In this paper, we study the solutions of the initial-value problem (IVP) for the quasi-geostrophic equations, namely ∂tθ + u.∇θ + κ (−∆) θ = 0, on R × ]0,+∞[ , θ (x, 0) = θ0(x), x ∈ R. Our goal is to establish the existence and uniqueness of regulars solutions for the two-dimentional dissipative quasi-geostrophic equation with initial data in a Sobolev space H satisfying suitable conditions wit...

To date, the only eeective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary diierential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoo (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and...

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as ∂tu + α∂ 3 xu + ∂ 5 xu + ∂ −1 x ∂ 2 yu + uux = 0, while α ∈ R. We introduce an interpolated energy space Es to consider the well-posedeness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0 < s ≤ 1. To obtain the local ...

The purpose of this paper is to use the Fuzzy Set Theory, specially fuzzy rule-based systems, as a mathematical tool for modeling dynamical systems given by differential equation, whose direction field is f (IVP-f ). As fuzzy systems methodology produces input-output systems, that is, it produces m outputs from n inputs, one can see it as fr : R → R, where r is the number of rules of the rule b...

Working in pressureless magnetohydrodynamics, we examine the consequences of some peculiar dispersive properties linear fast sausage modes (FSMs) one-dimensional cylindrical equilibria with a continuous radial density profile ($\rho_0(r)$). As recognized recently on solid mathematical grounds, cutoff axial wavenumbers may be absent for FSMs when $\rho_0(r)$ varies sufficiently slowly outside no...

Abstract: A new method for the optimal solutions is proposed. Originating from the continuous-time dynamics stability theory in the control field, the optimal solution is anticipated to be obtained in an asymptotically evolving way. By introducing a virtual dimension— —the variation time, a dynamic system that describes the variation motion is deduced from the Optimal Control Problem (OCP), and...

The theoretical background and the implementation of a new interval arithmetic approach for solving sets of differentialalgebraic equations (DAEs) are presented. The proposed approach computes guaranteed enclosures of all reachable states of dynamical systems described by sets of DAEs with uncertainties in both initial conditions and system parameters. The algorithm is based on VALENCIA-IVP,whi...

Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely connected to the notion of inference on latent variables in statistics. We describe a class of algorithms that formulate the solution to an IVP as inference on a l...

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, consider biharmonic exponential nonlinearity and Cahn–Hilliard equation. By using Fourier transform concept, generalized formula mild solution as well smoothing effects of resolvent operators are proved. For IVP associated first one, by Orlicz ...

ValEncIA-IVP is a verified solver for initial value problems for sets of ordinary differential equations which determines guaranteed enclosures of all reachable states. In this paper, we present its new features which allow for a wider application domain. They also improve the performance of ValEncIA-IVP. Especially for the simulation of asymptotically stable systems, we present a new exponenti...