The problem of heat conduction on networks of multiply connected rods is solved by providing an explicit solution of the one-dimensional heat equation in each domain. The size and connectivity of the rods is known, but neither temperature nor heat flux are prescribed at the interface. Instead, the physical assumptions of continuity at the interfaces are the only conditions imposed. This work generalizes that of Deconinck, Pelloni, and Sheils, 2014, for heat conduction on a series of one-dimensional rods connected end-to-end to the case of general configurations.
The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initial-boundary value problems, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel'fand and his co-authors.
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals introduced by one of the authors. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.
It is known that the unified transform method may be used to solve any well-posed initial-boundary value problem for a linear constant-coefficient evolution equation on the finite interval or the half-line. In contrast, classical methods such as Fourier series and transform techniques may only be used to solve certain problems. The solution representation obtained by such a classical method is known to be an expansion in the eigenfunctions or generalised eigenfunctions of the self-adjoint ordinary differential operator associated with the spatial part of the initial-boundary value problem. In this work, we emphasise that the unified transform method may be viewed as the natural extension of Fourier transform techniques for non-self-adjoint operators. Moreover, we investigate the spectral meaning of the transform pair used in the new method; we discuss the recent definition of a new class of spectral functionals and show how it permits the diagonalisation of certain non-self-adjoint spatial differential operators.
We give a characterisation of the spectral properties of certain linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and complemented with boundary conditions that may make the operator non-selfadjoint.
In particular, we associate the spectral properties of such an operator S with the form of the integral representation of the solution of a corresponding boundary value problem for the PDE qt(x,t)±iSq=0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we will use its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas, and studied extensively by one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.
We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series.
The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.
A new, unified transform method for boundary value problems on linear and integrable nonlinear partial differential equations was recently introduced by Fokas. We consider initial-boundary value problems for linear, constant-coefficient evolution equations of arbitrary order on a finite domain. We use Fokas' method to fully characterise well-posed problems. For odd order problems with non-Robin boundary conditions we identify sufficient conditions that may be checked using a simple combinatorial argument without the need for any analysis. We derive similar conditions for the existence of a series representation for the solution to a well-posed problem.
We also discuss the spectral theory of the associated linear two-point ordinary differential operator. We give new conditions for the eigenfunctions to form a complete system, characterised in terms of initial-boundary value problems.