Fractal oscillations of self-adjoint and damped linear differential equations of second-order

For a prescribed real number s ∈ [1, 2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y = y(x), y ∈ C²((0, T]) of the linear differential equation (p(x)y′)′ + q(x)y = 0 on (0, T], is bounded and fractal oscillatory near x = 0 with the fractal dimension equal to s. This means that y oscillates near x = 0 and the fractal (box-counting) dimension of the graph Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(y) is finite and strictly positive. It verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x = 0 like the chirp function y_ch(x) = a(x)S(φ(x)), which often occurs in the time-frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form y″ + (μ/x)y′ + g(x)y = 0, x ∈ (0, T]. In order to prove the main results, we state a new criterion for fractal oscillations near x = 0 of real continuous functions which essentially improves related one presented in [1].     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(x, t) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).     

Generalized Jordan algebras

We study commutative algebras which are generalizations of Jordan algebras. The associator is defined as usual by (x, y, z) = (x y)z − x(y z). The Jordan identity is (x², y, x) = 0. In the three generalizations given below, t, β, and γare scalars. ((x x)y)x + t((x x)x)y = 0, ((x x)x)(y x) − (((x x)x)y)x = 0, β((x x)y)x + γ((x x)x)y − (β + γ)((y x)x)x = 0. We show that with the exception of a few values of the parameters, the first implies both the second and the third. The first is equivalent to the combination of ((x x)x)x = 0 and the third. We give examples to show that our results are in some reasonable sense, the best possible.     

A finite difference solution to a two-dimensional parabolic inverse problem

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy u_t = u_xx + u_yy + p(t)u + ? in R × [0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(x^∗, y^∗, t) = E(t), 0 < t ? T, where E(t) is known and (x^∗, y^∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L_∞ norm. The convergence orders of u and p are of O(τ + h²). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.     

Duality for a Class of Multiobjective Control Problems

In this paper, a class of multiobjective control problems is considered, where the objective and constraint functions involved are f(t, x(t), ?(t), y(t), z(t)) with x(t) ∈ Rⁿ, y(t) ∈ Rⁿ, and z(t) ∈ R^m, where x(t) and z(t) are the control variables and y(t) is the state variable. Under the assumption of invexity and its generalization, duality theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems.     

Distributive equations of implications based on nilpotent triangular norms

In this paper, we explore the distributive equations of implications, both independently and along with other equations. In detail, we consider three classes of equations. (1) By means of the section of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(x, T(y, z)) = T(I(x, y), (x, z)) based on a nilpotent triangular norm T and an unknown function I, which indicates that there are no continuous solutions satisfying the boundary conditions of implications. Under the assumptions that I is continuous except the vertical section I(0, y), y ∈ [0, 1), we get its complete characterizations. (2) We prove that there are no solutions for the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z). (3) We obtain the sufficient and necessary conditions on T and I to be solutions of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, y) = I(N(y), N(x)).     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .     

Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

This paper presents a generalized Gaussian quadrature method for numerical integration over triangular, parallelogram and quadrilateral elements with linear sides. In order to derive the quadrature rule, a general transformation of the regions, R₁ = {(x, y)∣a ? x ? b, g(x) ? y ? h(x)} and R₂ = {(x, y)∣a ? y ? b, g(y) ? x ? h(y)}, where g(x), h(x), g(y) and h(y) are linear functions, is given from (x, y) space to a square in (ξ, η) space, S: {(ξ, η)∣0 ? ξ ? 1, 0 ? η ? 1}. Generlized Gaussian quadrature nodes and weights introduced by Ma et.al. in 1997 are used in the product formula presented in this paper to evaluate the integral over S, as it is proved to give more accurate results than the classical Gauss Legendre nodes and weights. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear sides. The performance of the method is illustrated for different functions over different two-dimensional regions with numerical examples.     

Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y²+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.     

On the polynomial limit cycles of polynomial differential equations

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution.     

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are

(x²+y)uxx+2xyuxy+y²uyy+gxux+g(y−1)uy=λu,     

Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order

Asymptotic and oscillatory behaviours near x=0 of all solutions y=y(x) of self-adjoint linear differential equation (Ppq): ^′(py^′)+qy=0 on (0,T], will be studied, where p=p(x) and q=q(x) satisfy the so-called Hartman-Wintner type condition. We show that the oscillatory behaviour near x=0 of (Ppq) is characterised by the nonintegrability of on (0,T). Moreover, under this condition, we show that the rectifiable (resp. unrectifiable) oscillations near x=0 of (Ppq) are characterised by the integrability (resp. nonintegrability) of on (0,T). Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturm?s comparison type theorem for the rectifiable oscillations is stated. Furthermore, previous results are used to establish such kind of oscillations for damped linear second-order differential equation y^″+g(x)y^′+f(x)y=0, and especially, the Bessel type damped linear differential equations are considered. Finally, some open questions are posed for the further study on this subject.     

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u_t = Δu + u^m(x, t)e^pv(0,t), v_t = Δv + u^q(0, t)e^nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u₀(v₀) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.     

On the enumeration of certain weighted graphs

We enumerate weighted simple graphs with a natural upper bound condition on the sum of the weight of adjacent vertices. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that the generating function for connected bipartite simple graphs is of the form p₁(x)/(1-x)^m+1. For nonbipartite simple graphs, we get a generating function of the form p₂(x)/(1-x)^m+1(1+x)^l. Here m is the number of vertices of the graph, p₁(x) is a symmetric polynomial of degree at most m, p₂(x) is a polynomial of degree at most m+l, and l is a nonnegative integer. In addition, we give computational results for various graphs.