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The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified…
Our work is about energy conserving fourth-order time discretizations of a three-field formulation of Maxwell's equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward…
We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators. These are approximations that generate strong numerical…
In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax…
In this work, we develop a space--time Chebyshev spectral collocation method for three-dimensional Maxwell's equations and combine it with tensor-network techniques in Tensor-Train (TT) format. Under constant material parameters, the…
The Maxwell equations for the electromagnetic potential, supplemented by the Lorenz gauge condition, are decoupled and solved exactly in de Sitter space-time studied in static spherical coordinates. There is no source besides the…
Because only two variables are needed to characterize a simple thermodynamic system in equilibrium, any such system is constrained on a 2D manifold. Of particular interest are the exact 1-forms on the cotangent space of that manifold, since…
An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that…
Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the…
An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using…
We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.
The generalized Maxwell equations with arbitrary gauge parameter are considered in the $11\times 11$-matrix form. The gauge invariance of such a model is broken due to the presence of a scalar field. The canonical and symmetrical Belinfante…
Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of…
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms…
A new formulation of the Maxwell equations based on two vector and two scalar potentials is proposed. The use of these potentials allows the electromagnetic field equations to be written in the form of a hyperbolic system. In contrast to…
We present a numerical method for solving the free-space Maxwell's equations in three dimensions using compact convolution kernels on a rectangular grid. We first rewrite Maxwell's Equations as a system of wave equations with auxiliary…
Taking the Noether gauge symmetry approach into account, we find spherically symmetric static black hole solutions of the non-minimal gauge-gravity Lagrangian of the $\mathcal{R}^\beta F^2$ model. At first, we consider a system of…
By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical…