Question

Uxx +Uyy = Q(x,y)
U= C (constant)
this is possions equation ..u are requested to solve it by splitting the problem into two parts.
u=v+w
v----> non-homogeneous equation with homogeneous boundary conditions
w----> homogeneous equation with non homogeneous boundary condition

Answer 1

Answer on Question # 82783, Math / Differential Equations

Question 1. Let $u_{xx} + u_{yy} = Q(x,y)$ and $u(x,y) = c$. Assume $u = v + w$, where $v \to$ nonhomogeneous equation with homogeneous boundary conditions, $w \to$ homogeneous equation with non homogeneous boundary condition.

Solution. Let $\Delta u(x,y) \equiv u_{xx}(x,y) + u_{yy}(x,y) = Q(x,y)$, $(x,y) \in \Omega$, where $\Omega$ is some region in the plane. If $Q(x,y) \neq 0$ this equation is called the Poisson equation.

Let $u = v + w$ where $\Delta v(x,y) = f(x,y)$, $(x,y) \in \Omega$ and $Bv(x,y) = 0$, $(x,y) \in \partial \Omega$; $\Delta w(x,y) = 0$, $(x,y) \in \Omega$ and $Bw(x,y) = g(x,y)$, $(x,y) \in \partial \Omega$. General form for boundary condition:

$$B u (x, y) \equiv \alpha (x, y) \frac {\partial u}{\partial n} (x, y) + \beta (x, y) u (x, y) = g (x, y), (x, y) \in \partial \Omega .$$

We have Dirichlet boundary condition: $Bu(x,y) = u(x,y) = c$.

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