Adam McBride

Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation

An initial-value problem modelling coagulation and fragmentation processes is studied. The results of earlier papers are extended to models where either one or both of the rates of coagulation and fragmentation depend on time. An abstract integral equation, involving the solution operator to the linear fragmentation part, is investigated via the contraction mapping principle. A unique global, non-negative, mass-conserving solution to this abstract equation is shown to exist. The latter solution is used to generate a global, non-negative, mass-conserving solution to the original non-autonomous coagulation and multiple-fragmentation equation.

Existence results for non‐autonomous multiple‐fragmentation models

We investigate an initial-value problem modelling fragmentation processes where particles split into two or more pieces at a rate, γ, that not only depends on the sizes of the particles involved but also on time. The existence of non-negative, mass-conserving solutions is established by considering a truncated version of an associated non-autonomous abstract Cauchy problem. The latter has solutions of the form u(t)=Un(t,t0)f, t⩾t0, where f is the known data at some fixed time t0⩾0 and {Un(t,s)} is a uniformly continuous evolution system. A limit evolution system {U(t,s)} is shown to exist. Depending on the form of the known data f at time t0, the scalar-valued function u, obtained from the limit evolution system via u(x, t)=[U(t, t0)f](x) for a.e. x>0, t⩾t0, is a solution of either the original initial-value problem or an integral version of this problem.

Fractional powers of a class of mellin multiplier transforms part iii

The paper is concerned with a class of Mellin multiplier transforms, mapping one weighted LP(0,) space into another, whose symbols are of a particular form. An expression is easily obtained for positive integral powers of such operators and this forms the basis of an extension to fractional powers, A rigorous framework for the analysis is described. Analogues of the index laws of ordinary algebra are established under stated conditions, Connections between powers of an operator and of its adjoint are explored. The theory is illustrated by means of simple integral operators. These examples serve to indicate some of the limitations of a classical setting and provide the stimulus for a distributional treatment which will be presented elsewhere.

A PHOTON TRANSPORT PROBLEM WITH A TIME-DEPENDENT POINT SOURCE

We consider a time-dependent problem of photon transport in an interstellar cloud with a point photon source modeled by a Dirac δ functional. The existence of a unique distributional solution to this problem is established by using the theory of continuous semigroups of operators on locally convex spaces coupled with a constructive approach for producing spaces of generalized functions.

Fractional transformations of generalised functions

A distributional theory of fractional transformations is developed. A constructive approach, based on the eigenfunction expansion method pioneered by Zemanian, is used to produce an appropriate space of test functions and corresponding space of generalized functions. The fractional transformations that are defined are shown to form an equicontinuous group of operators on the space of test functions and a weak* continuous group on the space of generalized functions. Integral representations for the fractional transformations are also obtained under certain conditions. The fractional Fourier transformation is considered as a particular case of our general theory.

Photon transport problems involving a point source

We consider both a direct and an inverse problem of photon transport in an interstellar cloud with a point photon source. By using a non-rigorous (but physically reasonable) procedure, we prove that the direct problem has a unique solution and that the inverse problem also has a unique solution, under the assumptions that a single value of the photon far-field is known and the scattering cross-section is suitably small. Finally, we show in a rigorous way that the direct problem has a unique distributional solution if the point source is modelled by a Dirac δ functional.

Fractional calculus of periodic distributions

Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Grünwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotient. The equivalence of the two approaches is established and an application to a fractional diffusion equation, posed in a space of periodic distributions, is also discussed.

Continuity and invertibility of distributional multidimensional fractional integrals

This paper continues the investigation started in [4]. Multidimensional fractional integrals and Riesz potentials are shown to have excellent mapping properties relative to the spaces of test functions and generalised functions introduced in [4].

Multidimensional fractional integrals and distributions

New spaces of generalized functions on Rn are introduced. These spaces generalize corresponding spaces on R+ studied previously by the first author. Basic properties are obtained in preparation for a study of fractional integrals, including Riesz potentials, in a companion paper.

Mathematics: The greatest subject in the world

Two prominent themes in the Presidential Address upon which this article is based were Enthusiasm and Challenge. Many, if not most, practitioners of Mathematics can trace their interest in the subject to an enthusiastic teacher at some stage of their education. In this day and age, it is all too easy to get bogged down in a morass of assessment and quality assurance. Yet it is vital that teachers of Mathematics at all levels retain their enthusiasm and pass this on to their students.