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We present a method that employs physics-informed deep learning techniques for parametrically solving partial differential equations. The focus is on the steady-state heat equations within heterogeneous solids exhibiting significant phase…
This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions (FAS) of the Lax operator L.…
In this paper we rigorously prove the existance of a non-trivial periodic orbit for the non-linear delay differential equation: $x'(t) = K \sin(x(t-1))$ for $K=1.6$. We show that the equations on the Fourier equations have a solution by…
A very simple and efficient local variational iteration method for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order…
We present a thermodynamically consistent theoretical framework for lyotropic liquid crystals (LCs) based on the GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) formalism. This formalism ensures…
One of the most straightforward ways to study thermal properties beyond linear response is to monitor the relaxation of an arbitrarily large left-right temperature gradient $T_L-T_R$. In one-dimensional systems which support ballistic…
Exact solutions are found for Euler's equations of rigid body motion for general asymmetrical bodies under the influence of torque by using Jacobi elliptic functions. Differential equations are determined for the amplitudes and the…
A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly…
Motivated by understanding the nonlinear gravitational dynamics of spacetimes admitting stably trapped null geodesics, such as ultracompact objects and black string solutions to general relativity, we explore the dynamics of nonlinear…
The calculation of one loop integrals at finite temperature requires the evaluation of certain series, which converge very slowly or can even be divergent. Here we review a new method, recently devised by the author, for obtaining…
Context: The solution of the nonlocal thermodynamical equilibrium (non-LTE) radiative transfer equation usually relies on stationary iterative methods, which may falsely converge in some cases. Furthermore, these methods are often unable to…
Several integration schemes exits to solve the equations of motion of the $N$-body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s this method was applied for…
Temperature stabilization of oil and gas wells is used to ensure stability and prevent deformation of a subgrade estuary zone. In this work, we consider the numerical simulation of thermal stabilization using vertical seasonal freezing…
We devote our studies to the subject of weakly nonintegrable dynamics of systems with a macroscopic number of degrees of freedom. Our main points of interest are the relations between the timescales of thermalization and the timescales of…
Thermal radiation is an abundant form of incoherent light. Generating coherent infrared light through incandescence promises a cheap alternative to the costly and epitaxially complex quantum cascade laser, however it remains a fundamental…
The averaged resonant equations of motion for the planar circular restricted three-body problem are solved on the linearization basis taking into account also non-gravitational effects. The averaged resonant equations are derived from…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
We propose an iterative finite element method for solving non-linear hydromagnetic and steady Euler's equations. Some three-dimensional computational tests are given to confirm the convergence and the high efficiency of the method.
This paper addresses an optimization problem in satellite observation mission planning, focusing on the challenges of decentralized decision-making among satellites, which is crucial for optimizing strategies in dynamic observation…
An analytic solution to a stationary heat conduction problem in 2D unbounded doubly periodic composite materials with temperature dependent conductivities of their components is given. Corresponding nonlinear boundary value problem is…