The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painlevé transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are reduced to the study of a nonlinear special function in particular limits. The universal object that these functions share is a Riemann-Hilbert representation: the nonlinear special function can be recovered from the solution of a Riemann-Hilbert problem (RHP). A RHP consists of finding a piecewiseanalytic function in the complex plane when the behavior of its discontinuities is specified. In this dissertation, the applied theory of Riemann-Hilbert problems, using both Hölder and Lebesgue spaces, is reviewed. The numerical solution of RHPs is discussed. Furthermore, the uniform approximation theory for the numerical solution of RHPs is presented, proving that in certain cases the convergence of the numerical method is uniform with respect to a parameter. This theory shares close relation to the method of nonlinear steepest descent for RHPs.The inverse scattering transform for the Korteweg-de Vries and Nonlinear Schrödinger equation is made effective by solving the associated RHPs numerically. This technique is extended to solve the Painlevé II equation numerically. Similar Riemann-Hilbert techniques are used to compute the so-called finite-genus solutions of the Korteweg-de Vries equation. This involves ideas from Riemann surface theory. Finally, the methodology is applied to compute orthogonal polynomials with exponential weights. This allows for the computation of statistical quantities stemming from random matrix ensembles. ACKNOWLEDGMENTS

Abstract. The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method also allows for the equally explicit solution of problems for which no classical approach exists. In addition, it is possible to elucidate which boundary-value problems are well posed and which are not. We provide examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, and others do not. For the former, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.

Recent advances in the numerical solution of Riemann-Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg-de Vries equation (KdV) and the defocusing modified Korteweg-de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accurate. The method is straightforward for the case of defocusing mKdV due to the lack of poles in the Riemann-Hilbert problem and the boundedness properties of the reflection coefficient. Solving KdV requires the introduction of poles in the Riemann-Hilbert problem and more complicated deformations. The introduction of a new deformation for KdV allows for the stable asymptotic computation of the solution in the entire (x, t)-plane. KdV and mKdV are dispersive equations and this method can fully capture the dispersion with spectral accuracy. Thus, this method can be used as a benchmarking tool for determining the effectiveness of future numerical methods designed to capture dispersion. This method can easily be adapted to other integrable equations with Riemann-Hilbert formulations, such as the nonlinear Schrödinger equation.

We solve the focusing and defocusing nonlinear Schrödinger (NLS) equations numerically by implementing the inverse scattering transform. The computation of the scattering data and of the NLS solution are both spectrally convergent. Initial conditions in a suitable space are treated. Using the approach of Biondini & Bui, we numerically solve homogeneous Robin boundary-value problems on the half line. Finally, using recent theoretical developments in the numerical approximation of Riemann-Hilbert problems, we prove that, under mild assumptions, our method of approximating solutions to the NLS equations is uniformly accurate in their domain of definition.