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This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.
The explicit evaluation of the partition function in the Schwarzian theory is presented.
We define a proper differential sequence of ordinary differential equations and introduce a method to derive an alternative sequence of integrals for such a sequence. We describe some general properties which are illustrated by several…
In this paper we study the existence and continuation of solution to general fractional differential equation with Hilfer fractional derivative. First we establish new local existence theorems. Then we derive the continuation theorems. With…
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…
We give an explicit description of the basic solutions of max-linear systems with two inequalities.
In this paper, we solve Laplace equation analytically by using differential transform method. For this purpose, we consider four models with two Dirichlet and two Neumann boundary conditions and obtain the corresponding exact solutions. The…
A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that…
Two results on the completeness of maximal solutions to first and second order ordinary differential equations (or inclusions) over complete Riemannian manifolds, with possibly time-dependent metrics, are obtained. Applications to…
Exact solutions of Dirac equation in two spatial dimensions in the Coulomb field are obtained. Equation which determines the so-called critical charge of the Coulomb field is derived and solved for a simple model.
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special…
Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discretely self-similar solutions.
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the…
The generalized Kuramoto-Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented.
In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector…
This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial conditions. The class of…
We consider new ways of obtaining series and integral representations for master integrals arising in the process of matching of QCD to NRQCD. The latter results are exact in space-time dimension $d$. In addition, we discuss series…
In this paper, we mainly investigate on the finite order transcendental entire solutions of two Fermat types delay-differential and one Fermat type c-shift equations, as these types were not considered earlier. Our results improve those of…