Pregunta: 5. Solve the problem: ∇2u=0,u(0,y)=0,u(x,0)=0,u(a,y)=0,u(x,b)=f(x),0

5. Solve the problem: $\begin{array}{lll}{\nabla }^{2}u=0,& & 0<x<a,0<y<b\\ u(0,y)=0,& u(a,y)=0,& 0<y<b\\ u(x,0)=0,& u(x,b)=f(x),& 0<x<a\end{array}$ where $f$ is the same as in the example. Sketch some level curves of $u(x,y)$. Example. Suppose $f_1$ and $f_2$ are both given by this function, whose graph is an isosceles triangle: $f_1(x)=f_2(x)=\{\begin{array}{ll}\frac{2x}{a},& 0<x<\frac{a}{2}\\ 2\left(\frac{a-x}{a}\right),& \frac{a}{2}<x<a.\end{array}$ Then $c_n=a_n=\frac{8}{{\pi }^2}\frac{\sin (n\pi /2)}{n^2}.$ The solution of the potential equation for these boundary conditions is $u(x,y)=\frac{8}{{\pi }^2}{\sum }_{n=1}^{\infty }\frac{\sin \left(\frac{n\pi }{2}\right)}{n^2}\frac{\sinh \left(\frac{n\pi }{a}y\right)+\sinh \left(\frac{n\pi }{a}(b-y)\right)}{\sinh \left(\frac{n\pi b}{a}\right)}\sin \left(\frac{n\pi x}{a}\right).$