Boundary value problems and medical imaging

• Published in: Journal of physics. Conference series 2014-01, Vol.490 (1), p.12017-7
• Main Authors: Fokas, Athanasios S, Kastis, George A
• Format: Article
• Language: English
• Subjects: Boundaries; Boundary conditions; Boundary value problems; Convergence; Domains; Exact solutions; Infinite series; Integrals; Inverse problems; Mathematical analysis; Mathematical models; Medical imaging; Mellin transforms; Partial differential equations; Physics; Positron emission; Tomography; Trigonometric functions
• ISSN: 1742-6596; 1742-6588; 1742-6596
• DOI: 10.1088/1742-6596/490/1/012017

The application of appropriate transform pairs, such as the Fourier, the Laplace, the sine, the cosine and the Mellin transforms, provides the most well known method for constructing analytical solutions to a large class of physically significant boundary value problems. However, this method has several limitations. In particular, it requires the given PDE, domain and boundary conditions to be separable, and also may not be applicable if the given boundary value problem is non-self-adjoint. Furthermore, it expresses the solution as either an integral or an infinite series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. Here, we review a method recently introduced by the first author which can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane which is uniformly convergent on the boundary of the domain. This method, which also suggests new numerical techniques, is illustrated for both evolution and elliptic PDEs. Athough this method was first applied to certain nonlinear PDEs called integrable and was originally formulated in terms of the so-called Lax pairs, it can actually be applied to linear PDEs without the need to analyse the associated Lax pair. The existence of Lax pairs is used here in order to motivate a related development, namely the emergence of a novel formalism for analysing certain inverse problems arising in medical imaging. Examples include PET and SPECT.