“There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”,
~ Nikolai Lobatchevsky (1792-1856)
- 9.1: Introduction
- The solutions of linear partial differential equations can be found by using the method of separation of variables to reduce solving partial differential equations to solving ordinary differential equations. We can also use transform methods to transform the given PDE into ODEs or algebraic equations. Solving these equations, we then construct solutions of the PDE using an inverse transform. We will describe in this chapter how one can use Fourier and Laplace transforms to this effect.
- 9.2: Complex Exponential Fourier Series
- Before deriving the Fourigr transform, we will need to rewrite the trigonometric Fourier series representation as a complex exponential Fourier series.
- 9.3: Exponential Fourier Transform
- Both the trigonometric and complex exponential Fourier series provide us with representations of a class of functions of finite period in terms of sums over a discrete set of frequencies.
- 9.4: The Dirac Delta Function
- The Dirac delta function, δ(x) this is one example of what is known as a generalized function, or a distribution. Dirac had introduced this function in the 1930′s in his study of quantum mechanics as a useful tool. It was later studied in a general theory of distributions and found to be more than a simple tool used by physicists. The Dirac delta function, as any distribution, only makes sense under an integral.
- 9.5: Properties of the Fourier Transform
- Before actually computing the Fourier transform of some functions, we prove a few of the properties of the Fourier transform.
- 9.6: The Convolution Operation
- In the list of properties of the Fourier transform, we defined the convolution of two functions, f(x) and g(x) to be the integral (f∗g)(x). In some sense one is looking at a sum of the overlaps of one of the functions and all of the shifted versions of the other function. The German word for convolution is faltung, which means "folding" and in old texts this is referred to as the Faltung Theorem. In this section we will look into the convolution operation and its Fourier transform.
- 9.7: The Laplace Transform
- Laplace made major contributions, especially to celestial mechanics, tidal ampliforms as one type of integral transform. The Fourier transform is useful sis, and probability. on infinite domains. However, students are often introduced to another integral transform, called the Laplace transform, in their introductory differential equations class. These transforms are defined over semi-infinite domains and are useful for solving initial value problems for ordinary differential equations.
- 9.8: Applications of Laplace Transforms
- Although the Laplace transform is a very useful transform, it is often encountered only as a method for solving initial value problems in introductory differential equations. In this section we will show how to solve simple differential equations. Along the way we will introduce step and impulse functions and show how the Convolution Theorem for Laplace transforms plays a role in finding solutions. However, we will first explore that the Laplace transform is useful in finding sums of infinite se
- 9.10: The Inverse Laplace Transform
- Inverse Laplace transform can be found by making use of Laplace transform tables and properties of Laplace transforms. One can do the same for Fourier transforms. However, in the case of Fourier transforms we introduced an inverse transform in the form of an integral. Does such an inverse integral transform exist for the Laplace transform? Yes, it does! In this section we will derive the inverse Laplace transform integral and show how it is used.
Thumbnail: Graph of a shifted Dirac delta function (1d) (CC By 4.0; Alexander Fufaev via Universaldenker)