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On space-time periodic solutions of the one-dimensional heat equation
1. | Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan |
2. | Department of Financial Engineering, Providence University, Taichung 43301, Taiwan |
$ u\left( x,t\right) $ |
$ u_{t} = u_{xx} $ |
$ u\left( x+a,t+b\right) = u\left( x,t\right) $ |
$ \left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), $ |
$ a\geq0,\ b\geq 0 $ |
$ a^{2}+b^{2}>0. $ |
$ u_{t} = u_{xx}+Au_{x}+Bu, $ |
$ A,\ B $ |
$ B>0 $ |
$ \cos\left( \sqrt{B}\left( x+At\right) \right) $ |
$ \sin\left( \sqrt{B}\left( x+At\right) \right). $ |
References:
[1] |
J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.
doi: 10.1017/CBO9781139086967. |
[2] |
D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.
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References:
[1] |
J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984.
doi: 10.1017/CBO9781139086967. |
[2] |
D. V. Widder, The Heat Equation, Pure and Applied Mathematics, Vol. 67. Academic Press, New York-London, 1975.
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