Related papers: Resolving an interface problem for the Dirac equat…
This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz-Ladik system posed on a…
We consider numerical methods for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients defined by a one parameter family of interface conditions at the discontinuity. We construct immersed…
In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar…
In this paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Such systems of coupled partial differential equations arise in, for example, the modeling…
This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface…
We consider the unified transform method, also known as the Fokas method, for solving partial differential equations. We adapt and modify the methodology, incorporating new ideas where necessary, in order to apply it to solve a large class…
We introduce a new abstraction for the representation and solution of multi-domain problems using finite element methods. This is an advance over previous work in that it achieves a single higher-level abstraction that represents…
We implement the Unified Transform Method of Fokas as a numerical method to solve linear partial differential equations on the half-line. The method computes the solution at any x and t without spatial discretization or time stepping. With…
We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an…
The vector-matrix Riemann boundary value problem for the unit disk with piecewise constant matrix is constructively solved by a method of functional equations. By functional equations we mean iterative functional equations with shifts…
For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior…
The differential transform method (DTM) is a relatively new technique that may be used to find a series solution to differential equations (both linear and nonlinear) through an iterative process. This brief manuscript is an initial effort…
We propose a new simple variant of Fast Gradient Method that requires only one projection per iteration. We called this method Triangle Method (TM) because it has a corresponding geometric description. We generalize TM for convex and…
This is Part II of the paper series on data-compatible T-matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series contains theory and here we present simulations for inverse scattering of…
We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of…
We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical…
We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
Transformers have demonstrated strong potential in offline reinforcement learning (RL) by modeling trajectories as sequences of return-to-go, states, and actions. However, existing approaches such as the Decision Transformer(DT) and its…
We adapt a finite difference method of solution of the two-dimensional massless Dirac equation, developed in the context of lattice gauge theory, to the calculation of electrical conduction in a graphene sheet or on the surface of a…