11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series

  • This is an improper integral.
  • We are interested in extending the method of separation of variables developed in Chapter 10 to a larger class of problems, such as differential equations, boundary conditions, or different geometrical regions.
    • Section 11.3 shows how to deal with a class of differential equations.
    • When the variables are separated, we focus on problems posed in various geometrical regions.
  • The method of separation of variables merits its place in the theory and application of partial differential equations because of its relative simplicity, as well as the considerable physical significance of many problems to which it is applicable.
  • The method has certain limitations that should not be forgotten.
    • The problem needs to be linear so that the principle of superposition can be invoked to construct additional solutions.
  • We need to be able to solve the ordinary differential equations in a convenient manner.
  • The geometry of the region involved in the problem is subject to severe restrictions.
    • On the other hand, a coordinate system must be used in which the variables can be separated, and the partial differential equation replaced by a set of ordinary differential equations.
    • There are about a dozen coordinate systems for Laplace's equation, and only rectangular, circular cylindrical, and spherical coordinates are likely to be familiar to most readers of this book.
    • 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.
    • At an elementary level, one is limited to regions with straight lines or circular arcs in two dimensions, or by planes, circular cylinders, circular cones, or spheres in three dimensions.
  • This fact is responsible for the intensive study that has been made of these equations.
    • Two of the three most important situations lead to singular, rather than regular, problems.
    • There are no exceptional problems and may be of even greater interest than regular ones.
    • An example involving an expansion of a function as a series of Bessel functions is the remainder of this section.
  • It's convenient to write Eq.
    • to study the motion of a circular membrane.

  • The allowable values of the separation constant are obtained from the roots of the equation.