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Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
We propose a new kind of stochastic absolute value equations involving absolute values of variables. By utilizing an equivalence relation to stochastic bilinear program, we investigate the expected value formulation for the proposed…
We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd…
This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with…
We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of…
We give a series of very general sufficient conditions in order to ensure the uniqueness of large solutions for --$\Delta$u + f (x, u) = 0 in a bounded domain $\Omega$ where f : $\Omega$ x R $\rightarrow$ R + is a continuous function, such…
An initial-boundary value problem for a generalized KdV equation posed on a half-line is considered. Existence and uniqueness of global regular solutions for arbitrary smooth initial data are established.
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a…
An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 =…
In this paper, we investigate global convergence properties of the inexact nonsmooth Newton method for solving the system of absolute value equations (AVE). Global $Q$-linear convergence is established under suitable assumptions. Moreover,…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
We study uniqueness of solutions to degenerate parabolic problems, posed in bounded domains, where no boundary conditions are imposed. Under suitable assumptions on the operator, uniqueness is obtained for solutions that satisfy an…
This paper is dedicated to present an exact solution for a nonlinear differential equation so-called Abel equation. This equation was known as one of the group of unsolvable differential equations. The present method is applicable for any…
We provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced. The boundary condition is allowed to be discontinuous and to depend…
The general solution to the Complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special…
We prove the existence of infinite energy global solutions of the cubic wave equation in dimension greater than 3. The data is a typical element on the support of suitable probability measures.
Many Engineering Problems could be mathematically described by Final Value Problem, which is the inverse problem of Initial Value Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the…