Related papers: Solutions of Nonlinear Differential Equations
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 define a generalization of convex functions, which we call $\delta$-convex functions, and show they must satisfy interior H\"older and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of…
A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations,…
A class of generalized Schr\"{o}dinger elliptic problems involving concave-convex and other types of nonlinearities is studied. A reasonable overview about the set of solutions is provided when the parameters involved in the equation assume…
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions,…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
In this paper, we introduce a class of nonlinear optimisation problems. Under mild assumptions, we obtain the existence of potential functions and show that the potential function is a generalised solution of a Monge-Amp\`ere type equation.…
Someone knowledgeable in nonstandard analysis may get the feeling that in the nonlinear theory of generalized functions, too often one works directly on the nets and spends effort to obtain results that should be clear from general…
This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a…
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…
This paper concerns exact differential equations. First, I define two types of functions which I have named Basic Function of Type One and Basic Function of Type Two. I then derive the property and theorems of these functions. Finally, by…
Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
We show that a linear differential equation whose coefficients are entire functions of completely regular growth may have an entire solution of finite order which is not of completely regular growth. This answers a question of Gol'dberg and…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…