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Differential Galois theory has played important roles in the theory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order nonlinear…
In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are…
In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic…
We give evaluations in closed form of certain non linear differential equations
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of…
Integral operators play an important role modeling various physical and biological processes. In this article we consider such a nonlinear integro-differential equation. We study several properties of equilibrium solutions of the operator…
In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.
We consider a family of higher-order Boussinesq equations with an arbitrary nonlinearity. We determine the classes of equations so that a certain type of Lie symmetry algebra is admitted in this family. In case of a quadratic nonlinearity…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be…
A gauge-invariant form of the nonlinear Hodge equations is studied.
The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…
We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate…
A system of nonlinear differential equations $x^{1+\gamma}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by…
In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.
In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest…
The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^{\nu}} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^{\nu}}, \] where $p, q \in\mathbb{R}^+$ and $\nu\in…
We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.
Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches…
A fifth--order nonlinear partial differential equation for the description of nonlinear waves in a liquid with gas bubbles is considered. Special solutions of this equation are studied. Some elliptic and simple periodic traveling waves…