A First Course in Partial Differential Equations

The following sections are included:

• Dedication
• Preface
• About the Authors
• Acknowledgments
• Contents

When modeling physical phenomena, often it is necessary to consider functions that depend on more than one variable. Therefore, if rates of changes are involved, partial derivatives of functions must be used and this leads to mathematical models that are partial differential equations. Loosely speaking, a partial differential equation is an equation that involves an unknown function of two or more variables and its partial derivatives. The following are some examples of partial differential equations:

This chapter is focused on first-order partial differential equations. This foray in first-order partial differential equations and their applications is self contained and independent of the remaining chapters of this text, thus it can be skipped or postponed without disrupting most other topics in the book. However, this chapter should be read before exploring the topics in Chap. 11 pertaining to nonlinear partial differential equations…

The heat equation was used to illustrate the method of separation of variables in Sec. 1.6. Separation of variables produced product solutions to the partial differential equation describing heat conduction in a uniform, one-dimensional rod. The product solutions were chosen to satisfy the Dirichlet boundary conditions selected for Example 1.12. To find a solution to a chosen initial condition the Principle of Superposition is employed. This necessitates expanding the initial condition function as a series of eigenfunctions (in the example, as a series of trigonometric sine functions). This produces a formal solution of an initial boundary value problem for the heat equation. As seen in the examples of Sec. 1.6, if the initial condition function is a finite linear combination of sine functions, then by equating coefficients of the sine functions, the solution to the initial boundary value problem of the one-dimensional heat equation is found in closed form. French mathematician and physicist/engineer, Jean Baptiste Joseph Fourier is known for initiating the investigation of the representation of functions as trigonometric series (now called Fourier series) and the application of this method to problems of heat transfer. In fact, Fourier essentially developed the procedure of separation of variables and presented many concrete examples of trigonometric expansions in connection with boundary value problems related to heat conduction problems. German mathematician Peter Dirichlet is believed to be the first person to describe sufficient conditions that guarantee the convergence of a Fourier series. The work of Fourier and Dirichlet leads to the important branch of mathematics known as Fourier analysis, or more generally, harmonic analysis today.

In this chapter solutions to one of the basic equations of mathematical physics will be presented. The previous material on separation of variables (Sec. 1.6) and Fourier series (Chap. 3) will be used extensively. The heat equation also arises in situations where a quantity (whether it is heat energy, a chemical, or something more abstract such as “information”) diffuses through a continuous medium, thus an understanding of its properties and solution is important to any applied mathematician or engineer. The sections of this chapter are organized according to the spatial domains and the boundary conditions imposed on the heat equation. The first section will treat the heat equation on a closed, bounded (i.e., compact) interval. After three different types of homogeneous boundary conditions on the closed, bounded interval have been studied, nonhomogeneous boundary conditions, such as boundary conditions which depend on time will be studied. One of the most important theoretical results related to the heat equation is that under relatively mild assumptions the maximum and minimum properties of the solution can be described and the uniqueness of the solution to the initial boundary value problem can be proved. Section 4.3 covering the Maximum Principle and the uniqueness of solutions can be skipped on a first reading. In Sec. 4.4 the boundedness of the spatial domain will be relaxed and solutions on infinite and semi-infinite intervals will be explored. The chapter will close with an extension of the one-dimensional results to solutions of the heat equation on a two-dimensional rectangular spatial domain and suggestions for further explorations of the heat equation.

In Chap. 1, the one-dimensional wave equation and the corresponding initial boundary value problems arising from various physical situations were described. This chapter will concentrate on the basic techniques of finding and interpreting the solutions of these initial boundary value problems. First, the method of separation of variables is used to solve the initial boundary value problem of Dirichlet type in Sec. 5.1. Next, Sec. 5.2 introduces d’Alembert’s solution as well as topics related to wave propagation. The rest of the chapter is devoted to nonhomogeneous initial boundary value problems and some theoretical results including the energy integral and uniqueness of solutions. Solutions of the wave equation possess very different properties compared to those of the heat equation. While the heat equation serves as an example from the family of parabolic equations, the wave equation is a member of the class of hyperbolic equations.

This chapter is devoted to the discussion of Laplace’s equation, which is an example of a second-order linear elliptic partial differential equation. Preliminary discussion of this type of equation was covered in Sec. 1.5. Laplace’s equation and the more general Poisson’s equation are of great importance in mathematics, mathematical physics, and various other areas. The steady-state solutions of the heat and wave equations satisfy Laplace’s equation (see Chap. 4 and Chap. 5).

A common theme runs through the process of solving most initial boundary value problems via the method of separation of variables. A solution is assumed to be a product of two of more functions each depending on a single variable. The functions which depend on the spatial variables (usually x, y, r, etc.) must satisfy the boundary conditions of the problem. This usually requires solving an ordinary differential equation with those associated boundary conditions, or as it has come to be called, a boundary value problem. See Eqs. (1.52)–(1.54) for a canonical example. The method of separation of variables usually introduces a constant which plays a role in the solution of the boundary value problem. Nontrivial solutions of the boundary value problem exist only for certain values of the constant. The values of the constant are called eigenvalues and the associated nontrivial solutions to the boundary value problem are called eigenfunctions. The boundary value problems encountered in the previous chapters on the heat, wave, and Laplace’s equation possess eigenfunctions forming orthogonal sets which, as a consequence of the results on Fourier series described in Chap. 3, allow a sufficiently smooth function, at least formally, to be represented as an infinite series whose terms are constant multiples of eigenfunctions. In many cases, this enabled the initial condition of the initial boundary value problem to be satisfied as well as the boundary conditions.

The reader must have noticed that performing the method of separation of variables in certain common domains such as one-dimensional intervals, rectangles, disks, sectors, and annuli results in being asked to solve similar types of ordinary differential equations and boundary value problems. So far eigenfunctions involving sine, cosine, exponential, and powers of an independent variable (in the case of Euler’s equation) have been common. However, a much wider variety of ordinary differential equations and boundary value problems occur frequently in applications of partial differential equations in the physical sciences and engineering. This chapter will review some of the differential equations arising in other domains and situations including Bessel’s equation, Legendre’s equation, and Laguerre’s equation. This chapter applies the Sturm-Liouville theory of Chap. 7, though many of the examples of the current chapter are singular Sturm-Liouville boundary value problems rather than regular problems. If the reader wishes, this chapter can be skimmed or skipped until one of these special ordinary differential equations arises in the course of carrying out the method of separation of variables. This section also serves as a review of series solutions of second-order linear ordinary differential equations near ordinary and regular singular points and the Method of Frobenius.

This chapter will explore some mathematical models of physical phenomena and will make use of the mathematical methods and solution techniques developed in earlier chapters. In particular the method of separation of variables and Fourier analysis will be used in several new situations. The applications of partial differential equation techniques to be presented here have been selected to illustrate the utility of some of the special functions introduced in Chap. 8. Many other applications could be explored as well and the interested reader is encouraged to pursue readings in electricity and magnetism, classical and quantum mechanics, and optics. The interested reader can refer to [Garrity (2015); Griffiths (1999); Rauch (2012)].

In previous chapters, particularly those exploring solution techniques for the heat and wave equations, some examples and exercises have included nonhomogeneous boundary conditions and ad hoc methods for finding the solutions to these initial boundary value problems were presented. The purpose of this chapter is to present a generally applicable method for solving nonhomogeneous initial boundary value problems. This method will systematically solve nonhomogeneous problems by breaking them into a collection of related initial boundary value problems in which the nonhomogeneous terms are isolated and separated in the various partial differential equations, boundary conditions, or initial conditions. The main result is driven by Duhamel’s Principle which is closely related to a method used to find particular solutions to linear, nonhomogeneous ordinary differential equations called variation of parameters.

So far this textbook has concentrated on linear partial differential equations except in Chap. 2 where the solutions to first order, quasilinear partial differential equations were found using the method of characteristics. In the real world, many physical phenomena are the result of complex interactions of many physical processes such as convection, diffusion, and reaction. To model such interactions, nonlinearity is unavoidable, and as a result, nonlinear partial differential equations arise. The subject of nonlinear partial differential equations is much more complicated and challenging than its linear counterpart. As seen in earlier chapters, the Principle of Superposition of solutions and the method of separation of variables play crucial roles in the process of finding solutions of boundary value problems for linear partial differential equations. However, these fundamental principles and methods are no longer applicable to most nonlinear partial differential equations. In fact, there is no general method that can be used to solve a large class of nonlinear partial differential equations. As Lawrence C. Evans [Lindenstrauss et al. (1994)] once put it: Keep in mind that there is in truth no central core theory of nonlinear PDE, nor can there be. The sources of PDE are so many — physical, probabilistic, geometric, etc. — that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear PDE by utterly different methods

The reader has reached another chapter of optional material intended to give a more complete picture of methods for finding solutions to partial differential equations. In this chapter, a numerical approach to approximating the solutions to partial differential equations will be introduced, the finite difference method. A proper study of this method would occupy the space of several textbooks and require a few semesters of study, so this chapter will merely introduce the method and illustrate its use on a sampling of examples. Readers interested in pursuing this topic further will find a wealth of printed and web-based resources. In particular the textbooks [Smith (1985)] and [Strikwerda (2004)] are good starting points. Source code files written in the Java programming language [Schildt (2014)] for the examples presented in this chapter are available for download from the website supporting this textbook. Java was chosen, not because of any special suitability for implementing finite difference methods, but because the language and tools required to develop, compile, debug, and run programs written in it are ubiquitous and generally available for free.

The following sections are included:

• Appendix A Complex Arithmetic and Calculus
  • Complex Numbers
  • Complex Functions
  • Complex Differentiation
  • Complex Integration
• Appendix B Linear Algebra Primer
  • Matrices and Their Properties
  • Determinant of a Square Matrix
  • Vector and Matrix Norms
  • Eigenvalues and Eigenvectors
  • Diagonally Dominant Matrices
  • Positive Definite Matrices

The following sections are included:

• Bibliography
• Index