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

∇²φ = −k²φ

where k is a constant. The Generalized Helmholtz equation is that equation with k being a function of the independent variable(s).

The One Dimensional Case

In one dimension the Helmholtz equation is

(d²φ/dx²) = −k²φ(x)

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.

Change of Variable

The sinusoidal solution being a function of kx suggests that the solution at the generalized equation may a function of

X=∫₀^xk(z)dz.
hence
dX=k(x)dx
and
(dX/dx) = k(x)

Then

(dφ/dx) = (dφ/dX)(dX/dx) = (dφ/dX)k(x)
and
(d²φ/dx²) = (d²φ/dX²)k²(x) + (dφ/dX)(dk/dX)k

Since (d²φ/dx²) is equal to −k²φ the above equation can be reduced to

(d²φ/dX²) + (dφ/dX)(dk/dX)/k = −φ

A Matric Equation

Let (dφ/dX) be denoted as ψ and (dk/dX)/k as γ. Then

(dφ/dX) = ψ
(dψ/dX) = −φ − γψ

In matric form

(dΦ/dX) = −MΦ

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:

(dy/dx) = −μ(x)y
 
(1/y)(dy/dx) = −μ(x)
Inegrating
from 0 to x
gives
ln(y(x)) − ln(y(0)) = −∫₀^xμ(z)dz
hence
y(x) = exp(−∫₀^xμ(z)dz)y(0)

This suggests that the solution to the matrix equation is

Φ(X) = exp(−∫₀^XMdZ)Φ(0)

The integral of the matrix M is the following matrix

∫0XM(Z)dZ =	|   0	−X |
	| X	∫0Xγ(z)dz |

The solution is therefore

| φ(X) |		|   0	X         |	| φ(0) |
|	| = exp	{		}
| ψ(X) |		| −X	−∫0Xγ(z)dz     |	|ψ(0) |

(To be continued.)