Problems to Section 5.3

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Problems to Section 5.3

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4

Problem 1.

  1. Consider Dirichlet problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty < x < \infty, y>0, \\ &u|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

  2. Consider Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x< \infty, y>0, \\ &u_y|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f$?

Problem 2.

  1. Consider Dirichlet problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty<x<\infty, 0<y<1, \\ &u|_{y=0}=f(x),\quad u|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

  2. Consider Dirichlet-Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty<x<\infty, 0<y<1, \\ &u|_{y=0}=f(x), \quad u_y|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

  3. Consider Neumann problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x<\infty, 0< y< 1, \\ &u_y|_{y=0}=f(x), \quad u_y|_{y=1}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f,g$?

Problem 3.

Consider Robin problem \begin{align} &u_{xx}+u_{yy}=0,\qquad -\infty< x<\infty, y>0, \\ &(u_y+\alpha u)|_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition (if any) must satisfy $f$?

Hint. Consider separately $\alpha>0$ and $\alpha<0$.

Problem 4.

  1. Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty< x<\infty, y>0, \\ &u|_{y=0}=f(x),\quad &u_y|_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral.

  2. Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty< x<\infty, y>0, \\ &u_{yy}|_{y=0}=f(x),\quad &\Delta u_{y}|_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f,g$?


$\Uparrow$  $\uparrow$  $\Rightarrow$