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Helmholtz Equation of One Dimension |
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The Helmholtz equation arises in many contexts in the attempt to give a mathematical explanation of the physical world. These range from Alan Turing's explanation of animal coat patterns to Schrödinger's time-independent equation in quantum theory.
The Helmholtz equation per se is
where k is a constant. The Generalized Helmholtz equation is that equation with k being a function of the independent variable(s).
In one dimension the Helmholtz equation is
It just has the sinusoidal solution of φ(x) = A·sin(kx)+B·cos(kx). In one dimension the Generalized Helmholtz equation has a sinusoidal-like solution of varying amplitude and wavelength.
The sinusoidal solution being a function of kx suggests that the solution at the generalized equation may a function of
Then
Since (d²φ/dx²) is equal to −k²φ the above equation can be reduced to
Let (dφ/dX) be denoted as ψ and (dk/dX)/k as γ. Then
In matric form
where
Φ = | | φ | |
| ψ | |
M = | | 0 | −1 | |
| 1 | γ | |
Note that γ is a function of X and hence so is the matrix M.
For the analogous scalar differential equation the solution would go as follows:
This suggests that the solution to the matrix equation is
The integral of the matrix M is the following matrix
∫_{0}^{X}M(Z)dZ = | | 0 | −X | |
| X | ∫_{0}^{X}γ(z)dz | |
The solution is therefore
| φ(X) | | | 0 | X | | | φ(0) | | |||||
| | | = exp | { | } | |||||
| ψ(X) | | | −X | −∫_{0}^{X}γ(z)dz | | |ψ(0) | |
(To be continued.)
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