An invariant imbedding method for computation of eigenlengths of singular two-point boundary-value problems

In this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular TPBVP with the singularity at the origin. The invariant imbedding yields a first-order nonlinear equation called a Riccati equation and also gives the initial conditions at the origin for this equation. With the aid of Theorem 8 in Section 3 we numerically integrate the Riccati equation to “blowup” which gives our computed eigenlength.In closing, we would like to comment on the numerical merits of the integration-to-blowup technique. On the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results. The feature of a variable step size is essential as the value of z approaches the actual eigenlength. However, it is desirable to have a priori estimate or bounds of the eigenlength similar to those of Boland and Nelson [2] for the nonsingular case. The singular system, however, presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds. Hopefully an investigation of the matrix R(z) will yield these results.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

Analytic Solutions and Singularity Formation for the Peakon b-Family Equations

This paper deals with the well-posedness of the b-family equation in analytic function spaces. Using the Abstract Cauchy-Kowalewski theorem we prove that the b-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to H ^s with s>3/2, and the momentum density u ₀?u _0,xx does not change sign, we prove that the solution stays analytic globally in time, for b≥1. Using pseudospectral numerical methods, we study, also, the singularity formation for the b-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity approaches the real axis, estimating the rate of decay of the Fourier spectrum.     

Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.     

A bound on the dimension of interval orders

We make use of the partially ordered set (I(0, n), <) consisting of all closed intervals of real numbers with integer endpoints (including the degenerate intervals with the same right- and left-hand endpoints), ordered by [a, b] < [c, d] if b < c, to show that there is no bound on the order dimension of interval orders. We then turn to the problem of computing the dimension of I(0, n), showing that I(0, 10) has dimension 3 but I(0, 11) has dimension 4. We use these results as initial conditions in obtaining an upper bound on the dimension of I(0, n) as a logarithmic function of n. It is our belief that this example is a “canonical” example for interval orders, so that the computation of its dimension should have significant impact on the problem of computing the dimension of interval orders in general.     

Application of the internal time operators for the Renyi map

Recently, the internal time operator for the Renyi map has been constructed (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals). It corresponds to a phase space given by the interval [0,1] and to the invariant Lebesgue measure. In this paper, following the idea of (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals), we construct the time operator for a dynamical system with an arbitrary invariant measure μ and an arbitrary phase space X=[a,b] with a and b finite or infinite. We illustrate also the action of such an operator on a fixed initial state.     

Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation

In this paper, a piecewise constant time-stepping discontinuous Galerkin method combined with a piecewise linear finite element method is applied to solve control constrained optimal control problem governed by time fractional diffusion equation. The control variable is approximated by variational discretization approach. The discrete first-order optimality condition is derived based on the first discretize then optimize approach. We demonstrate the commutativity of discretization and optimization for the time-stepping discontinuous Galerkin discretization. Since the state variable and the adjoint state variable in general have weak singularity near t =?0and t = T, a time adaptive algorithm is developed based on step doubling technique, which can be used to guide the time mesh refinement. Numerical examples are given to illustrate the theoretical findings.     

Invariant imbedding and multiple shooting for the solution of unstable linear boundary value problems

Invariant imbedding has been used to solve unstable linear boundary value problems for a few years. First this method is derived using the theory of characteristics; there the boundary value problem has to be imbedded in a problem of double dimension. If the corresponding Riccati equation has a critical length, one has to repeat the algorithm. A relation between this repeated invariant imbedding and multiple shooting is shown. In examples invariant imbedding, repeated invariant imbedding, multiple shooting and the superposition principle are compared.     

A nonasymptotic method for singular perturbation problems

In this paper, an approximate method for the numerical integration of singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval is presented. The method is distinguished by the following fact: the original second-order differential equation is replaced by an approximate first-order differential equation with a small deviating argument and is solved efficiently by employing the Simpson rule, coupled with the discrete invariant imbedding algorithm. The proposed method is iterative on the deviating argument. Several numerical examples have been solved to demonstrate the applicability of the method.     

On the computational complexity of best Chebyshev approximations

The best Chebyshev approximation of degree n to a continuous function f on [0, 1] is the unique polynomial ϕ of degree less than or equal to n such that the maximum difference of f and ϕ on [0, 1] is minimized. On the basis of a formal model of computation, it is shown that the question of whether the best Chebyshev approximations of polynomial-time computable functions on [0, 1] are always polynomial-time computable depends on the relationship among well-known discrete complexity classes. In particular, P = NP implies that these best approximations are polynomial-time computable, and EXP ≠ NEXP implies that these best approximations are not polynomial-time computable. It is also pointed out that the fact that the popular Remes algorithm converges fast does not conflict with the above result, since the Remes algorithm requires, in each iteration, the finding of maximal points of continuous functions on an interval [a, b], which is, in general, provably intractable.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

A factorization method for elliptic problems in a circular domain

We present a method to factorize a second order elliptic boundary value problem in a circular domain, in a system of uncoupled first order initial value problems. We use a space invariant embedding technique along the radius of the circle, in a decreasing way. This technique is inspired in the temporal invariant embedding used by J.-L. Lions for the control of parabolic systems. The singularity at the origin for the initial value problems is studied. A formal calculation for more general star-shaped domains is presented. To cite this article: J. Henry et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Numerical treatment of singularly perturbed two point boundary value problems

We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort.     

Approximating a real number by a rational number with a limited denominator: A geometric approach

We consider the problem of determining the rational number which best approximates the real number a and such that its denominator belongs to an interval [b,b^′]. There is a related geometric problem consisting in finding the integer point lying in the vertical domain D of the form {(x,y)∈R²∣b≤x≤b^′} such that the straight line passing through the origin and through this point best approximates the straight line L of slope a passing through the origin. The computation of this point is interlinked with the computation of both the convex hulls of the integer points located above and below the straight line L respectively and lying in the vertical domain D. In the literature, many general convex hull algorithms exist, as the gift wrapping algorithm for example. However, we focus on two interesting approaches to compute these convex hulls which are especially appropriated in our special configuration. The first one mainly uses number theory and runs in O(log(b^′)) time. The other is in line with computational geometry as the method proposed in 1999 by Balza-Gomez et al. [H. Balza-Gomez, J.-M. Moreau, D. Michelucci, Convex hull of grid points below a line or a convex curve, in: DGCI ’99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, Springer, Marne-la-Vallée, France, 1999, pp. 361-374] which runs in O(log(b^′−b)) time. We propose a new method for the computation of these convex hulls which combines number theory and computational geometry. Our method preserves the optimal time complexity and is the first being output sensitive. Indeed, we compute the convex hulls in time linear in their vertex number. Moreover, the resulting algorithm is very simple and so is suitable for implementation.     

Spatial curve singularities and the monster/semple tower

The Monster tower ([MZ01], [MZ10]), known as the Semple Tower in Algebraic Geometry ([Sem54], [Ber10]), is a tower of fibrations canonically constructed over an initial smooth n-dimensional base manifold. Each consecutive fiber is a projective n — 1 space. Each level of the tower is endowed with a rank n distribution, that is, a subbundle of its tangent bundle. The pseudogroup of diffeomorphisms of the base acts on each level so as to preserve the fibration and the distribution. The main problem is to classify orbits (equivalence classes) relative to this action. Analytic curves in the base can be prolonged (= Nash blown-up) to curves in the tower which are integral for the distribution. Prolongation yields a dictionary between singularity classes of curves in the base n-space and orbits in the tower. This dictionary yielded a rather complete solution to the classification problem for n = 2 ([MZ10]). A key part of this solution was the construction of the ‘RVT’ classes, a discrete set of equivalence classes built from verifying conditions of transversality or tangency to the fiber at each level ([MZ10]). Here we define analogous ‘RC’ classes for n > 2 indexed by words in the two letters, R (for regular, or transverse) and C (for critical, or tangent). There are 2 ^k?1 such classes of length k and they exhaust the tower at level k. The codimension of such a class is the number of C’s in its word. We attack the classification problem by codimension, rather than level. The codimension 0 class is open and dense and its structure is well known. We prove that any point of any codimension 1 class is realized by a curve having a classical A _2k singularity (k depending on the type of class). Following ([MZ10]) we define what it means for a singularity class in the tower to be “tower simple”. The codimension 0 and 1 classes are tower simple, and tower simple implies simple in the usual sense of singularity. Our main result is a classification of the codimension 2 tower simple classes in any dimension n. A key step in the classification asserts that any point of any codimension 2 singularity is realized by a curve of multiplicity 3 or 4. A central tool used in the classification are the listings of curve singularities due to Arnol’d ([Arn99], Bruce-Gaffney ([BG82]), and Gibson-Hobbs ([GH93]). We also classify the first occurring truly spatial singularities as subclasses of the codimension 2 classes. (A point or a singularity class is “spatial” if there is no curve which realizes it and which can be made to lie in some smooth surface.) As a step in the classification theorem we establish the existence of a canonical arrangement of hyperplanes at each point, lying in the distribution n-plane at that point. This arrangement leads to a coding scheme finer than the RC coding. Using the arrangement coding we establish the lower bound of 29 for the number of distinct orbits in the case n = 3 and level 4. Finally, Mormul ([Mor04], [Mor09]) has defined a different coding scheme for singularity classes in the tower and in an appendix we establish some relations between our coding and his.     

An iterative method for a system of linear complementarity problems with perturbations and interval data

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^∗],[y^∗]) which includes the solution set for the system of linear complementarity problems defined by any fixed A∈[A],B∈[B],b∈[b] and c∈[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.     

Optimal control of capital injections by reinsurance in a diffusion approximation

In this paper we consider a diffusion approximation to a classical risk process, where the claims are reinsured by some reinsurance with deductible b ∈ [0,b?], where b = b? means “no reinsurance” and b = 0 means “full reinsurance”. The cedent can choose an adapted reinsurance strategy (b _t ) _{t ≥0}, i.?e. the deductible can be changed continuously. In addition, the cedent has to inject fresh capital in order to keep the surplus positive. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton–Jacobi–Bellman approach. Some examples illustrate the method.     

A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.     

For the stationary compressible viscous Navier-Stokes equations with no-slip condition on a convex polygon

Our concern is with existence and regularity of the stationary compressible viscous Navier-Stokes equations with no-slip condition on convex polygonal domains. Note that [u,p]=[0,c], c a constant, is the eigenpair for the singular value λ=1 of the Stokes problem on the convex sector. It is shown that, except the pair [0,c], the leading order of the corner singularities for the nonlinear equations is the same as that of the Stokes problem. We split the leading corner singularity from the solution and show an increased regularity for the remainder. As a consequence the pressure solution changes the sign at the convex corner and its derivatives blow up.     

Invariant imbedding applied to homogeneous two-point boundary-value problems with a singularity of the first kind

It is shown that previous work of Elder can be used to extend the version of invariant imbedding due to M. R. Scott to homogeneous (vector) differential systems having a singularity of the first kind. The boundary conditions considered consist of existence (finite) at the singularity and specified values for some subset of the dependent variables at a second point. The important special case of a second-order equation is discussed in some detail. Computational considerations are discussed and numerical examples are presented.