Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.     

Zero product determined Jordan algebras,I

We show that the Jordan algebra 𝒮 of symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that if a bilinear map {.,?.} from 𝒮?×?𝒮 into a vector space X satisfies {x, y}?=?0 whenever x?○?y?=?0, then there exists a linear map T : 𝒮?→?X such that {x,?y}?=?T(x?○?y) for all x, y?∈?𝒮 (here, x?○?y?=?xy?+?yx).     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

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.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

Rearrangements of incidence tables

Given integers 0 < k ? n and a permutation α mapping the set of integers from 1 to n onto itself, there exists a permutation β mapping the set of unordered pairs of integers from 1 to n onto itself in such a way that whenever 0 < |a ? b| ? k and {aα, bα}β = {x, y}, then also 0 < |x ? y| ? k and x and y are equal to or between aα and bα.     

On a conjecture of R.F. Scott (1881)

A formula is given for the permanent of a general Cauchy matrix ((x_i?y_j)^?1). In a special case, where the x_i and the y_j are the distinct nth roots of 1 and ?1 respectively, a formula for the permanent, conjectured by R. F. Scott, is proved by computing the eigenvalues of related circulants.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Symmetric positive systems in corner domains

Consider the symmetric positive system of n equations in m + 2 variables,

$\text{A}\text{?u}\text{?x}\text{+ B}\text{?u}\text{?y}\text{+}\text{∑}\text{i=1}\text{m}\text{C}{}_{\mathrm{i}}\text{?u}\text{?z}{}_{\mathrm{i}}\text{+ Du = ?}$

in the corner domain x > 0, y > 0, ? ∞ < z_i < ∞, with homogeneous data on x = 0 and y = 0. The n × n matrices A, B, C_i are symmetric and D is sufficiently positive. On the boundary surfaces the matrix coefficients A, B, C_i satisfy certain “torsion” conditions. For ? with square integrable first-order derivatives, the strong solution with first-order strong derivatives is derived for the boundary value problem. For less restricted ?, the partially differentiable strong solution is established, provided more severe torsion conditions are satisfied on the boundaries. Also, the partially differentiable strong solution is obtained for the case that the torsion conditions are satisfied on one side of the boundary only.     

Constructive existence and uniqueness for two-point boundary-value problems with a linear gradient term

Constructive existence and uniqueness theorems are established for nonlinear two-point boundary-value problems governed by the equation y″=f(x, y)+p(x)y′, 0?x?1. The existence and uniqueness is established for functions y(x) that satisfy a certain constraint, i.e., that sfncy(x)sfnc is bounded by a known function or that 0?y(x)?M for some M. Applications to scientific problems in the literature are given.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

An optimal filtering method for stable analytic continuation

In this paper, we consider the problem of numerical analytic continuation of an analytic function f(z)=f(x+iy) on a strip domain Ω⁺={z=x+iy∈C∣x∈R,0<y<y₀}, where the data is given approximately only on the real axis y=0. This problem is severely ill-posed: the solution does not depend continuously on the given data. A novel method (filtering) is used to solve this problem and an optimal error estimate with Hölder type is proved. Numerical examples show that this method works effectively.     

Asymptotic error expansions for hypersingular integrals

This paper presents quadrature formulae for hypersingular integrals $\int_a^b\frac{g(x)}{|x-t|^{1+\alpha }}\mathrm{d}x$ , where a?<?t?<?b and 0?<?α?≤?1. The asymptotic error estimates obtained by Euler–Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h ^2μ ) for α?=?1 and O(h ^2μ???α ) for 0?<?α?<?1, where μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.     

Counting singular plane curves via Hilbert schemes

We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singularities for which our results suffice include the topological type with local equation x^a+y^b with ?a?3b. We work out the example of curves with the analytic type of singularity with local equation x²+yⁿ for 1<n<9.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

A fourth-order finite difference method based on uniform mesh for singular two-point boundary-value problems

We present a new fourth-order finite difference method based on uniform mesh for the (weakly) singular two-point boundary value problem: (x^αy′)′ = f(x, y), 0 < x ⩽ 1, y(0) = A, y(1) = B, 0 < α < 1. Our method provides O(h⁴)-convergent approximations for all α ∈ (0, 1); for α = 0 it reduces to the well-known fourth-order method of Numerov for $\text{y″ = ƒ(x, y)}$.