Related papers: Numerical realization of the variational method fo…
We discuss Rayleigh-Ritz variational calculations with nonorthogonal basis sets that exhibit the correct asymptotic behaviour. We construct the suitable basis sets for general one-dimensional models and illustrate the application of the…
The flow of Ree-Eyring and Casson non-Newtonian fluids is investigated using a variational principle to optimize the total stress. The variationally-obtained solutions are compared to the analytical solutions derived from the…
Applications of variational methods are typically restricted to conservative systems. Some extensions to dissipative systems have been reported too but require ad hoc techniques such as the artificial doubling of the dynamical variables.…
This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of…
Travelling solitary waves in the one-dimensional discrete nonlinear Schr\"{o}dinger equation (DNLSE) with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in an analytical form.…
We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally…
We develop the technique of the variational approximation for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms can be discarded for various reasons. For…
We formulate a novel numerical method suitable for the solution of topology optimization problems in solid mechanics. The most salient feature of the new approach is that the space and time discrete equations of the numerical method can be…
We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural…
The Rayleigh-Ritz procedure for determining bound-states of the Schr\"{o}dinger equation relies on spectral representation of the solution as a linear combination of the basis functions. Several possible extensions of the method to…
We apply the variational approach to solitons in highly nonlocal nonlinear media in $D=1,2,3$ dimensions. We compare results obtained by the variational approach with those obtained in the accessible soliton approximation, by considering…
We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative…
We investigate bright solitons in the one-dimensional Schr\"odinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between…
This paper is presented to give numerical solutions of some cases of nonlinear wave-like equations with variable coefficients by using Reduced Differential Transform Method (RDTM). RDTM can be applied most of the physical, engineering,…
We investigate the problem of shaping radially symmetric annular beams into desired intensity patterns along the optical axis. Within the Fresnel approximation, we show that this problem can be expressed in a variational form equivalent to…
A theoretical method for treating collisions in the presence of multiple potentials is developed by employing the Schwinger variational principle. The current treatment agrees with the local (regularized) frame transformation theory and…
The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A…
We introduce a model of media with the cubic attractive nonlinearity concentrated along a single or double stripe in the two-dimensional (2D) plane. The model can be realized in terms of nonlinear optics (in the spatial and temporal domains…
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6]…
We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms…