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We consider the problem of deciding whether a common solution to a multivariate polynomial equation system is isolated or not. We present conditions on a given truncated Puiseux series vector centered at the point ensuring that it is not…
In this paper, we concentrate on the Novikov equation. We provide a description of the solution in a neighborhood of each singular point.
We investigate the presence of localized solutions in models described by a single real scalar field with generalized dynamics. The study offers a method to solve very intricate nonlinear ordinary differential equations, and we illustrate…
The present paper deals with the existence and uniqueness of global classical solutions to the continuous coagulation and nonlinear multiple fragmentation equations for large classes of unbounded coagulation, collision and breakup kernels.…
This paper is concerned with generalising the results for Lie $CT$-algebras to Leibniz algebras. In some cases our results give a generalisation even for the case of a Lie algebra. Results on $A$-algebras are used to show every Leibniz…
A general Riccati equation is integrated in quadratures in case one of its coefficients is an arbitrary function and two others are expressed through it.
A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the…
In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schr\"{o}dinger problem \begin{align*} \varepsilon^{2s}(-\Delta)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \…
We derive some existence results for the solutions of the Tzitz\'eica equation \begin{equation*} -\Delta u + h_1(x)e^{Au} + h_2(x)e^{-Bu}=0 \end{equation*} and the generalized Tzitz\'eica equation \begin{equation*} -\Delta u +…
In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schr\"{o}dinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -\Delta u+V(x)u+\lambda…
In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive,…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves…
In this paper we review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using…
We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic…
We study radial solutions of the semilinear elliptic equation $\Delta u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a…
We consider the Goursat problem for linear partial differential equations with constant coefficients in two complex variables. We find the conditions for summable solutions of the Goursat problem in the case when the Newton polygon has…
In this paper we give an explicit representation of the solutions of a characteristic Cauchy problem for a class of PDEs with singular coefficients. We give the explicit solutions in terms of the Gauss hypergeometric functions, which enable…