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1. The numerical unified transform method for initial-boundary value problems on the half-line

   https://doi.org/10.1093/imanum/drab007  

   Deconinck, Bernard; Trogdon, Thomas; Yang, Xin (March 2021, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We implement the unified transform method of Fokas as a numerical method to solve linear evolution partial differential equations on the half-line. The method computes the solution at any $$x$$ and $$t$$ without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method’s complexity does not increase for large $x,t$ and the method is more accurate as $x,t$ increase (absolute errors are smaller, relative errors are bounded). Our goal is to make no assumptions on the functional form of the initial or boundary functions beyond some decay and smoothness, while maintaining high accuracy in a large region of the $(x,t)$ plane. 

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2. Localized Model Reduction for Nonlinear Elliptic Partial Differential Equations: Localized Training, Partition of Unity, and Adaptive Enrichment

   https://doi.org/10.1137/22M148402X  

   Smetana, Kathrin; Taddei, Tommaso (June 2023, SIAM Journal on Scientific Computing)

   We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations. CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomized data compression algorithm based on oversampling for the construction of the components' reduced spaces: the approach exploits random boundary conditions of controlled smoothness on the oversampling boundary. We further propose an adaptive residual-based enrichment algorithm that exploits global reduced-order solves on representative systems to update the local reduced spaces. We prove exponential convergence of the enrichment procedure for linear coercive problems; we further present numerical results for a two-dimensional nonlinear diffusion problem to illustrate the many features of our methodology and demonstrate its effectiveness. 

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3. Riemann–Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line

   https://doi.org/https://doi.org/10.1016/j.amc.2018.03.049  

   Hu, Bei-Bei; Xia, Tie-Cheng; Ma, Wen-Xiu (September 2018, Applied mathematics and computation)

   In this work, we investigate the two-component modified Korteweg-de Vries (mKdV) equation, which is a complete integrable system, and accepts a generalization of 4 × 4 matrix Ablowitz–Kaup–Newell-Segur (AKNS)-type Lax pair. By using of the unified transform approach, the initial-boundary value (IBV) problem of the two-component mKdV equation associated with a 4 × 4 matrix Lax pair on the half-line will be analyzed. Supposing that the solution {u1(x, t), u2(x, t)} of the two-component mKdV equation exists, we will show that it can be expressed in terms of the unique solution of a 4 × 4 matrix Riemann–Hilbert problem formulated in the complex λ-plane. Moreover, we will prove that some spectral functions s(λ) and S(λ) are not independent of each other but meet the global relationship. 

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4. Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

   https://doi.org/10.1111/sapm.12642  

   Alkın, Aykut; Mantzavinos, Dionyssios; Özsarı, Türker (January 2024, Studies in Applied Mathematics)

   Abstract We establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator. 

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5. Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case

   https://doi.org/10.1088/1751-8121/ac761d  

   Barraquand, Guillaume; Krajenbrink, Alexandre; Le Doussal, Pierre (June 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift − v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t , it is natural to rescale parameters as ( u , v ) = t −1/3 ( a , b ) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ( a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b ⩾ 0, we determine exactly the large time distribution F a , b stat of the height function h (0, t ). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1 / t 2 → 1. 

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