Second-order Quasilinear Differential Equations Research Articles

Second-order quasilinear oscillation with impulses

Sufficient conditions for oscillation of all solutions of a class of second-order quasilinear differential equations with fixed moments of impulse effect are found.

Interval oscillation criteria for second-order quasi-linear nonhomogeneous differential equations with damping

By using two inequalities due to Hölder and Hardy, Littlewood and Polya as well as averaging functions, several interval oscillation criteria are established for the second-order quasi-linear differential equation with forced term of the form ( r( t)| y ′( t)| α−1 y ′( t)) ′+ p( t)| y ′( t)| α−1 y ′( t)+ q( t)| y( t)| β−1 y( t)= e( t) that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0,∞), rather than on the whole half-line, where β> α>0. Our results make use of the oscillatory properties of the forcing term and extend a recent result which is obtained by means of a Picone identity.

Oscillation criteria for second-order quasilinear functional differential equations

Some oscillation criteria for the second-order quasilinear functional differential equations ▪ are obtained. Our results generalize and improve some known results of both differential equations and delay differential equations.

Stability and higher integrability of derivatives of solutions in double obstacle problems

Higher integrability of the derivatives of solutions to double obstacle problems associated with the second-order quasilinear elliptic differential equation ∇· A( x,∇ u)=0 is obtained under natural assumptions on obstacles. This result is used to prove a stability result for solutions to double obstacle problems for varying equations.

Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations with sub-homogeneity

We give asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations. We obtain further the uniqueness of positive decaying solutions.

Some nonlinear equations reducible to diffusion-type equations

The problem of the reducibility of a system of second-order quasi-linear parabolic differential equations to diffusion-type equations is considered. An effective solution algorithm is suggested for this problem in the nondegenerate case.

Solving Second-Order Differential Equations with Lie Symmetries

Lie"s theory for solving second-order quasilinear differential equations based on its symmetries is discussed in detail. Great importance is attached to constructive procedures that may be applied for designing solution algorithms. To this end Lie"s original theory is supplemented by various results that have been obtained after his death one hundred years ago. This is true above all of Janet"s theory for systems of linear partial differential equations and of Loewy"s theory for decomposing linear differential equations into components of lowest order. These results allow it to formulate the equivalence problems connected with Lie symmetries more precisely. In particular, to determine the function field in which the transformation functions act is considered as part of the problem. The equation that originally has to be solved determines the base field, i.e. the smallest field containing its coefficients. Any other field occurring later on in the solution procedure is an extension of the base field and is determined explicitly. An equation with symmetries may be solved in closed form algorithmically if it may be transformed into a canonical form corresponding to its symmetry type by a transformation that is Liouvillian over the base field. For each symmetry type a solution algorithm is described, it is illustrated by several examples. Computer algebra software on top of the type system ALLTYPES has been made available in order to make it easier to apply these algorithms to concrete problems.

The oscillation and asymptotically monotone solutions of second-order quasilinear differential equations

We offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation (a(t)|y′| α−1y′)′ + Q(t, y) = P(t, y, y′) as well as for the existence of a positive monotone solution of the damped differential equation (a(t)|y′| α−1y′)′ + b(t)|y′| α−1y′ + H(t, y) = 0, where α > 0. Examples that dwell upon the importance of our results are also included.

Analysis of Munch theory

This paper extends the theoretical results of Tyree, Christy, and Ferrier on sucrose translocation in the sieve tubes of plants. Simple algebraic expressions are obtained for the order of magnitude of sucrose fluxes, sucrose velocities, and the maximum distance of translocation predicted by Munch theory. It is shown that for steady state Munch transport the sucrose velocity satisfies a second-order quasilinear ordinary differential equation. Order-of-magnitude calculations suggest that over short distances Munch theory transports solute by utilizing the hydrostatic and osmotic pressure gradients, while for long-distance transport gravitational effects play an important role.

On the optimal control of systems governed by quasilinear integro-partial differential equations of parabolic type

Recently, M. N. Oğuztöreli presented certain results on the existence and uniqueness of solutions of systems governed by a linear integro-partial differential equation of parabolic type with delayed arguments. Since his results admit only smooth coefficients, they could not be used directly in the study of the optimal control problems with bounded measurable control variables appearing in the coefficients of the system equations. In this paper, we consider a class of systems described by second-order quasilinear parabolic integro-partial differential equations with all but the second-order coefficients assumed bounded measurable. Our principal results are: Theorem 3.5, which establishes the existence and uniqueness of solutions of this class of systems (with controls in the coefficients), and Theorem 4.4, which gives a necessary condition for optimality for the corresponding controlled system.

Calculation of wave amplitudes by ray tracing

A method of calculating wave amplitudes by ray tracing is presented. It consists of an integration of six first-order differential equations, and three second-order quasilinear differential equations. The six first-order equations are the ray equations, and an equation for an additional Euler angle ψ. Two of the second-order equations are equations for two scale factors, h2 and h3, of an oblique system a2, a3 defined on the wavefront. The third second-order equation determines the angle χ between a2 and a3.

Expansion of a hole in a disk with kinematic hardening

A complete solution is obtained for the enlargement of a circular hole in a disk of initial thickness h 0 = αr n 0 , where α and n are constants, using Prager's kinematic hardening law with Tresca's yield function and its associated flow rule. The governing equations are reduced to a second-order quasilinear differential equation. Detailed results are given for the cases when n = 0 and n = 1, and comparison is made with the results obtained for the isotropic hardening law.