On space-time periodic solutions of the one-dimensional heat equation. (English) Zbl 1439.35188
Summary: We look for solutions \( u\left( x,t\right) \) of the one-dimensional heat equation \( u_t = u_{xx} \) which are space-time periodic, i.e. they satisfy the property \( u\left( x+a,t+b\right) = u\left( x,t\right)\) for all \(\left( x,t\right) \in\left( -\infty,\infty\right) \times\left( -\infty,\infty\right), \) and derive their Fourier series expansions. Here \( a\geq0, b\geq 0 \) are two constants with \( a^2+b^2>0.\) For general equation of the form \( u_t = u_{xx}+Au_x+Bu, \) where \(A, B \) are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when \( B>0\) and is given by a linear combination of \( \cos\left( \sqrt{B}\left( x+At\right) \right)\) and \(\sin\left( \sqrt{B}\left( x+At\right) \right)\).
Keywords:
heat equation; time-periodic solution; space-periodic solution; space-time periodic solution; travelling wave solutionReferences:
[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. · Zbl 0567.35001 |
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