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Über dieses Buch

Overview The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. The core consists of solution methods, mainly separation of variables, for boundary value problems with constant coeffi­ cients in geometrically simple domains. Too often an introductory course focuses exclusively on these core problems and techniques and leaves the student with the impression that there is no more to the subject. Questions of existence, uniqueness, and well-posedness are ignored. In particular there is a lack of connection between the analytical side of the subject and the numerical side. Furthermore nonlinear problems are omitted because they are too hard to deal with analytically. Now, however, the availability of convenient, powerful computational software has made it possible to enlarge the scope of the introductory course. My goal in this text is to give the student a broader picture of the subject. In addition to the basic core subjects, I have included material on nonlinear problems and brief discussions of numerical methods. I feel that it is important for the student to see nonlinear problems and numerical methods at the beginning of the course, and not at the end when we run usually run out of time. Furthermore, numerical methods should be introduced for each equation as it is studied, not lumped together in a final chapter.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract
In this section we will introduce some ideas and notations which we will use often in the study of partial differential equations. You will have seen most of this before in calculus. Details and proofs of these results can be found in most calculus books or in an advanced calculus book such as [Bar].
Jeffery Cooper

Chapter 2. First-Order Equations

Abstract
A partial differential equation (PDE) for the scalar function u(x, y, z) is an equation
$$F\left( {x,y,z,u,u_x ,u_y ,u_z , \ldots ,D^m u, \ldots } \right) = 0.$$
Jeffery Cooper

Chapter 3. Diffusion

Abstract
In this chapter we investigate equations which model diffusion processes, such as heat flow in a solid, or the spread of dye in water. There is an important difference between a diffusion process and the wave motion studied in Chapter 2. In diffusion, initial data is smeared and smoothed out; there are no sharp fronts. We shall see that this is a result of the constitutive relation (or diffusion law). In the latter part of this chapter we will combine nonlinear wave motion and diffusion.
Jeffery Cooper

Chapter 4. Boundary Value Problems for the Heat Equation

Abstract
Up to now we have studied fairly broad qualitative properties of solutions of the heat equation. We have not attempted to solve problems with boundary conditions assigned at both ends of a bar of finite length. In this chapter we study these problems and exploit the theory of linear operators to give a unified treatment of the many possible combinations of boundary conditions.
Jeffery Cooper

Chapter 5. Waves Again

Abstract
In this chapter we shall study the linear wave equation in one space variable. Solutions of this equation have many features in common with the first-order equation u t + cu x = 0 that we studied in Chapter 2. The linear wave equation arises in many contexts which include gas dynamics (acoustics), vibrating solids, and electromagnetism.
Jeffery Cooper

Chapter 6. Fourier Series and Fourier Transform

Abstract
In this chapter we look at some of the eigenfunction expansions in terms of Fourier series. We develop the Fourier transform and use it to solve the heat equation again. We also give a brief treatment of the discrete Fourier transform (DFT) and the fast Fourier transform (FFT).
Jeffery Cooper

Chapter 7. Dispersive Waves and the Schrödinger Equation

Abstract
After a short section in which we discuss the method of stationary phase, we treat dispersive waves. Then we turn to quantum mechanics, using the Fourier transform to establish the Heisenberg uncertainty principle. Finally we derive the Schrödinger equation. We solve the free Schrödinger equation and the Schrödinger equation with a square well potential.
Jeffery Cooper

Chapter 8. The Heat and Wave Equations in Higher Dimensions

Abstract
We shall use x to denote points in R2 or R3 with components (x, y) or (x, y, z), and the Euclidean length of a vector is denoted \(\left| x \right| = \sqrt {x^2 + y^2 } \) or \(\left| x \right| = \sqrt {x^2 + y^2 + z^2 } \). The Laplace operator (Laplacian) in R2 or R3 is
$$\Delta u = u_{xx} + u_{yy} {\text{ or }}\Delta u = u_{xx} + u_{yy} + u_{zz} .$$
Jeffery Cooper

Chapter 9. Equilibrium

Abstract
In Chapter 8, we have seen how the Laplace and Poisson equations occur in steady-state heat flow. In this chapter we investigate properties of solutions of these equations and representations of the solutions in terms of integrals. We characterize these solutions as minimizers of certain variational problems, and again arrive at the notion of weak solutions.
Jeffery Cooper

Chapter 10. Numerical Methods for Higher Dimensions

Abstract
In this chapter we investigate a finite difference scheme for the Laplace equation. Then we turn our attention to the finite element method for the Poisson equation. Finally we look at the Galerkin method for a time-dependent problem.
Jeffery Cooper

Chapter 11. Epilogue: Classification

Abstract
In Section 2.1 we indicated that there is a way of grouping second order PDE’sinto classes,such that the solutions of PDE’s in the same class have similar qualitative properties. We shall discuss this classification now and relate it to our physical classification into equations which describe diffusion, wave propagation, and equilibrium situations. We let
$$Lu = au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y + fu$$
(11.1)
be the general second-order operator in two variables with constant coefficients. For convenience in this discussion, we shall assume that a > 0. The characteristic polynomial associated with L is
$$q\left( {x,y} \right) = ax^2 + 2bxy + cy^2 + dx + ey + f.$$
(11.2)
Jeffery Cooper

Backmatter

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