Wilson Lamb

A fractional power approach to fractional calculus

Abstract A theory of fractional powers of operators on an arbitrary Frechet space is discussed. As special cases, multivariable fractional integrals and derivatives defined on certain spaces of test functions and generalised functions are obtained. In particular, properties of the two-dimensional Riesz fractional integral are determined and used to solve a distributional initial value problem involving the wave operator.

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.

A Volterra integral type method for solving a class of nonlinear initial‐boundary value problems

It is observed that the one-dimensional heat equation with certain nonlinear boundary conditions can be reformulated as a system of coupled Volterra integral equations. A product trapezoidal scheme is proposed for the numerical solution of this integral equation system, and some numerical experiments are given to compare the performances of this integral equation approach and the Crank-Nicholson method applied to the original initial-boundary value problem.

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.

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.

On relating two approaches to fractional calculus

Abstract In this paper, two methods are discussed for defining fractional powers of the nth order ordinary differential expression L = x a 1 Dx a 2 D ··· x a n Dx a n + 1 (D = d dx ) , where each ai (i = 1, 2, …, n + 1) is a complex number and x is real and positive. An important role is played by the number m = ¦∑ i = 1 n + 1 a i − n¦ . When m = 0, spectral theory is used to obtain a suitable formula for Lα (α ϵ C ) while the case m>0, considered in a previous paper by A. C. McBride (Proc. London Math. Soc. (3) 45 (1982), 519–546), is dealt with by expressing Lα in terms of Erdelyi-Kober operators. We give accounts of each approach and go on to consider how the results for m = 0 can be obtained as limiting cases of the results for m>0.

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.

Spectral theory and functional calculus for operators on spaces of generalized functions

Abstract The generalized eigenfunction expansion theory of Zemanian for a differential operator with discrete spectrum is extended to an arbitrary self-adjoint operator on a separable Hilbert space. The extension of the operator to an appropriately defined space of distributions is shown to have a spectral resolution in that space and a functional calculus is developed. Applications to the solution of distributional initial-boundary value problems via semigroup theory are considered.

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.