Quasilinear Differential Equations Research Articles

Uniform estimates for positive solutions to quasi-linear differential equations of even order

The existence of uniform estimates for positive solutions with the same domain to the even-order differential equation $$y^{(n)} + \sum\limits_{i = 0}^{n - 1} {a_i (x)y^{(i)} } + p(x)\left| y \right|^{k - 1} y = 0$$ with k > 1 is proved. The estimates for solutions depend on those for the continuous coefficients p(x) > 0 and ai(x), not on the coefficients themselves.

On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases

A singularly perturbed system of second-order quasilinear ordinary differential equations with a small parameter multiplying the second derivatives is examined in the case where the coefficient matrix of the first derivatives is singular and does not depend on the unknown functions.

Existence of positive solutions of higher-order quasilinear ordinary differential equations

In this paper the even-order quasilinear ordinary differential equation \( D(\alpha _{n})D(\alpha _{n-1})\cdots D(\alpha _{1})x + p(t)|x|^{\beta -1}x=0 \)is considered under the hypotheses that n is even, D(α i )x = (|x|αi−1 x)′, α i > 0(i = 1,2,…, n), β > 0, and p(t) is a continuous, nonnegative, and eventually nontrivial function on an infinite interval [a, ∞), a > 0. The existence of positive solutions of (1.1) is discussed, and basic results to the classical equation \( x^{(n)} + p(t)|x|^{\beta -1}x=0 \)are extended to the more general equation (1.1). In particular, necessary and sufficient integral conditions for the existence of positive solutions of (1.1) are established in the case α 1α2⋅s α n ≠ β.

Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type , where and are positive constants, and are positive continuous functions on , and is a continuously differentiable function such that , . We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.

Existence and asymptotic behavior of positive solutions of higher‐order quasilinear ordinary differential equations

AbstractThe existence of non‐extreme positive solutions of n th‐order quasilinear ordinary differential equations is discussed. In particular, necessary and sufficient integral conditions for the existence of non‐extreme positive solutions are established for a certain class of equations. Basic results to a classical ordinary differential equation are generalized. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.

Modeling of modifier‐solute peak interactions in chromatography

AbstractThe behavior of a chromatographic column where a single solute is fed in the presence of a suitable modifier is analyzed in the context of equilibrium theory. In particular, the pulse propagation of the solute, whose retention depends on the modifier concentration, has been analyzed as a full‐fledged binary problem by the method of hodograph transformation applied to the pair of quasilinear differential equations. It has been shown that all possible operating conditions can be summarized with respect to the modifier concentration into three cases. These have been solved analytically by constructing the solution first in the hodograph plane, and then mapping it onto the physical plane using the method of characteristics. The peculiar properties of the derived hodograph plane were explained, and their impact on the pulse propagation for the three different cases has been elucidated. For the cases of intermediate adsorptivity of the modifier, the occurrence of solute peaks with infinitely high concentrations; that is, solute impulse, has been proven as well as the phenomena of double solute peaks. © 2005 American Institute of Chemical Engineers AIChE J, 2006

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.

On oscillatory solutions of quasilinear differential equations

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.

On an extension of the Lyapunov criterion of stability for quasi-linear systems via integral inequalities methods

In this article, we concern ourselves with a new concept for comparing the stability degree of two dynamical systems. By using the integral inequality method, we give a criterion which allows us to compare the growth rate of two Itô quasi-linear differential equations. It can be viewed as an extension of the Lyapunov criterion to the stochastic case.

Singularity-induced bifurcations in lumped circuits

A systematic analysis of singular bifurcations in semistate or differential-algebraic models of electrical circuits is presented in this paper. The singularity-induced bifurcation (SIB) theorem describes the divergence of one eigenvalue through infinity when an operating point or equilibrium locus of a parameterized differential-algebraic model crosses a singular manifold. The present paper extends this result to cover situations in which several eigenvalues diverge; we prove a multiple SIB theorem which states that a minimal rank (resp. index) change makes it possible to compute the number of diverging eigenvalues in terms of an index (resp. rank) change in the matrix pencil characterizing the linearized problem. The scope of the work comprises quasi-linear ordinary differential equations, semiexplicit index-1 differential-algebraic equation (DAEs), and Hessenberg index-2 DAEs, describing different electrical configurations. The electrical features from which singularities and, specifically, singular bifurcations stem are extensively discussed. Examples displaying simple, double, and triple SIB points illustrate different ways in which the spectrum may diverge.

Positive Solutions of Quasilinear Elliptic Equations

This paper is concerned with existence theorems for positive solutions of the Dirichlet problem for quasilinear elliptic differential equation containing a gradient term. Using the shooting method and the a priori estimates for the first zero, we obtain sufficient conditions for the existence of classical positive solutions of the problem in the ball.

Implicit difference methods for parabolic functional differential equations

AbstractWe present a new class of numerical methods for the solution of quasilinear parabolic functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better that the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Results obtained in the paper can be applied to differential integral problems and equations with retarded variables.

Nonexistence of solutions to Hele-Shaw equations

We study nonexistence of the solutions to quasilinear elliptic differential equations arising from nonisothermal, non-Newtonian Hele-Shaw flows. The proof is based on the trial function method developed by Pohozaev without recourse to comparison theorems and to the maximum principle.

Asymptotic behavior of solutions of second order quasilinear differential equations with delay depending on the unknown function

The asymptotic behavior of the nonoscillatory solutions of quasilinear differential equations of second order with delay depending on the unknown function is considered. The main results given by [Bainov et al. (J. Comput. Appl. Math. 91 (1998) 87–96) and Wong (Funkcial. Ekvac. 11 (1968) 207–234)] are improved and generalized.

Oscillation and nonoscillation criteria for second order quasilinear difference equations

New oscillation and nonoscillation theorems are obtained for the second order quasilinear difference equation Δ ( | Δ x n − 1 | ρ − 1 Δ x n − 1 ) + p n | x n | ρ − 1 x n = 0 , where ρ > 0 is a constant and { p n } n = 1 ∞ is a real sequence with p n ⩾ 0 .

On Asymptotic Properties of Continuously Differentiable Solutions of Quasilinear Functional Differential Equations

We establish new properties of C1 [τ(1), + ∞)-solutions of the quasilinear functional differential equation $$\dot x(t) = ax(t) + bx(\tau (t)) + c\dot x(\tau (t)) + f(x(t)),x(\tau (t))),\quad \mathop {\lim }\limits_{t \to + \infty } \;\tau (t) = + \infty ,$$ in the neighborhood of the singular point t = +∞.

Geometrical numerical algorithms for a plasticity model with Armstrong-Frederick kinematic hardening rule under strain and stress controls

In this paper, we approach the numerical integration problem of a plasticity model with the Armstrong-Frederick kinematic hardening rule on back stress through a combination of the techniques of integral representation and geometrical integrator. First, the internal symmetry group of the constitutive model is investigated. Then, we develop two geometrical integrators for strain control and stress control, respectively. These integrators are obtained by a discretization of the integral representation of the constitutive equations and an exponential approximation of the quasi-linear differential equations system for the relative stress, which guarantee to retain the consistency condition exactly without the need for any iterations. Some numerical examples are used to assess the performance of the new algorithms. The measures in terms of stress relative errors and also isoerror maps confirm that our schemes are superior to the classical radial return methods.

Minimization Methods for Quasi-Linear Problems with an Application to Periodic Water Waves

Penalization and minimization methods are usedto give an abstract "semiglobal" result on the existence of nontrivial solutions of parameter-dependent quasi-linear differential equations in variational form. A consequence is a proof of existence, by infinite-dimensional variational means, of bifurcation points for quasi-linear equations which have a line of trivial solutions. The approach is to penalize the functional twice. Minimization gives the existence of critical points of the resulting problem, and a priori estimates show that the critical points lie in a region unaffected by the leading penalization. The other penalization contributes to the value of the parameter. As applications we prove the existence of periodic water waves, with and without surface tension.

Two-Sided Approximation of Solutions of a Multipoint Problem for an Ordinary Differential Equation with Parameters

We construct an algorithm for the two-sided approximation of a solution of a multipoint boundary-value problem for a quasilinear differential equation under assumptions that are two-sided analogs of the Pokornyi B-monotonicity of the right-hand side of the equation. We establish conditions for the monotonicity of successive approximations and their uniform convergence to a solution of the problem.