# Preface

This section gives some examples of solving the Helmholtz equation \( \nabla^2 u + k^2 u = 0 . \)

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Introduction to Linear Algebra with *Mathematica*

## Glossary

# Helmholtz equation

We demonstrate application of the separation of variables in solving the Helmholtz equation \( \nabla^2 u + k^2 u = 0 . \) The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

Cartesian Coordinates

*k*² is the eigenvalue. We split Eq.\eqref{EqHelmholtz.1} into a set of ordinary differential equations by considering

*u*=

*X Y Z*and rearranging terms, we get

*x*alone, whereas the right-hand side depends only on

*y*and

*z*and not on

*x*. But

*x, y*, and

*z*are all independent coordinates. The equality of both sides depending on different variables means that the behavior of

*x*as an independent variable is not determined by

*y*and

*z*. Therefore, each side must be equal to a constant, a constant of separation. We choose

*y*equated to a function of

*z*, as before. We resolve it, as before, by equating each side to another constant of separationm, -

*m*²:

*n*² by \( \displaystyle k^2 = \lambda^2 + m^2 + n^2 \) to produce a symmetric set of equations. Now we have three ODEs \eqref{EqHelmholtz.3},\eqref{EqHelmholtz.6}, and \eqref{EqHelmholtz.7}.

Our solution should be labeled according to the choice of our constants λ, *m*, and *n*; that
is,

*k*² = λ² +

*m*² +

*n*², we may choose λ,

*m*, and

*n*as we like, and formula \eqref{EqHelmholtz.8} will still be a solution of the Helmholtz equation, provided

*X*

_{λ}(

*x*) is a solution of Eq.\eqref{EqHelmholtz.3}, and so on. We may develop the most general solution of Eq.\eqref{EqHelmholtz.1} by taking a linear combination of solutions

*u*

_{λ,m,n}:

*c*

_{λ,m}are finally chosen to permit

*u*(

*x, y, z*) to satisfy the boundary conditions of the problem, which, as a rule, lead to a discrete set of values λ,

*m*.

Circular Cylindrical Coordinates

*u*(

*x, y, z*) dependent on ρ, φ, and

*z*, the Helmholtz equation becomes

*u*:

*u*and moving the

*z*derivative to the right-hand side yields

*z*on the right appears to depend on a function of ρ and φ on the left. We resolve this by setting each side of equal to the same constant. Let us choose -λ². Then

*k*² + λ² =

*n*², multiplying by ρ², amd rearranging terms, we obtain

*m*² and

The original Helmholtz equation, a three-dimensional PDE, has been replaced by three ODEs, Eqs. \eqref{EqHelmholtz.11}, \eqref{EqHelmholtz.12}, and \eqref{EqHelmholtz.13}. A solution of the Helmholtz equation is

Spherical Polar Coordinates

*k*² constant, in spherical polar coordinates

*u*, we get

*r*² sin²θ, we can isolate \( (1/\Phi)\,({\text d}^2 \Phi /{\text d}\phi^2 ) \) to obtain

*r*and θ alone. Since

*r*, θ, and φ are independent variables, we equate each side of the equation above to a constant. In almost all physical problems φ will appear as an azimuth angle. This suggests a periodic solution rather than an exponential. With this in mind, let us use -

*m*² as the separation constant, which, then, must be an integer squared. Then

*r*² and rearranging terms, we obtain

*k*² be a constant is unnecessarily severe. The separation process will still be possible for

*k*² as general as

*f, g*, and

*h*;

*p*is another parameter. In the hydrogen atom problem, one of the most important examples of the Schrödinger wave equation with a closed form solution is

*k*² =

*f*(

*r*), with

*k*²independent of θ, φ. Equation \eqref{EqHelmholtz.23} for the hydrogen atom becomes the associated Laguerre equation. The great importance of this separation of variables in spherical polar coordinates stems from the fact that the case

*k*² =

*k*²(r) covers a tremendous amount of physics: a great deal of the theories of gravitation, electrostatics, and atomic, nuclear, and particle physics. And with

*k*² =

*k*²(

*r*), the angular dependence is isolated in \eqref{EqHelmholtz.21} and \eqref{EqHelmholtz.22} that can be solves explicitely.

Finally, as an illustration of how the constant m in Eq.\eqref{EqHelmholtz.21} is restricted, we note that φ in cylindrical and spherical polar coordinates is an azimuth angle. If this is a classical problem, we shall certainly require that the azimuthal solution Φ(φ) be single-valued; that is,

*m*must be an integer. Which integer it depends on the details of the problem. If the integer |

*m*| > 1, then Φ will have the period 2π/

*m*. Whenever a coordinate corresponds to an axis equation \eqref{EqHelmholtz.21} must be holt for any φ, the azimuth angle, and

*z*, an axis of translation of the cylindrical coordinate system. The solutions, of course, are sin(𝑎

*z*) and cos(𝑎

*z*) and the corresponding hyperbolic function (or exponentials) sinh(𝑎

*z*) and cosh(𝑎

*z*) for +𝑎².

- Grigoryan, V, Parial Differential Equations, 2010, University of California, Santa Barbara.

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