Separation of variables for solutions of partial differential equation

In summary, the method of separation of variables works for boundary valued problems as long as the problem is linear, the boundary consists of co-ordinate curves or surfaces, and the equation is separable in those co-ordinates. This is possible because the solution can be written as a sum of solutions that satisfy mostly homogeneous boundary conditions, making it applicable for Sturm-Liouville theory.
  • #1
jamesb1
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Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?
 
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  • #2
jamesb1 said:
Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?

You don't need the boundary conditions to be homogeneous for the method to work, and conversely homogeneity of the boundary conditions does not guarantee that the method will work.

What you do need is for the problem to be linear, the boundary to consist of co-ordinate curves or surfaces, and for the equation to be separable in those co-ordinates.

Since the problem is linear, its solution can be written as a sum of solutions satisfying mostly homogeneous boundary conditions, which is what one needs in order to apply Sturm-Liouville theory.
 

Related to Separation of variables for solutions of partial differential equation

1. What is separation of variables for solutions of partial differential equations?

Separation of variables is a method used to solve partial differential equations (PDEs) by breaking them down into simpler equations that can be solved independently. This method is based on the assumption that the solution to a PDE can be written as the product of two or more simpler functions, each of which depends on only one of the variables in the equation.

2. When is separation of variables used for solving PDEs?

Separation of variables is typically used for solving linear PDEs, where the dependent variable and its partial derivatives appear in the equation in a linear fashion. This method is also most effective when the PDE has homogeneous boundary conditions, meaning that the boundary conditions do not depend on the dependent variable.

3. What are the steps involved in separation of variables?

The first step is to assume that the solution can be written as the product of two or more simpler functions. Then, the equation is rearranged to place all terms with the dependent variable on one side and all terms with the independent variables on the other side. The next step is to equate each of these sides to a constant, resulting in a set of ordinary differential equations. Finally, these separate ODEs are solved and combined to obtain the solution to the PDE.

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

While separation of variables is a powerful method for solving PDEs, it is not applicable to all types of PDEs. It is most effective for linear PDEs with homogeneous boundary conditions. Additionally, finding the separate solutions to the ODEs can be challenging and may not always be possible.

5. Can separation of variables be used for non-linear PDEs?

No, separation of variables is only applicable to linear PDEs. Non-linear PDEs require more advanced techniques, such as numerical methods, to find solutions.

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