Superposition on continuty condition

[smoger]

I'm confused about the superposition when the B.C(boundary condition) is continuity condition between two domain.

The laplace equation(or possion) expressed as (in circular annulus),

∂²A/∂r² + 1/r ∂A/∂r + 1/r² ∂²A/∂θ² = f(r,θ)


Assume that there are two domains where the boundary of each domain is :

1. (a<r<b, 0<θ<2pi)
with B.C : A₁(b,θ)=A₂(b,θ) , and etc.. on elsewhere(r=a, or θ=somewhere)

2. (b<r<c, 0<θ<2pi)
with B.C : ∂A₁/∂θ=∂A₂/∂θ (at r=b)
, and etc.. on elsewhere(r=c, or θ=somewhere)


Is that right to solve the equation by adopting super position theory?
for example, In domatin 1,

A₁(r,θ)=A₁₁(r,θ)+A₁₂(r,θ) (1)

that,
A₁₁(r,θ) satisfy A₁₁(b,θ)=0 (2)

A₁₂(r,θ) satisfy A₁₂(b,θ)=A₂(b,θ) (3)


Can i make the continuty condition 0 as a general non homogeneous B.C condition to solve the equation by
super position? thanks in advance

 

[jvicens]

Thank you for your question. Superposition is a very useful tool in solving boundary value problems, including those involving continuity conditions between two domains. In your case, it is possible to use superposition to solve the Laplace or Poisson equation in the circular annulus described in your post.

First, let's briefly review what superposition is. Superposition is a mathematical principle that allows us to combine solutions to simpler problems to obtain a solution to a more complex problem. In the case of boundary value problems, this means that we can solve for the solution in one domain and then add it to the solution in another domain to obtain the overall solution.

Now, let's apply this to your specific problem. In domain 1, we can solve for A11(r,θ) using the boundary condition A11(b,θ)=0 and the Laplace/Poisson equation. Similarly, in domain 2, we can solve for A12(r,θ) using the boundary condition A12(b,θ)=A2(b,θ) and the Laplace/Poisson equation. Then, we can add these two solutions together to obtain the overall solution A1(r,θ) in domain 1, as shown in equation (1) in your post.

The key here is that each of these solutions satisfies the Laplace/Poisson equation and the boundary conditions in their respective domains. By adding them together, we are able to satisfy the continuity condition between the two domains.

To answer your question, yes, you can use a general non-homogeneous boundary condition as a part of the superposition method. As long as each individual solution satisfies the Laplace/Poisson equation and the boundary conditions in their respective domains, the overall solution will also satisfy the continuity condition between the two domains.

I hope this helps to clarify your confusion about superposition and its application to boundary value problems. Please let me know if you have any further questions or need more clarification.