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In this chapter, we briefly discuss the Fourier transform and show how this transformation can be used to solve differential equations where the spatial domain is all of ℝ. [ABSTRACT FROM AUTHOR]

Reaction-diffusion equations arise as mathematical models in a series of important applications, e.g. in models of superconducting liquids, flame propagation, chemical kinetics, biochemical reactions, predator-prey systems in ecology and so on. Both numerical and mathematical analysis of reaction-diffusion equations are currently very active fields of research. Obviously, we cannot study the subject at an advanced level in the present text, but we can get a general feeling of what these problems are about. Our aim is merely to present some simple models and to explore some of their properties using finite difference schemes and energy estimates. Further examples can be found in the exercises.1 [ABSTRACT FROM AUTHOR]

The two previous chapters have been devoted to Fourier series. Of course, the main motivation for the study of Fourier series was their appearance in formal analytic solutions of various partial differential equations like the heat equation, the wave equation, and Poisson's equation. [ABSTRACT FROM AUTHOR]

Let f be a piecewise continuous function defined on [-1, 1] with a full Fourier series given by $$ \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos \left( {k\pi x} \right) + b_k \sin \left( {k\pi x} \right)} \right).} $$ . converge to the function f ?" If we here refer to convergence in the mean square sense, then a partial answer to this question is already established by Theorem 8.2. At least we have seen that we have convergence if and only if the corresponding Parseval's identity holds. However, we like to establish convergence under assumptions which are easier to check. Also, frequently we are interested in notions of convergence other than convergence in the mean. [ABSTRACT FROM AUTHOR]

In the previous chapters Fourier series have been the main tool for obtaining formal solutions of partial differential equations. The purpose of the present chapter and the two following chapters is to give a more thorough analysis of Fourier series and formal solutions. The Fourier series we have encountered in earlier chapters can be thought of as examples of a more general class of orthogonal series, and many properties of Fourier series can be derived in this general context. In the present chapter we will study Fourier series from this point of view. The next chapter is devoted to convergence properties of Fourier series, while we return to partial differential equations in Chapter 10. There the goal is to show that the formal solutions are in fact rigorous solutions in a strict mathematical sense. [ABSTRACT FROM AUTHOR]

Poisson's equation is a fundamental partial differential equation which arises in many areas of mathematical physics, for example in fluid flow, flow in porous media, and electrostatics. We have already encountered this equation in Section 6.4 above, where we studied the maximum principle for harmonic functions. As a corollary of the maximum principle we have in fact already established that the Dirichlet problem for Poisson's equation has at most one solution (see Theorem 6.8). [ABSTRACT FROM AUTHOR]

The purpose of this chapter is to study maximum principles. Such principles state something about the solution of an equation without having to solve it. [ABSTRACT FROM AUTHOR]

In the previous chapter we derived a very powerful analytical method for solving partial differential equations. By using straightforward techniques, we were able find an explicit formula for the solution of many partial differential equations of parabolic type. By studying these analytical solutions, we can learn a lot about the qualitative behavior of such models. This qualitative insight will also be useful in understanding more complicated equations. [ABSTRACT FROM AUTHOR]

The purpose of this chapter is to study initial-boundary value problems for the wave equation in one space dimension. In particular, we will derive formal solutions by a separation of variables technique, establish uniqueness of the solution by energy arguments, and study properties of finite difference approximations. [ABSTRACT FROM AUTHOR]

The historical paths of mathematical physics, mathematical analysis, and methods for solving partial differential equations are strongly interlaced, and it is often difficult to draw boundaries between them. In particular, this is the case in the field of Fourier analysis. This field was initiated by Joseph Fourier (1768-1830), a French physicist who studied heat conduction. In his analysis of this problem, he invented the most influential method for solving partial differential equations to this day. For over 200 years his work has been the foundation of certain areas of mathematical analysis. Any student of engineering or of the natural sciences has to master his techniques. [ABSTRACT FROM AUTHOR]

In Chapter 1 above we encountered the wave equation in Section 1.4.3 and the heat equation in Section 1.4.4. These equations occur rather frequently in applications, and are therefore often referred to as fundamental equations. We will return to these equations in later chapters. Another fundamental equation is Poisson's equation, given by $$ - \sum\limits_{j = 1}^n {\frac{{\partial ^2 u}} {{\partial x_j^2 }} = f,} $$ where the unknown function u is a function of n spatial variables X1,..., xn. [ABSTRACT FROM AUTHOR]

You are embarking on a journey in a jungle called Partial Differential Equations. Like any other jungle, it is a wonderful place with interesting sights all around, but there are also certain dangerous spots. On your journey, you will need some guidelines and tools, which we will start developing in this introductory chapter. [ABSTRACT FROM AUTHOR]