Superposition in separation method of variables

In summary, the conversation discusses the use of different boundary conditions to solve a problem using superposition. It also touches on the concept of creating a square pulse using two step functions and the potential solution to the given problem. @kuruman provides a helpful hint and encourages further exploration and discussion on the topic.
  • #1
Celso
33
1
Homework Statement
In a cube, the specified boundary counditions are V = ##V_{0}## in the ##z = 0## and ##z = d## planes and ##V = 0## for the other sides. Knowing the solution for when only one of the z planes is kept at ##V = V_{0}## how is it possible to use superposition to know ##V(r)## in this situation?
Relevant Equations
##\nabla^2 V = 0##
Each different boundary condition means a different charge configuration, how can this problem be solved using superposition?
 
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  • #2
Seems very unlikely to me. Anyone says it can be ?
 
  • #3
Let ##\Phi_0(x,y,z)## = solution when only the face at z = 0 is at ##V_0##.
Let ##\Phi_d(x,y,z)## = solution when only the face at z = d is at ##V_0##.
Assuming that the side of the cube is ##d##, isn't it true that ##\Phi_d(x,y,z)=\Phi_0(x,y,d-z)##?
 
  • #4
Can you create a square pulse using two step functions? That is the crux of the question. I won't give the answer, as it is homework.
 
  • #5
BvU said:
Seems very unlikely to me. Anyone says it can be ?
Ah! I get it: the exercise text as it was given to you says so o:) . Then it is most likely true, hmm ?
And @kuruman is giving you (much) more than just a hint !
 
  • #6
BvU said:
Ah! I get it: the exercise text as it was given to you says so o:) . Then it is most likely true, hmm ?
And @kuruman is giving you (much) more than just a hint !
It's easy to assume that the problem is asking for the solution. Perhaps I was overly generous when I posted but I wanted to point OP in the right direction and there is still quite a bit to be said to complete a formal answer.
 
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Related to Superposition in separation method of variables

1. What is superposition in separation method of variables?

Superposition in separation method of variables is a mathematical technique used to solve differential equations by breaking them down into simpler equations that can be solved separately. It involves using the principle of linear combination, where the solutions of the simpler equations are added together to find the solution of the original equation.

2. How does superposition work in separation method of variables?

In superposition, the original equation is divided into two or more simpler equations, each with its own set of variables. The solutions of these equations are then combined using linear combination to find the solution of the original equation.

3. What are the benefits of using superposition in separation method of variables?

Superposition allows for the solution of complex differential equations by breaking them down into simpler equations. This makes it easier to solve and understand the behavior of the system. It also allows for the use of different boundary conditions for each equation, which can provide more accurate solutions.

4. Are there any limitations to using superposition in separation method of variables?

Superposition can only be used for linear differential equations, meaning that the variables must be raised to the first power. It also requires the equations to be homogeneous, meaning that all terms must have the same degree. Additionally, the variables must be independent, meaning that they cannot be multiplied or divided by each other.

5. Can superposition be used in other scientific fields?

Yes, superposition is a widely used technique in various scientific fields such as physics, engineering, and chemistry. It is used to solve differential equations, but it can also be applied to other types of equations, such as partial differential equations and boundary value problems.

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