Related papers: Singularity methods for meromorphic solutions of d…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection…
The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this…
We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in $R$ and $R^2$ are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic…
Most evolution equations %or wave equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron…
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.
In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise,…
We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as…
Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and…
We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use to them stabilize numerical calculations. Our method follows classical analysis for first-order systems and…
This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov-Serrin and a…
This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are…
In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations…
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations \label{GDMZabstract} \frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial u}{\partial…
In this work, a new technique has been presented to find approximate solution of linear integro-differential equations. The method is based on modified orthonormal Bernoulli polynomials and an operational matrix thereof. The method converts…
Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to…
In the paper, we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation \[P_k\big(z,f,f^{(1)},\ldots, f^{(k)}\big)=f^{(1)}(f-\mathscr{L}_k(f))-\varphi (f-a)(f-b)=0,\]…
We consider a singular fractional differential equation involving generalized Katugampola derivative and obtain the existence and uniqueness of its solution. A scheme for uniformly approximating solution is constructed by using Picard…