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The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of…
The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the…
The spurious states found in numerical implementations of envelope function models for semiconductor heterostructures and nanostructures have been shown to be readily removed by employing a first-order difference scheme. This approach is…
We study the Cauchy problem for the Whitham modulation equations for monotone increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is…
We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a…
In the constraint programming framework, state-of-the-art static and dynamic decomposition techniques are hard to apply to problems with complete initial constraint graphs. For such problems, we propose a hybrid approach of these techniques…
We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite…
This work aims at presenting a Discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use of DG methods is motivated…
Improving electronic structure calculations for practical and technologically-important materials has been a never-ending pursue. This is especially true for transition and post-transition metal oxides for which the current first-principles…
We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of…
We review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of one-parameter families of vector fields, developed by the authors during the last years. We review, in particular, the…
The aim of this paper is to apply a high-order discontinuous-in-time scheme to second-order hyperbolic partial differential equations (PDEs). We first discretize the PDEs in time while keeping the spatial differential operators…
In this work, the Haar wavelet method for the coupled non-linear transient PDE-ODEs system with the Neumann boundary condition has been proposed. The capability of the method in handling multiple jump discontinuities in the coefficients and…
Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion…
We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data…
We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…
A new approach is introduced for deriving a mixed variational formulation for Kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary parts. Based on a regular decomposition of…
We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is…
There are two kinds of solutions of the Cauchy problem of first order, the viscosity solution and the more geometric minimax solution and in general they are different. The aim of this article is to show how they are related: iterating the…