Introduction, Analytical Solution of PDEs, Mathematical Methods

Mathematical Methods (10/24.539)

X. Analytical Solution of PDEs

Introduction

The topic of PDEs is probably the most important subject in applied engineering math. This is true from the physical viewpoint because PDEs result from the mathematical modeling of real engineering/physical systems. From the pure math perspective, the solution of PDEs brings together much of one’s knowledge of advanced mathematics - ODEs, eigenvalue problems, orthogonality, Laplace and Fourier Transforms, linear algebra, numerical methods, etc.. In addition, there are a rich variety of interesting applications from which to select examples to illustrate the various methods. Thus, not only is this subject of fundamental importance, it can also be quite rewarding when all your mathematics background comes together to yield interesting solutions to challenging real-world problems.

In our relatively brief overview of PDE solution methods (note that one could easily spend a full semester or more on this subject), we will find that certain techniques are appropriate for certain classes of problems (this is especially true for the numerical solution techniques discussed in Section XI). Thus, it is often useful to classify various PDEs into one or more categories, and then discuss solution methods appropriate for the various classes that have been defined. The most common classification scheme in use deals with 2^nd order Quasi-Linear systems that are defined by

where u(x,y) is the dependent variable, x and y are the two independent variables, the system can have variable coefficients, A, B, and C, and the RHS function, , can be just about any function of interest (linear or nonlinear). The term quasi-linear is used because the left hand side (LHS) of eqn. (10.1) in linear in the dependent variable, but the RHS function may not be.

Equation (10.1) is often written using a shorthand notation for the partial derivatives, or

where, in general, the coefficients A, B, and C are functions of x and y, and the u_xx, u_xy, and u_yy second-order partial derivatives are defined explicitly by

We will use this shorthand notation throughout our treatment of PDE solution methods.

The most common PDE classification scheme identifies the PDE as either a hyperbolic, parabolic, or elliptic equation depending on the sign of the term (which can vary with x and y). In particular, we have the following classification scheme:

These types of systems give rise to significantly different characteristic behavior and, as mentioned above, the solution scheme for each method can also differ. An example of each type of PDE is summarized below:

Application	Differential Equation	Value of Coefficients	Sign of	PDE Class
Wave Equation			positive	hyperbolic
Diffusion Equation			zero	parabolic
Poisson’s Equation			negative	elliptic

Methods for solution of PDEs are best demonstrated by examples. This statement is probably applicable for most subjects in science, math, and engineering courses, but it is particularly true here. Thus, our approach, after a very brief overview of the theory, will be to simply do, in detail, as many problems as possible.

With this overall direction in mind, the organization for the remainder of this section of notes is given below:

The Separation of Variables Method

The Classical Separation of Variables Method

Treatment of Inhomogeneous Equations and BCs

The Eigenfunction Expansion Method

Analytical Solutions using Separation of Variables

Example 10.1 - Heat Transfer -- Finite Bar with Homogeneous BCs

Example 10.2 - Heat Transfer -- Finite Bar with Fixed End Temperatures

Example 10.3 - Heat Transfer -- Finite Bar with Time Dependent BCs

Example 10.4 - Heat Transfer -- Cylindrical Bar with Homogeneous BCs

10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated November 1998).

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