Separation of variables for solutions of partial differential equation

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SUMMARY

The method of separation of variables is applicable to linear partial differential equations (PDEs) with boundary conditions that are not necessarily homogeneous. The key requirements for this method to be effective include the linearity of the problem, the boundary being defined by coordinate curves or surfaces, and the equation being separable in those coordinates. The use of Sturm-Liouville theory is essential, as it allows for the construction of solutions from a sum of individual solutions that satisfy homogeneous boundary conditions.

PREREQUISITES
  • Understanding of linear partial differential equations (PDEs)
  • Familiarity with boundary value problems
  • Knowledge of Sturm-Liouville theory
  • Concept of separable equations in coordinate systems
NEXT STEPS
  • Study the application of Sturm-Liouville theory in solving linear PDEs
  • Explore examples of boundary value problems with non-homogeneous conditions
  • Learn about coordinate systems and their role in separable equations
  • Investigate the implications of linearity in PDE solutions
USEFUL FOR

Mathematicians, physicists, and engineers working with partial differential equations, particularly those involved in boundary value problems and seeking to deepen their understanding of separation of variables and Sturm-Liouville theory.

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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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.
 

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