Solving the PDE Wave Equation - A_n & B_n Terms

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SUMMARY

The discussion focuses on solving the Partial Differential Equation (PDE) wave equation, specifically addressing the A_n and B_n terms in the context of Fourier analysis. It emphasizes the use of countably infinite coefficients in linear PDE solutions, which reflects the infinite dimensionality of the solution space. The standard procedure for solving homogeneous linear PDEs involves separating the equation, obtaining a basis, constructing an infinite sum, and applying initial or boundary conditions to determine the coefficients using generalized Fourier techniques.

PREREQUISITES
  • Understanding of Fourier analysis and its application to linear PDEs
  • Familiarity with homogeneous linear PDEs and their properties
  • Knowledge of boundary and initial conditions in differential equations
  • Concept of infinite dimensional vector spaces in mathematics
NEXT STEPS
  • Study the method of separation of variables for PDEs
  • Learn about constructing bases for solution spaces in linear PDEs
  • Explore generalized Fourier series and their applications
  • Investigate the implications of boundary conditions on PDE solutions
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Mathematicians, physicists, and engineering students focusing on differential equations, particularly those interested in wave phenomena and Fourier analysis techniques.

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Putting a countable infinity of A's and B's instead of only 2 values A and B is a standard technique in Fourier analysis, which is particulary well suited for linear ODE's/PDE's. That is you rightfully assume that, because the linear/vector space of solutions is infinite dimensional, there is one (generally different) coefficient for each basis vector of the solution space, hence the infinite summation.
 
Which part of it do you not understand.
This is how a homogenous linear PDE is usually solved:
1.separate equation (possible for many of the usual PDEs)
2.Solve the equations to get a "basis"
3.Write an infinite sum with them
4.Use the imposed initial/boundary conditions to solve for the coefficients of the linearly independent solutions with the generalized Fourier's trick

If none of that made sense to you, then you are not ready for that book.
 

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