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use the method of separation of variables to solve the following nonhomogeneous initial-Neumann problem: Hint: write...

use the method of separation of variables to solve the following nonhomogeneous initial-Neumann problem:

\left\{\begin{matrix} u_{t}-u_{xx}=tx & 0<x<L,t>0\\ u(x,0=1)&0\leq x\leq L \\ u _{x}(0,t)=u_x(L,t)=0{_{}}&t>0 \end{matrix}\right.

Hint: write the candidate solution as u(x,t)=\sum _{k\geq 0}c_{k}(t)v_{k}(x) $ where v_{k} $ are the eigenfunctionsof the eigenvalue problem associated with the homogeneous equation.

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Answer #1

→ kau Write the candidate Solution as Wixit)- E cksti v cə) --K20 where Uk are the eigerfunctions of the eigenvalue problem a2 è (antiyht -4 Canti(t) = nzo solutions are respectively, S coltı- LX + Can (0) - O tue (21+134 t (2071)2 F Therefore, the s

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