A PDE and Linear operator questions.

[Karlisbad]

Let be L and G 2 linear operators so they have the same set of Eigenvalues, then:

$$L[y]=-\lambda _{n} y$$ and $$G[y]=-\lambda _{n} y$$

then i believe that either L=G or L and G are related by some linear transform or whatever, in the same case it happens with Matrices having the same eigenvalues.

2. The second question is..if we have the Linear operator defining a PDE:

$$F(U_xx , U_yy , U_xy ,x,y)=0$$

Let's suppose we know the function U(x,y) that satisfy the PDE above, then what's the best method to solve:

$$F(U_xx , U_yy , U_xy ,x,y)=g(x,y)$$

knowing the solution of the case where g(x,y)=0

 

[Feynman]

you can apply the weak formulation on the 2 equations and compare it

 

[HallsofIvy]

Let be L and G 2 linear operators so they have the same set of Eigenvalues, then:

$$L[y]=-\lambda _{n} y$$ and $$G[y]=-\lambda _{n} y$$

Did you mean to assert that they have the same eigenvectors too?

then i believe that either L=G or L and G are related by some linear transform or whatever, in the same case it happens with Matrices having the same eigenvalues.

Any two linear operators are "related by some linear transformation". I think you meant L= MGM^-1 for some invertible matrix M. Yes, that's true. If two linear operators have the same eigenvalues, even if they don't have the same eigenvectors, they are "equivalent" in that sense.

2. The second question is..if we have the Linear operator defining a PDE:

$$F(U_xx , U_yy , U_xy ,x,y)=0$$

Let's suppose we know the function U(x,y) that satisfy the PDE above, then what's the best method to solve:

$$F(U_xx , U_yy , U_xy ,x,y)=g(x,y)$$

knowing the solution of the case where g(x,y)=0

Green's function is designed for that case. If G(x,y,t) is the Green's Function of F (and some boundary conditions) then the solution to the inhomogeneous differential equation (with the same boundary conditions) is $\int G(x,y,t)g(t)dt$. Look up "Green's Function" or "Green's Function Method"