egin{document}$ u_{t} = u_{xx} $end{document} which are space-time periodic, i.e. they satisfy the property egin{document}$ uleft( x+a,t+b
ight) = uleft( x,t
ight) $end{document} for all egin{document}$ left( x,t
ight) inleft( -infty,infty
ight) imesleft( -infty,infty
ight), $end{document} and derive their Fourier series expansions. Here egin{document}$ ageq0, bgeq 0 $end{document} are two constants with egin{document}$ a^{2}+b^{2}>0. $end{document} For general equation of the form egin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $end{document} where egin{document}$ A, B $end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when egin{document}$ B>0 $end{document} and is given by a linear combination of egin{document}$ cosleft( sqrt{B}left( x+At
ight)
ight) $end{document} and egin{document}$ sinleft( sqrt{B}left( x+At
ight)
ight). $end{document}
We look for solutions egin{document}$ uleft( x,t
ight) $end{document} of the one-dimensional heat equation egin{document}$ u_{t} = u_{xx} $end{document} which are space-time periodic, i.e. they satisfy the property egin{document}$ uleft( x+a,t+b
ight) = uleft( x,t
ight) $end{document} for all egin{document}$ left( x,t
ight) inleft( -infty,infty
ight) imesleft( -infty,infty
ight), $end{document} and derive their Fourier series expansions. Here egin{document}$ ageq0, bgeq 0 $end{document} are two constants with egin{document}$ a^{2}+b^{2}>0. $end{document} For general equation of the form egin{document}$ u_{t} = u_{xx}+Au_{x}+Bu, $end{document} where egin{document}$ A, B $end{document} are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when egin{document}$ B>0 $end{document} and is given by a linear combination of egin{document}$ cosleft( sqrt{B}left( x+At
ight)
ight) $end{document} and egin{document}$ sinleft( sqrt{B}left( x+At
ight)
ight). $end{document}