Anthony C. L Ashton

A novel method of solution for the fluid-loaded plate

We study the equations governing a fluid-loaded plate. We first reformulate these equations as a system of two equations, one of which is an explicit non-local equation for the wave height and the velocity potential on the free surface. We then concentrate on the linearized equations and show that the problems formulated either on the full or the half-line can be solved by employing the unified approach to boundary value problems introduced by one of the authors in the late 1990s. The problem on the full line was analysed by Crighton & Oswell (Crighton & Oswell 1991 Phil. Trans. R. Soc. Lond. A335, 557–592 (doi:10.1098/rsta.1991.0060)) using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in Ldt1(R+) and furthermore yields a simpler solution representation that immediately implies that the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, which, however, involves two unknown functions. The main difficulty with the half-line problem is the characterization of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex-valued integral equation of the convolution type. This equation can be solved in a closed form using the Laplace transform. By prescribing the initial data η0 to be in H⟨5⟩5(R+), or equivalently four times continuously differentiable with sufficient decay at infinity, we show that the solution depends continuously on the initial data, and, hence, the problem is well-posed.

The Fundamental k-Form and Global Relations

In (Proc. Roy. Soc. London Ser. A 453 (1997), no. 1962, 1411-1443) A.S. Fokas introduced a novel method for solving a large class of boundary value problems associated with evolution equations. This approach relies on the construction of a so-called global relation: an integral expression that couples initial and boundary data. The global relation can be found by constructing a differential form dependent on some spectral parameter, that is closed on the condition that a given partial differential equation is satisfied. Such a differential form is said to be fundamental (Quart. J. Mech. Appl. Math. 55 (2002), 457- 479). We give an algorithmic approach in constructing a fundamentalk-form associated with a given boundary value problem, and address issues of uniqueness. Also, we extend a result of Fokas and Zyskin to give an integral representation to the solution of a class of boundary value problems, in an arbitrary number of dimensions. We present an extended example using these results in which we construct a global relation for the linearised Navier-Stokes equations.

Laplace's equation on convex polyhedra via the unified method

We provide a new method to study the classical Dirichlet problem for Laplaces equation on a convex polyhedron. This new approach was motivated by Fokas’ unified method for boundary value problems. The central object in this approach is the global relation: an integral equation which couples the known boundary data and the unknown boundary values. This integral equation depends holomorphically on two complex parameters, and the resulting analysis takes place on a Banach space of complex analytic functions closely related to the classical Paley–Wiener space. We write the global relation in the form of an operator equation and prove that the relevant operator is bounded below using some novel integral identities. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realization of the fundamental principle of Ehrenpreis.

Elliptic equations with low regularity boundary data via the unified method

We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner singularities.

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.