We analyze the convergence behavior of collocation schemes applied to approximate solutions of BVPs in nonlinear index 1 DAEs, which exhibit a critical point at the left boundary. Such a critical point of the DAE causes a singularity in the inherent nonlinear ODE system. In particular, we focus on the case when the inherent ODE system is singular with a singularity of the first kind and apply p...

We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel’s nonlinear differential equations ODEs of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind Neumann functions , as well as of the free member of the considered ODE; the parameter ν being intr...

The impetus of this study is to investigate the chaotic behavior of a size-dependent nano-beam with double-sided electrostatic actuation, incorporating surface energy effect and intermolecular interactions. The geometrically nonlinear beam model is based on Euler-Bernoulli beam assumption. The influence of the small-scale and the surface energy effect are modeled by implementing the consistent ...

In this paper we study the nonchaotic and chaotic behavior of all 3D conservative quadratic ODE systems with five terms on the right-hand side and one nonlinear term (5-1 systems). We prove a theorem which provides sufficient conditions for solutions in 3D autonomous systems being nonchaotic. We show that all but five of these systems:(3.8a,b), (3.11b), (3.34)(A = ∓1), (4.1b),and (4.7a,b) are n...

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics [21], Pour-El and Richards asked, “What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?” Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this pa...

This paper introduces a methodology for the synthesis of nonlinear finite-dimensional output feedback controllers for systems of quasi-linear parabolic partial Ž . differential equations PDEs , for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinitedimensional stable fast complement. Combination of Galerkin’s method ...

In this paper, axisymmetric flows over stretching surfaces in a porous medium in the presence of a magnetic field with the navier boundary condition have been considered. The governing partial differential equation (PDE) has been reduced to nonlinear ordinary differential equation (ODE) by using suitable similarity transformations. The collocation methods based on Multiquadric (MQ) Radial basis...

The study of dynamical models of HIV infection, based on a system of nonlinear ordinary differential equations (ODE), has considerably improved the knowledge of its pathogenesis. While the first models used simplified ODE systems and analyzed each patient separately, recent works dealt with inference in non-simplified models borrowing strength from the whole sample. The complexity of these mode...

Feedback control for the monodomain equations is studied. The dynamics of interest are governed by a coupled PDE-ODE reaction diffusion system with non-monotone nonlinearity of FitzHugh-Nagumo type. A localized distributed control is used to locally stabilize the nonlinear system. This is achieved by a Riccati-based feedback law, determined by the linearized system. It is shown that the Riccati...

Abstract. The first order nonlinear ODE φ̇(t) + sinφ(t) = q(t), q(t) = B +A cosωt, where A, B, ω are real constants, is considered, the transformation converting it to a second order linear homogeneous ODE with polynoimial coefficients is found. The latter is identified as a particular case of the double confluent Heun equation. The series of algebraic constraints on the constant parameters is f...