Ch 11.5: Further Remarks on Separation of Variables: A Bessel Series Expansion In this chapter we are interested in extending the method of separation of variables developed in Chapter 10 to a larger class of problems – to problems involving more general differential equations, more general boundary conditions, or different geometrical regions. We indicated in Section 11.3 how to deal with a class of more general differential equations or boundary conditions. In this section we concentrate on problems posed in various geometrical regions, with emphasis on those leading to singular Sturm-Liouville problems when the variables are separated.

Separation of Variables: Linear Problems Because of its relative simplicity, as well as the considerable physical significance of many problems to which it is applicable, the method of separation of variables merits its important place in the theory and application of partial differential equations. However, this method does have certain limitations that should not be forgotten. In the first place, the problem must be linear so that the principle of superposition can by invoked to construct additional solutions by forming linear combinations of the fundamental solutions of an appropriate homogeneous problem.

Separation of Variables: Solving Resulting Ordinary Differential Equations We must also be able to solve the ordinary differential equations, obtained after separating the variables, in a reasonably convenient manner. In some problems to which separation of variables may be applied in principle, it is of limited practical value due to a lack of information about the solutions of the ordinary differential equations that appear.

Separation of Variables: Geometry of Region The geometry of the region involved in the problem is subject rather severe restrictions. On the one hand, a coordinate system must be employed in which the variables can be separated, and the partial differential equation replaced by a set of ordinary equations. For Laplace’s equation there are many such coordinate systems, including rectangular, cylindrical, and spherical. On the other hand, the boundary of the region of interest must consist of coordinate curves or surfaces – that is, curves or surfaces on which one variable remains constant. Thus, at an elementary level, we are limited to regions bounded by straight lines or circular arcs in two dimensions, or by planes, cylinders, cones, or spheres in three dimensions.

Separation of Variables: Singular Problems In three dimensional problems, the separation of variables in Laplace’s operator uxx + uyy + uzz leads to X'' + X = 0 in rectangular coordinates, to Bessel’s equation in cylindrical coordinates, and to Legendre’s equation in spherical coords. It is this fact that is largely responsible for the intensive study that has been made of these equations and the functions defined by them. It is also noteworthy that two of the three most important situations lead to a singular, rather than regular, Sturm-Liouville problems. Thus singular problems are by no means exceptional and may even be of greater interest than regular ones.

The Vibrations of a Circular Elastic Membrane (1 of 8) In Section 10.7, we noted that the transverse vibrations of a thin elastic membrane are governed by the wave equation To study the motion of a circular membrane it is convenient to write this equation in polar coordinates: We will assume that the membrane has unit radius, that it is fixed securely around its circumference, and that initially it occupies a displaced position independent of the angular variable , from which it is released at time t = 0.

Boundary Value Problem (2 of 8) Because of the circular symmetry of the initial and boundary conditions, we assume that u is also independent of , that is, u is a function of r and t only. Thus our boundary value problem is where f (r) describes the initial configuration of the membrane. For consistency, we also require that f (1) = 0. Finally, we require u(r,t) to be bounded for 0  r  1.

Separation of Variables Method (3 of 8) We begin by assuming u(r,t) = R(r)T(t). Substituting this into our differential equation we obtain where  > 0 is a constant. It follows that

Bessel’s Equation of Order Zero (4 of 8) To solve we introduce the variable  = r, from which we obtain This is Bessel’s equation of order zero, and hence or As in Section 11.4, J0 and Y0 are Bessel functions of the first and second kinds, respectively, of order zero.

Eigenfunctions (5 of 8) Thus we have The boundedness condition on u(r,t) requires that R remains bounded as r  0. Since J0(0) = 1 and Y0(x)  - as x  0, we choose c2 = 0. The boundary condition u(1,t) = 0 then requires that J0() = 0. Recall from Section 11.4 that J0() = 0 has an infinite set of discrete positive zeros 1 < 2 < … < n < …. The functions J0(nr) are eigenfunctions of a singular Sturm-Liouville problem and can be used as the basis of a series expansion for the given function f.

Fundamental Solutions (6 of 8) The fundamental solutions of this problem, satisfying the partial differential equation and boundary condition, as well as the boundedness condition, are Next we assume that u(r,t) can be expressed as an infinite linear combination of the fundamental solutions:

Coefficients (7 of 8) From the previous slide, we have The initial conditions require that From Equation 26 of Section 11.4, we obtain

Solution (8 of 8) Thus the solution to our boundary value problem describing the transverse vibrations of a thin elastic membrane, is given by