Related papers: Solutions of Nonlinear Differential Equations
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…
Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of…
In this work, we give the general solution sequential linear conformable fractional differential equations in the case of constant coefficients for {\alpha}(\in)(0,1]. In homogeneous case, we use a fractional exponential function which…
As a significant strengthening of properties of earlier algebras of generalized functions, here are presented classes of such algebras which can deal with dense singularities. In fact, the cardinal of the set of singular points can be…
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 systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think…
For any $n\in\mathbb{N}$ a nonlinear ordinary differential equation with Lie algebra of point symmetries isomorphic to $\frak{sl}(2,\mathbb{R})$ is given.
In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove…
We introduce a hybrid Cole-Hopf-Darboux transformation to relate solutions of nonlinear and linear second order differential equations and derive a sufficient condition for this correspondence. In particular we show that solutions of some…
As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to…
The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
This article is devoted to the study of solutions of non-homogenous linear differential equations having entire coefficients. We get all non-trivial solutions of infinite order of equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots…
This paper introduces the concept of renormalized solution for a general class of non-coercive nonlinear parabolic problems, including both singularities and unbounded lower order terms. We prove existence and uniqueness of renormalized…