Boundary Value Problems/Problem Lookup

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A directory/database of BVP problems.

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