Partial Differential Equations

The following sections are included:

• Dedication
• Preface to the First Edition
• Preface to the Second Edition
• Contents

This is a preparatory chapter where basic notions and terminologies used throughout the book are introduced. They are the order of equations, the definition of solutions, and linear and nonlinear equations. Specifically, we learn basic terminologies such as order of equations and solutions in Section 1.1 and in Section 1.2 the linearity, especially the superposition principle, which plays an important role in PDEs, is introduced. In this book most PDEs we study are linear and either the first or the second order. Basic aspects of these equations are discussed throughout this chapter. For the first order linear equations we learn a method called the method of characteristics in Section 1.3. This method is used in Section 1.4 and later chapters, especially in Chapters 3 and 5, where we study the wave equation and both linear and nonlinear first order equations, respectively. For the second order equations, the classification of the second order linear equations is treated in Section 1.4, which motivates Chapters 2, 3, and 4, where we study three fundamental second order linear equations. We also introduce the concept of well-posedness in Section 1.5.

In Section 2.1 we derive the heat equation from the basic principle called the conservation law. This law shows up in many places and it is important to know how the heat equation is derived. This principle states that the rate of change of the heat energy in a region is equal to the heat flux across the boundary of the region. The heat flux is the heat energy crossing the boundary per unit area per unit time. From Sections 2.2 to 2.5 we study the initial boundary value problems. We consider both homogeneous and non-homogeneous boundary conditions. There are several goals for this chapter. One is to learn the solution method called the separation of variables. This technique is applied for various partial differential equations including the wave equation and the Laplace equation. It is important to become familiar with the method. Another goal is to learn how the boundary conditions affect the solutions. We discuss the several well-known boundary conditions. Also, we see how the superposition principle is applied for these problems…

In this chapter we study the wave equation in detail. This equation or its extension explains the phenomena of propagation and oscillations (or vibrations). We all know that the sounds propagate in air or in other media. We also know that the lights propagate through the space. The Internet uses fiber optics with which the information (actually pulses carrying information) propagates at the speed of light. Equations modeling these phenomena are more complicated than the wave equation that we study in this chapter. Nevertheless, by studying the wave equation, we are able to learn the basic idea of how the information such as sound propagates…

In this chapter we consider the Laplace and the Poisson equations. Motivations for this chapter are explained in Section 4.1. In Section 4.2 we continue the separation of variables and we apply the method to the boundary value problems of the Laplace equation. We discuss the cases where the domains are rectangular and circular. In Section 4.3 we study the fundamental solutions. This is a solution to the Laplace equation which depends only on the radial variable and is used to construct the Green’s function. This is discussed in Section 4.4 and we have solution representation of the Laplace equation by the Green’s functions when the domains are simple. Solutions of the Laplace equation are called the harmonic functions and their properties are discussed in Section 4.5 including the maximum principle and the mean value property. In Section 4.6 one of the well-posedness issues will be studied.

This chapter is a continuation of Chapter 1 where we studied the first order linear constant coefficient equations. We study the linear, semi-linear, quasilinear, and fully nonlinear equations. If a given equation is linear in the highest order derivatives, it is called the semi-linear or quasilinear equation and if it is nonlinear in the highest order derivatives, it is called the fully nonlinear equation…

In the separation of variables discussed in earlier chapters, we expanded a given function in the Fourier sine or cosine series. It was mentioned that they are the special cases of the Fourier series. In this chapter we study the Fourier series in details and discuss the relations among these series. In Section 6.2 we introduce the Fourier series on the interval [−L, L] and then derive the Fourier sine and cosine series for the interval [0, L]. We will see that we have the Fourier sine or cosine series depending on a function being odd or even. Section 6.1 is a review section on even, odd and periodic functions. In Section 6.3 we discuss the convergence of the Fourier series. Specifically, we study three types of convergence, the mean-square (or L²), the pointwise, and the uniform convergence. The relation between the term by term differentiation of the Fourier series of f and the Fourier series of f′ is discussed in Section 6.4. This gives a justification for the eigenfunction expansions that we study in Chapter 7. In Section 6.5 we discuss, a related topic, the eigenvalue problems. In one-dimension it is called the Sturm-Liouville Problems.

In this chapter we study the separation of variables in the multi-dimensional problems. We discuss the heat and wave equations in two or three dimensions and the Laplace equation in three dimensions. The key issue in this chapter is how the separation of variables we studied in one-dimensional problems is extended to the multi-dimensional problems. One thing we have to cope with is the geometry of the domains. Typical geometries discussed in the textbooks are rectangles, circles, circular cylinders, and spheres. In this chapter we mainly confine our attention to the rectangular coordinates. We learn that the eigenfunctions are composed of the eigenfunctions for one-dimensional problems. We repeatedly solve the one-dimensional problems. We also discuss the eigenvalue problems for higher dimensions. In Section 7.2 we briey explain the eigenvalue problems and then the Rayleigh quotient, which relates the eigenvalue problems and the minimization problems, will be discussed. In Section 7.3 we discuss an extension of separation of variables, the eigenfunction expansion, to more complex problems such as the ones involving the non-homogeneous boundary conditions and the source terms.

In this chapter we continue the separation of variables in higher dimensions and study the effects of geometry. In this chapter we treat the cases where the domains are circular, cylindrical, and spherical. These are the domains more complex than the rectangular ones, yet the separation of variables is applicable. ODEs we obtain are not so simple as in the rectangular case. For the circular, cylindrical, and spherical cases we need to introduce the Bessel’s equation to handle the radial components in the separation of variables. Also, in the spherical case we introduce the associated Legendre equation to handle one of the angular components.

The Fourier transform is a useful tool in linear PDEs. In this chapter we study the basics of the Fourier transform and its applications. We first introduce the delta function in two different ways. In Section 9.1.1 we use a classical way and define it as a limit of a sequence of functions whose peak approaches infinity and whose area is one. In Section 9.1.2 we use a modern way and define it as a special functional called the distribution. In Section 9.2 we define the Fourier transform and its inverse. The properties of Fourier transform are discussed in Section 9.3. The applications of the Fourier transform are studied in Section 9.4. We see how the Fourier transform is applied to the heat, wave and Laplace equations.

The Laplace transform is another transformation that is useful for solving DEs. In this chapter we apply the Laplace transform to solve PDEs. The Laplace and the inverse transforms are defined in Section 10.1. In Section 10.2 we discuss the properties of the Laplace transform including the relation with the Fourier transform. The applications to DEs are discussed in Section 10.3.

In Chapter 2 we studied the Green’s function for the initial value problem of the heat equation and in Chapter 4 we did the Green’s functions for the Laplace equation. In this chapter with the help of the delta functions we study the Green’s functions in more details. The solution to a system of linear equations Ax = b is given by x = A⁻¹ b, where A⁻¹ is the inverse matrix provided that it exists. With the aid of the delta function and the Green’s identity we will see that the Green’s function plays a similar role in PDEs to the inverse matrices in systems of linear equations, i.e., if we find the Green’s function, we have a solution representation for the problem. This is one of the reasons why we study the Green’s functions. In Section 11.1 we revisit the Green’s functions for the Laplace Equation and construct the Green’s function using the eigenfunction expansion. In Sections 11.2 and 11.3 we study the Green’s function for the heat and wave equations, respectively. We use the eigenfunction expansions for the initial boundary value problems and the Duhamel’s principle or the Fourier transform for the initial value problems to construct the Green’s functions.

This chapter is a supplemental chapter where we treat basic results concerning two well-known nonlinear PDEs in fluid dynamics and a celebrated equation in finance. In Section 12.1 we study systems of first order quasi-linear equations. This section can be thought of as an extension of Sections 5.2 and 3.2. In Section 12.2 we consider the viscous Burgers equation. This is a model equation to study the combined effects of convection and diffusion. We also study the most celebrated equation in finance called the Black-Scholes-Merton equation. This is a linear convection diffusion equation and perhaps the most important equation for option pricing.

The following sections are included:

• Appendix A
  • Maximum, Minimum, Supremum, and Infimum
  • Norm
  • Continuity and Discontinuity
    • Pointwise Continuity and Uniform Continuity
    • Discontinuity
  • Exchanging the Order of Integration and Differentiation
  • Infinite Series
  • Lebesgue Integral
  • Useful Formulas in ODEs
    • First Order Linear Equations
    • Bernoulli Equations
    • Second Order Linear Constant Coefficient Equations
    • Variation of Parameters Formula
  • Linear Algebra
    • Solutions to Systems of Linear Equations
    • Eigenvalues, Eigenvectors, and Diagonalization
• Bibliography
• Index