Related papers: Formal exact operator solutions to nonlinear diffe…
We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The particular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the…
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases…
Problem for the first order differential equation with an unbounded operator coefficient in Banach space and nonlinear nonlocal condition is considered. A numerical method is proposed and justified for the solution of this problem under…
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 the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace…
Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.
We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…
We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that…
The Adomian decomposition method is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The aim of this paper is to apply Adomian decomposition method to obtain approximate solutions of nonlinear…
The non-commutative strategy developed by Bagarello (see Int. Jour. of Theoretical Physics, 43, issue 12 (2004), p. 2371 - 2394) for the analysis of systems of ordinary differential equations (ODEs) is extended to a class of partial…
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.
We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively…
At present, only some special differential equations have explicit analytical solutions. In general, no one thinks that it is possible to analytically find the exact solution of nonlinear equations. In this article based on the idea that…
Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a…
The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this…