Decay of solutions of some nonlinear parabolic inequalities associated with time-dependent convexes

A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: lim_{t → ∞} ∝^t₀a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that lim_{t → ∞} ∝^t₀a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established.     

Integral comparison theorems and extremal points for linear differential equations

Comparison theorems of integral type are developed for linear differential equations of the form (1) L_ny + p(t)y = 0 and (2) L_ny + q(t)y = 0. Under the assumption that (2) has no nontrivial solution satisfying given homogeneous two point boundary conditions it follows that the same is true for (1), provided certain sign and integral conditions hold. The criteria may be thought of as rather general extensions of the Hille-Wintner type and also include and extend recent results of Elias.     

On the oscillation of a second-order delay equation

The oscillatory nature of two equations (r(t) y′(t))′ + p₁(t)y(t) = f(t), (r(t) y′(t))′ + p₂(t) y(t ? τ(t))= 0, is compared when positive functions p₁ and p₂ are not “too close” or “too far apart.” Then the main theorem states that if h(t) is eventually negative and a twice continuously differentiable function which satisfies (r(t) h′(t))′ + p₁(t) h(t) ? 0, then this inequality is necessary and sufficient for every bounded solution of (r(t) y′(t))′ + p₂(t) y(t ? τ(t)) = 0 to be nonoscillatory.     

Minimum norm solutions of single stiff linear analytic differential equations

The $\text{l}$₂-norm of the infinite vector of the terms of the Taylor series of an analytic function is used to measure the “unsmoothness” of the function. The sets of solutions to the scalar differential equations y′(t) = λy(t) + f(t) and y′(t) = q(t)y(t) + f(t) are analyzed with respect to this norm. A number of results on the particular solution with minimum norm are given.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

On the Contractivity and Asymptotic Stability of Systems of Delay Differential Equations of Neutral Type

In this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y(t) = Ly(t) + M(t)y(t – (t)) + N(t)y(t – (t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented.     

Asymptotic stability of a class of integro-differential equations

We examine the asymptotic stability of the zero solution of the first-order linear equation x′(t) = Ax(t) + ∝₀^tB(t ? s) x(s) ds, where B(t) is integrable and does not change sign on [0, ∞). The results are applied to an examination of the stability of equilibrium of some nonlinear population models.     

Large time behavior of differential equations with drifted periodic coefficients modeling carbon storage in soil

This paper is concerned with the linear ODE in the form y′(t) = λρ(t)y(t) + b(t), λ < 0 which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function ρ(t), a linear drift in the coefficient b(t) involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

One-step 5-stage Hermite-Birkhoff-Taylor ODE solver of order 12

A one-step 5-stage Hermite-Birkhoff-Taylor method, HBT(12)5, of order 12 is constructed for solving nonstiff systems of differential equations y^′=f(t,y), y(t₀)=y₀, where y∈Rⁿ. The method uses derivatives y^′ to y⁽⁹⁾ as in Taylor methods combined with a 5-stage Runge-Kutta method. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 12 leads to Taylor- and Runge-Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. HBT(12)5 has a larger interval of absolute stability than Dormand-Prince DP(8, 7)13M and Taylor method T12 of order 12. The new method has also a smaller norm of principal error term than T12. It is superior to DP(8, 7)13M and T12 on the basis the number of steps, CPU time and maximum global error on common test problems. The formulae of HBT(12)5 are listed in an appendix.     

The Hukuhara-Kneser property for parabolic systems with nonlinear boundary conditions

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Control problems for parabolic equations on controlled domains

Given the one-dimensional heat equation v_t = v_xx on the controlled domain Q(y) = {(t, x); 0 < x < y(t), 0 < t < T} subject to some initial-boundary conditions, we study the problem of optimally selecting y(·) from some admissible class so as to maximize a given payoff of fixed duration. Q(y) is thus a controlled domain. We also study the problem in which the heat equation holds in Q(y, z) = {z(t) < x < y(t), 0 < t < T}; z minimizing, y maximizing, i.e., the differential game. The principle techniques involved are (i) transforming the controlled domain to an uncontrolled domain and then (ii) using the method of lines for parabolic equations to enable us to use known results for control systems governed by ordinary differential equations. Sufficient conditions for existence in an admissible class is given and the method of lines allows numerical techniques to be applied to determine the optimal control in our class.     

A note on singular nonlinear boundary value problems for the one-dimensional p-Laplacian

Existence theorems about the positive solution for the singular equation (ϕ_p(y′))′ + f(t,y) = 0, y(0) = y(1) = 0 are established. The results are obtained by using a fixed-point theorem in cones.     

Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

A one-step 7-stage Hermite-Birkhoff-Taylor ODE solver of order 11

A one-step 7-stage Hermite-Birkhoff-Taylor method of order 11, denoted by HBT(11)7, is constructed for solving nonstiff first-order initial value problems y^′=f(t,y), y(t₀)=y₀. The method adds the derivatives y^′ to y⁽⁶⁾, used in Taylor methods, to a 7-stage Runge-Kutta method of order 6. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 11 leads to Taylor- and Runge-Kutta-type order conditions. These conditions are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. The new method has a larger scaled interval of absolute stability than the Dormand-Prince DP87 and a larger unscaled interval of absolute stability than the Taylor method, T11, of order 11. HBT(11)7 is superior to DP87 and T11 in solving several problems often used to test higher-order ODE solvers on the basis of the number of steps, CPU time, and maximum global error. Numerical results show the benefit of adding high-order derivatives to Runge-Kutta methods.     

Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping

In this paper, we are concerned with the oscillation of third order nonlinear delay differential equations of the form

(r₂(t)(r₁(t)y^′)′)′+p(t)y^′+q(t)f(y(g(t)))=0.     

Comparison theorems for the third order trinomial differential equations with delay argument

In this paper we study asymptotic properties of the third order trinomial delay differential equation (*) y‴(t) − p(t)y′(t) + g(t)y(τ(t)) = 0 by transforming this equation to the binomial canonical equation. The results obtained essentially improve known results in the literature. On the other hand, the set of comparison principles obtained permits to extend immediately asymptotic criteria from ordinary to delay equations. Research was supported by S.G.A. No.1/003/09.