# Boundary Value Problems/Problem Lookup

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A directory/database of BVP problems.

PDE BCs ICs Click for link to solution
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${\displaystyle u_{xx}={\frac {1}{k}}u_{t}}$ ${\displaystyle u(0,t)=0{\mbox{ }}u(L,t)=0}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_hom_dirichlet_hom
${\displaystyle u_{xx}={\frac {1}{k}}u_{t}}$ ${\displaystyle u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_hom_dirichlet_non_hom
${\displaystyle u_{xx}={\frac {1}{k}}u_{t}}$ ${\displaystyle u_{x}(0,t)=0{\mbox{ }}u(L,t)=T_{1}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_hom_neumann_dirichlet
${\displaystyle u_{xx}={\frac {1}{k}}u_{t}}$ ${\displaystyle u_{x}(0,t)=-k(u(0,t)-A){\mbox{ }}u(L,t)=T_{2}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_hom_dirichlet_non_hom
${\displaystyle u_{xx}+g(x)={\frac {1}{k}}u_{t}}$ ${\displaystyle u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_hom_dirichlet_non_hom
${\displaystyle u_{xx}={\frac {1}{k}}u_{t}+k(u(x,t)-A)}$ ${\displaystyle u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_1D_heat_nonhom_dirichlet_non_hom
${\displaystyle u_{xx}={\frac {1}{k}}u_{t}}$ ${\displaystyle u(0,t)=T_{1}{\mbox{ }}u(L,t)=T_{2}}$ ${\displaystyle u(x,0)=f(x)}$ BVP_2D_Laplaces_hom_BCs_non_hom
${\displaystyle \displaystyle \nabla ^{2}u=0}$ ${\displaystyle u(0,y)=\pi -y}$
${\displaystyle u(1,x)=0}$
${\displaystyle u_{y}(x,0)=x}$
${\displaystyle u_{y}(x,\pi )=x}$
none
${\displaystyle \displaystyle u_{xx}=u_{tt}+10}$ ${\displaystyle u(0,t)=0}$
${\displaystyle u(1,t)=0}$
${\displaystyle u(x,0)=0}$
${\displaystyle u_{t}(x,0)=x(1-x)}$
${\displaystyle \displaystyle \nabla ^{2}u={\frac {1}{4}}u_{t}}$ ${\displaystyle u(x,0,t)=0}$
${\displaystyle u(x,5,t)=0}$
${\displaystyle u(0,y,t)=0}$
${\displaystyle u(3,y,t)=0}$
${\displaystyle u(x,y,t)=sin({\frac {\pi x}{3}})sin({\frac {\pi y}{5}})}$
${\displaystyle \displaystyle \nabla ^{2}u={\frac {1}{4}}u_{t}}$+ F(x,y,t) ${\displaystyle u(x,0,t)=0}$
${\displaystyle u(x,M,t)=0}$
${\displaystyle u(0,y,t)=0}$
${\displaystyle u(L,y,t)=0}$
${\displaystyle u(x,y,t)=sin({\frac {\pi x}{L}})sin({\frac {\pi y}{M}})}$