B. van Brunt

The Lebesgue-Stieltjes Integral

We now proceed to formulate the definition of the integral that we are going to study. It results from combining the ideas of two people. The French mathematician Henri Lebesgue (1875-1941), building on earlier work by Emile Borel (1871-1956) on the measure of a set, succeeded in defining an integral (the Lebesgue integral) that applied to a wider class of functions than did the Riemann integral, and for which the convergence theorems were much simpler. The Dutch mathematician Thomas Stieltjes (1856-1894) was responsible for the notion of integrating one function with respect to another function. His ideas were originally developed as an extension of the Riemann integral, known as the Riemann-Stieltjes integral. The subsequent combination of his ideas with the measure-theoretic approach of Lebesgue has resulted in a very powerful and flexible concept of integration.

A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model

In this paper we study the probability density function solutions to a second-order pantograph equation with a linear dispersion term. The functional equation comes from a cell growth model based on the Fokker–Planck equation. We show that the equation has a unique solution for constant positive growth and splitting rates and construct the solution using the Mellin transform.

Solutions to an advanced functional partial differential equation of the pantograph type

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

Conservation laws for second-order parabolic partial differential equations

Conservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Frechet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

The Lebesgue Spaces Lp

There are many mathematical problems for which the solution is a function of some kind, and it is often whole real line has the useful property that sums and constant multiples of functions in the set are also in the s both possible and convenient to specify in advance the set of functions within which the solution is to be sought. For example, the solution to a first-order differential equation might be specified as being differentiable on the whole real line. The set of functions differentiable on the et. In fact, this set of functions has the structure of a vector space, where the “vectors” are functions.

Double and Repeated Integrals

Lebesgue-Stieltjes integrals of functions of more than one variable can be defined using the same approach as was used in Section4.5 for functions of one variable. For the sake of simplicity we will discuss only functions of two variables. The process for functions of more than two variables is completely analogous.

Properties of the Integral

In this chapter we will examine some of the essential properties of the Lebesgue-Stieltjes integral, culminating in the convergence theorems that (as remarked in Chapter 3) are among the most important features of the Lebesgue-Stieltjes theory.

Hilbert Spaces and L2

Hilbert spaces are a special class of Banach spaces. Hilbert spaces are simpler than Banach spaces owing to an additional structure called an inner product. These spaces play a significant role in functional analysis and have found widespread use in applied mathematics. We shall see at the end of this section that the Lebesgue space L2 (and its complex relative H2) is a Hilbert space. In this and the next section, we introduce some basic definitions and facts concerning Hilbert spaces of immediate interest to our discussion of the space L2. Further details and proofs of the results presented in these sections can be found in most books on functional analysis, e.g., [25].

Some Analytic Preliminaries

Before we can develop the theory of integration, we need to revisit the concept of a sequence and deal with a number of topics in analysis involving sequences, series, and functions.