Related papers: Recurrence formula for some higher order evolution…
We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a…
This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has emerged according to which the general solution may be…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary…
We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution…
A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic…
We adapt the classical theory of local well-posedness of evolution problems to cases in which the nonlinearity can be accurately quantified by two different norms. For ordinary differential equations, we consider $\dot{x} = f(x,x)$ for a…
This paper is devoted to the study of generalised time-fractional evolution equations involving Caputo type derivatives. Using analytical methods and probabilistic arguments we obtain well-posedness results and stochastic representations…
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We…
In this Chapter, using Riccati equation as our main example, we tried to demonstrate at least some of the ideas and notions introduced in Chapter 1 - integrability in quadratures, conservation laws, etc. Regarding transformation group and…
This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, $\alpha$ and $\alpha_m$, satisfy the conditions $1<\alpha\le 2$ and…
Motivated by a recent work of Tr\"umper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application…
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a…
Results of a previous paper [Commun. Contemp. Math., 09 (2007) 217-251] on the existence of solutions to a nonlinear evolution equation in an abstract Lebesgue space, arising from kinetic theory, are re-obtained in the more general setting…
The Riccati equation method is used to establish a new comparison theorem for systems of two linear first order ordinary differential equation. This result is based on a, so called, concept of "null-classes", and is a generalization of…
We consider the Cauchy problem for stochastic fractional evolution equations with Caputo time fractional derivative of order $1<\alpha<2$ and space variable coefficients on an unbounded domain. The space derivatives that appear in the…
We show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order $\delta >1$, where $\delta \to 1+0$.
This paper presents an algebraic approach to characterizing higher-order differential operators. While the foundational Leibniz rule addresses first-order derivatives, its extension to higher orders typically involves identities relating…