Laplace's Equation Solution in 2D

[neutrino2063]

I am using separation of variables and superposition to solve:

$$u_{xx}+u_{yy}=0; <br /> for (x,y) \in (0,L)$$ X $$(0,H)$$
$$u(0,y)=f(y);<br /> u(L,y)=0;<br /> u(x,0)=g(x);<br /> u(x,H)=0$$

Is it correct to assume that I can write my solution as:
$$u=u_1+u_2$$

Where:
$$u_1$$
is the solution with BC
$$u(0,y)=0;<br /> u(L,y)=0;<br /> u(x,0)=g(x);<br /> u(x,H)=0$$

And:
$$u_2$$
is the solution with BC
$$u(x,0)=0;<br /> u(x,H)=0;<br /> u(0,y)=f(y);<br /> u(L,y)=0$$

 

[gabbagabbahey]

I am using separation of variables and superposition to solve:

$$u_{xx}+u_{yy}=0; <br /> for (x,y) \in (0,L)$$ X $$(0,H)$$
$$u(0,y)=f(y);<br /> u(L,y)=0;<br /> u(x,0)=g(x);<br /> u(x,H)=0$$

Is it correct to assume that I can write my solution as:
$$u=u_1+u_2$$

Where:
$$u_1$$
is the solution with BC
$$u(0,y)=0;<br /> u(L,y)=0;<br /> u(x,0)=g(x);<br /> u(x,H)=0$$

And:
$$u_2$$
is the solution with BC
$$u(x,0)=0;<br /> u(x,H)=0;<br /> u(0,y)=f(y);<br /> u(L,y)=0$$

Yes, your assumption looks fine; because Laplace's equation is linear, any solution to it will obey the superposition principle. This means that if you find one solution that satisfies one set of boundary conditions, and another solution that satisfies a different set of boundary conditions, The superposition of the two solutions will satisfy the sum of the two sets of boundary conditions.