Solving Differential Equations

In this chapter, standard methods are described for solving differential equations, which are usually encountered in physics. The Laplace equation, the diffusion equation and the wave equation are solved assuming suitable boundary conditions. The Sturm-Louiville problem, the method of separation of variables, and Green’s functions are discussed in detail—volume and surface Green’s functions, etc. The content of this chapter is more or less standard, although the methods are not. To fully appreciate boundary value problems, a Python code is given which seeks to find the energy eigenvalues and eigenfunctions for the quantum simple harmonic oscillator numerically—the student must guess the correct value of energy—only the correct energy eigenvalues lead to the boundary conditions being satisfied.