[Solution Library] Consider the partial differential equation (partial u)/(partial t)=(partial^2 u)/(∂ x^2) By assuming a solution of the form u(x,

Question: Consider the partial differential equation

\[\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}\]

By assuming a solution of the form $u(x, t)=X(x) T(t)$, deduce that

\[u_{\alpha}(x, t)=\left(A_{\alpha} \cos (\alpha x)+B_{\alpha} \sin (\alpha x)\right) e^{-\alpha^{2} t}\]

where $A_{\alpha}$ and $B_{\alpha}$ are constants, is a solution for any constant $\alpha$. Show that if we now impose the boundary conditions $u(0, t)=0, u(\pi, t)=0$, then this reduces the possible solutions to those of the form

\[u_{n}(x, t)=B_{n} \sin (n x) e^{-n^{2} t}\]

for $n=0,1,2, \ldots$ and where $B_{n}$ is a constant. Hence or otherwise find the solution of the

problem

\[\begin{aligned} & \frac{\partial u}{\partial t}=\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}},\quad u(0,t)=0,\quad u(\pi ,t)=0 \\ & u(x,0)={{\sin }^{2}}(x),\quad 0

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[Solution Library] Consider the partial differential equation (partial u)/(partial t)=(partial^2 u)/(∂ x^2) By assuming a solution of the form u(x,

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