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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…
In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are…
Grain growth in polycrystals is often simulated using orientation-field models, which employ a field to represent the local orientation of the crystal lattice. These models can be challenging to represent a realistic misorientation…
The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.
We revisit the proof of the liquid-vapor phase transition for systems with finite-range interaction by Lebowitz, Mazel and Presutti and extend it to the case where we additionally include a hard-core interaction to the Hamiltonian. We…
Mechanical metamaterials exhibit exotic properties at the system level, that emerge from the interactions of many nearly rigid building blocks. Determining these emergent properties theoretically has remained an open challenge outside of a…
We prove the existence and uniqueness of the solution to the doubly nonlinear parabolic systems with mixed boundary conditions. Due to the unilateral constraint the problem comes as a variational inequality. We apply the penalty method and…
We consider an optimal control problem constrained by a parabolic partial differential equation (PDE) with Robin boundary conditions. We use a well-posed space-time variational formulation in Lebesgue--Bochner spaces with minimal…
The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is…
Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often…
A new scheme of realizing the nonadiabatic conditional geometric phase shift via a noncoplanar (and coiled) fiber system is presented in this Letter. It is shown that the effective Hamiltonian that describes the interaction of polarized…
We consider a system of PDEs of Monge-Kantorovich type that, in the isotropic case, describes the stationary configurations of two-layers models in granular matter theory with a general source and a general boundary data. We propose a new…
This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward - forward systems of equations of mean field games. We present local well-posedness, global existence and some…
We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give…
A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method [10]. This quantum model is then associated with a stochastic…
The phase field method is an effective tool for modeling microstructure evolution in materials. Many efficient implicit numerical solvers have been proposed for phase field simulations under uniform and time-invariant model parameters. We…
The existence problem for {C}ahn--{H}illiard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point…
We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in…
We present a solution of the problem of a free massless scalar field on the half line interacting through a periodic potential on the boundary. For a critical value of the period, this system is a conformal field theory with a non-trivial…
The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination…