check-that-the-function-u-x-t-exp-n-2-2-pi-2-L-2-t-sin-npix-L-n-1-2-satisfy-the-heat-equation-heat-equation-2-2-u-x-2-u-t-0-lt-x-lt-L-

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Question Number 168977 by MikeH last updated on 23/Apr/22

check that the function u(x,t) = exp{−((n^2 α^2 π^2 )/L^2 )t} sin((nπx)/L) n = 1,2,... satisfy the heat equation heat equation α^2 (∂^2 u/∂x^2 ) = (∂u/∂t), 0 < x < L

$$\mathrm{check}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function} \\ $$$${u}\left({x},{t}\right)\:=\:\mathrm{exp}\left\{−\frac{{n}^{\mathrm{2}} \alpha^{\mathrm{2}} \pi^{\mathrm{2}} }{{L}^{\mathrm{2}} }{t}\right\}\:\mathrm{sin}\frac{{n}\pi{x}}{{L}} \\ $$$${n}\:=\:\mathrm{1},\mathrm{2},…\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{heat}\:\mathrm{equation} \\ $$$$\boldsymbol{\mathrm{heat}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\alpha^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }\:=\:\frac{\partial{u}}{\partial{t}},\:\mathrm{0}\:<\:{x}\:<\:{L} \\ $$

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check-that-the-function-u-x-t-exp-n-2-2-pi-2-L-2-t-sin-npix-L-n-1-2-satisfy-the-heat-equation-heat-equation-2-2-u-x-2-u-t-0-lt-x-lt-L-

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