Related papers: Formal exact operator solutions to nonlinear diffe…
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,…
In this letter we apply a method recently devised in \cite{aapla03} to find precise approximate solutions to a certain class of nonlinear differential equations. The analysis carried out in \cite{aapla03} is refined and results of much…
Using the principle of structural analogy of solutions, approaches have been developed for constructing exact solutions of complex nonlinear PDEs, including PDEs with delay, based on the use of special solutions to auxiliary simpler related…
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational…
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…
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
The logistic function is shown to be solution of the Riccati equation, some second-order nonlinear ordinary differential equations and many third-order nonlinear ordinary differential equations. The list of the differential equations having…
We propose an operational method for the solution of differential equations involving vector products. The technique we propose is based on the use of the evolution operator, defined in such a way that the wealth of techniques developed…
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution…
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal nonlinear drift term is reduced to a similar problem for the correspondent linear equation. The relation between symmetry operators of the linear and nonlinear…
The general abstract arbitrary order (N) Cauchy problem was solved in a closed form as a sum of exponential propagator functions. The infinite sparse exponential series was solved with the aid of a homogeneous differential equation. It…