NONTRIVIAL SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR FOURTH-ORDER DIFFERENTIAL EQUATIONS

The singular boundary value problems for fourth-order differential equations are considered under some conditions concerning the first eigenvalues of the relevant linear operators. Sufficient conditions which guarantee the existence of nontrivial solutions are obtained. We use the topological degree to prove our main results.     

Eigenvalue problems for fractional ordinary differential equations

The eigenvalue problems are considered for the fractional ordinary differential equations with different classes of boundary conditions including the Dirichlet, Neumann, Robin boundary conditions and the periodic boundary condition. The eigenvalues and eigenfunctions are characterized in terms of the Mittag–Leffler functions. The eigenvalues of several specified boundary value problems are calculated by using MATLAB subroutine for the Mittag–Leffler functions. When the order is taken as the value 2, our results degenerate to the classical ones of the second-ordered differential equations. When the order α satisfies 1 < α < 2 the eigenvalues can be finitely many.     

An existence and uniqueness criterion for solutions of boundary value problems

In this paper a technique is developed for the study of the existence and uniqueness of solutions to nth order ordinary differential equations satisfying n-point boundary conditions. Liapunov-like functions are employed to determine the existence and uniqueness of solutions to linear equations satisfying the boundary conditions, and these solutions are in turn used to determine existence for the general nonlinear case. A by-product of this technique is a matching technique for linear equations by which solutions of certain k-point boundary value problems (k < n) can be matched to extend the interval of existence for solutions to the n-point problem.     

A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional ε(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h→0) by applying suitably chosen extension operators p_h. A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of ε(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided.Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples.     

Convergence of rank-type equations

Convergence results are presented for rank-type difference equations, whose evolution rule is defined at each step as the kth largest of p univariate difference equations. If the univariate equations are individually contractive, then the equation converges to a fixed point equal to the kth largest of the individual fixed points of the univariate equations. Examples are max-type equations for k = 1, and the median of an odd number p of equations, for k = (p + 1)/2. In the non-hyperbolic case, conjectures are stated about the eventual periodicity of the equations, generalizing long-standing conjectures of G. Ladas.     

A note on order of convergence of numerical method for neutral stochastic functional differential equations

In this paper, we study the order of convergence of the Euler-Maruyama (EM) method for neutral stochastic functional differential equations (NSFDEs). Under the global Lipschitz condition, we show that the pth moment convergence of the EM numerical solutions for NSFDEs has order p/2 − 1/l for any p ? 2 and any integer l > 1. Moreover, we show the rate of the mean-square convergence of EM method under the local Lipschitz condition is 1 − ε/2 for any ε ∈  (0, 1), provided the local Lipschitz constants of the coefficients, valid on balls of radius j, are supposed not to grow faster than log j. This is significantly different from the case of stochastic differential equations where the order is 1/2.     

Nordsieck methods with computationally verified algebraic stability

We describe the search for algebraically stable Nordsieck methods of order p = s and stage order q = p, where s is the number of stages. This search is based on the theoretical criteria for algebraic stability proposed recently by Hill, and Hewitt and Hill, for general linear methods for ordinary differential equations. These criteria, which are expressed in terms of the non-negativity of the eigenvalues of a Hermitian matrix on the unit circle, are then verified computationally for the derived Nordsieck methods of order p ? 2.     

Stability of Taylor–Couette and Dean flow: A semi-analytical study

An alternative method is presented for solving the eigenvalue problem that governs the stability of Taylor–Couette and Dean flow. The eigenvalue problems defined by the two-point boundary value problems are converted into initial value problems by applying unit disturbance method developed by Harris and Reid [27] in 1964. Thereafter, the initial value problems are solved by differential transform method in series and the eigenvalues are computed by shooting technique. Critical wave number and Taylor number for Taylor–Couette flow are computed for a wide range of rotation ratio (μ), −4 ? μ ? 1 (first mode) and −2 ? μ ? 1 (second mode). The radial eigenfunction and cell patterns are presented for μ = −1, 0, 1. Also, we have computed critical wave number and Dean number successfully.     

Fredholm property of boundary value problems for a fourth-order elliptic differential-operator equation with operator boundary conditions

In a Hilbert space H, we study the Fredholm property of a boundary value problem for a fourth-order differential-operator equation of elliptic type with unbounded operators in the boundary conditions. We find sufficient conditions on the operators in the boundary conditions for the problem to be Fredholm. We give applications of the abstract results to boundary value problems for fourth-order elliptic partial differential equations in nonsmooth domains.     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

Limit points of eigenvalues of (di)graphs

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D, the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set D of digraphs (graphs) is a limit point of eigenvalues of a set of bipartite digraphs (graphs), where consists of the double covers of the members in D. 4. Every limit point of eigenvalues of a set D of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in D. 5. If M is a limit point of the largest eigenvalues of graphs, then −M is a limit point of the smallest eigenvalues of graphs.     

On a method to transform algebraic differential equations into universal equations

The paper introduces an algorithm which transforms homogeneous algebraic differential equations into universal differential equations (in the sense of L. A. Rubel) havingC ⁿ (ℝ)-solutions. By applications of the algorithm to different initial equations some new universal differential equations are found, and all the known equations due to R. J. Duffin are rediscovered with this method. Assuming weak conditions one can find Cⁿ(ℝ)-solutionsy of the differential equation close to any continuous function such that 1, with 0 ≤k ₁ <k ₂ < .... <k _s ≤n are linearly independent over the field of real algebraic numbers at the rational points q1,...,qs.     

A global characterization of the Fu?ík spectrum for a system of ordinary differential equations

The Fu?ík spectrum for systems of second order ordinary differential equations with Dirichlet or Neumann boundary values is considered: it is proved that the Fu?ík spectrum consists of global C¹ surfaces, and that through each eigenvalue of the linear system pass exactly two of these surfaces. Further qualitative, asymptotic and symmetry properties of these spectral surfaces are given. Finally, related problems with nonlinearities which cross asymptotically some eigenvalues, as well as linear-superlinear systems are studied.     

On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.     

Numerical similarity reductions of the (1+3)-dimensional Burgers equation

We consider the (1+3)-dimensional Burgers equation u_t = u_xx + u_yy + u_zz + uu_x which has considerable interest in mathematical physics. Lie symmetries are used to reduce it to certain ordinary differential equations. We employ numerical methods to solve a number of these ordinary differential equations.     

Property (X) for Second-Order Nonlinear Differential Equations of Generalized Euler Type

In this paper the generalized nonlinear Euler differential equation t²k(tu′)u″ + t(f(u)+ k(tu′))u′ + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X⁺), which is very important for the existence of periodic solutions and oscillation theory.     

Minimal Positive Solutions to Some Singular Second-Order Differential Equations

The existence of a minimal C¹[0, 1] positive solution is established for some second-order singular boundary value and initial value problems by new schemes, which are related to x′. Our nonlinearity may be singular at t = 0, 1, x = 0, or x′ = 0.