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A non-perturbative approach to the solution of the time-dependent, two-center Dirac equation is presented with a special emphasis on the proper treatment of the potential of the nuclei. In order to account for the full multipole expansion…
We investigate the electrostatics of a partially formed, charged spherical shell in a salt solution. We solve the problem numerically at the Poisson-Boltzmann level and analytically in the Debye-Huckel regime. From the results on energetics…
We study a class of nonlinear eigenvalue problems of Schr\"{o}dinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and…
This paper investigates abstract integro-differential hyperbolic equations, focusing on the probabilistic representation of their solutions. Our analysis is based on fractional derivatives and non-local operators, which are powerful tools…
We address a numerical framework for the stability and bifurcation analysis of nonlinear partial differential equations (PDEs) in which the solution is sought in the function space spanned by physics-informed random projection neural…
Analytic first and second derivatives of the energy are developed for the fragment molecular orbital method interfaced with molecular mechanics in the electrostatic embedding scheme at the level of Hartree-Fock and density functional…
We present a spectrally accurate embedded boundary method for solving linear, inhomogeneous, elliptic partial differential equations (PDE) in general smooth geometries, focusing in this manuscript on the Poisson, modified Helmholtz, and…
The accuracy of applying density functional theory to noncovalent interactions is hindered by errors arising from low-density regions of interaction-induced change in the density gradient, error compensation between correlation and exchange…
This work considers charged systems described by the modified Poisson--Nernst--Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the steady-state modified PNP…
We develop an exact electrostatic formulation for a finite-length conducting cylindrical shell of finite thickness separating two dielectric media with arbitrary permittivity contrast. The boundary-value problem is reduced to a coupled…
Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the…
The description of a conducting medium in thermal equilibrium, such as an electrolyte solution or a plasma, involves nonlinear electrostatics, a subject rarely discussed in the standard electricity and magnetism textbooks. We consider in…
The modeling of many phenomena in various fields such as mathematics, physics, chemistry, engineering, biology, and astronomy is done by the nonlinear partial differential equations (PDE). The hyperbolic telegraph equation is one of them,…
Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper concerns an inverse acoustic-elastic interaction problem, which is to determine the…
The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to…
A novel method is developed for constructing periodic solutions of a model equation describing nonlocal Josephson electrodynamics. This method consists of reducing the equation to a system of linear ordinary differential equations through a…
An algorithm for fast calculation of the Coulombic forces and energies of point particles with free boundary conditions is proposed. Its calculation time scales as N log N for N particles. This novel method has lower crossover point with…
Analytical solutions of the Bohr Hamiltonian are obtained in the $\gamma$-unstable case, as well as in an exactly separable rotational case with $\gamma\approx 0$, called the exactly separable Morse (ES-M) solution. Closed expressions for…
We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic…
This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic…