A PDE and Linear operator questions.

In summary: PDEs.In summary, two linear operators with the same set of eigenvalues are equivalent and can be related by a linear transformation. For solving a PDE with a known solution for the case of g(x,y)=0, Green's function method can be used to find the solution for the inhomogeneous case.
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

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

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"

1. What is a PDE?

A PDE, or partial differential equation, is an equation that involves multiple variables and their partial derivatives. It is commonly used in mathematical modeling to describe physical phenomena such as heat transfer, fluid dynamics, and quantum mechanics.

2. How is a PDE different from an ODE?

A PDE involves multiple variables and their partial derivatives, while an ODE (ordinary differential equation) involves only one variable and its derivatives. PDEs are used to describe systems in multiple dimensions, while ODEs are used for systems in one dimension.

3. What are some common types of PDEs?

Some common types of PDEs include the heat equation, wave equation, and Laplace's equation. These equations have different forms and are used to model different physical phenomena. Other types of PDEs include the diffusion equation, Schrödinger equation, and Navier-Stokes equations.

4. What is a linear operator?

A linear operator is a mathematical function that maps one vector space to another, preserving the vector space's structure. In the context of PDEs, linear operators are used to define the relationship between the dependent and independent variables in the equation. They are essential in solving PDEs and can be represented by matrices or differential operators.

5. How are PDEs solved?

PDEs can be solved using various analytical and numerical methods. Analytical methods involve finding an exact solution to the equation using mathematical techniques such as separation of variables or Fourier transforms. Numerical methods, on the other hand, use computer algorithms to approximate the solution to the PDE. Some common numerical methods include finite difference, finite element, and spectral methods.

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