A generalized mixed type of quartic–cubic–quadratic–additive functional equations

We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part II

In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value. Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

On dual integral equations in the semiconservative case

A dual integral equations on the whole real axis with an unknown function f is considered. It is supposed that the kernel functions of the equations are even and representable as superposition of exponents. The equation is transferred to a system of integral equations on the positive semi-axis with two unknown functions: f ₁(x) = f(x) and f ₂(x) = f(−x). Applying a factorization method and using the solution of Ambartsumian equation, a system of Laplace transforms α ₁, α ₂ of functions f ₁, f ₂ is obtained. Under some conditions on the free term, the existence and uniqueness of the solution of that system is proved in the semi-conservative case. A construction of the functions α ₁, α ₂ is given by means of successive approximations, and a construction method of the solution (f ₁, f ₂) by α ₁, α ₂ is described.     

On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

We consider complex-valued functions f∈L ¹(ℝ₊), where ℝ₊:=[0,∞), and prove sufficient conditions under which the sine Fourier transform [^(f)]_s\hat{f}_{s} and the cosine Fourier transform [^(f)]_c\hat{f}_{c} belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.     

Quasi-<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm orthogonal Galerkin expansions in sums of Jacobi polynomials

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x) ^α (1 + x) ^β . However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an L ^p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.     

Continuum cardinality of the set of solutions of one class of equations that contain the function of frequency of ternary digits of a number

We study the equation ν ₁(x) = x, where ν ₁(x) is the function of frequency of the digit 1 in the ternary expansion of x. We prove that this equation has a unique rational root and a continuum set of irrational solutions. An algorithm for the construction of solutions is proposed. We also describe the topological and metric properties of the set of all solutions. Some additional facts about the equations ν _i (x) = x, i = 0, 2, are given. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1414–1421, October, 2008.     

The Lambert W-functions and some of their integrals: a case study of high-precision computation

The real-valued Lambert W-functions considered here are w ₀(y) and w _− 1(y), solutions of we ^w  = y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision of these functions and of the integrals ò₁^￥ [-w₀(-xe^-x)]^a x^-bx\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x, α > 0, β ∈ ℝ, and ò₀¹ [-w_-1(-x e^-x)]^a x^-bx\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β.     

Approximation of potential integral by radial bases for solutions of Helmholtz equation

For a Helmholtz equation Δu(x) + κ ² u(x) = f(x) in a region of R ^s , s ≥ 2, where Δ is the Laplace operator and κ = a + ib is a complex number with b ≥ 0, a particular solution is given by a potential integral. In this paper the potential integral is approximated by using radial bases with the order of approximation derived.      

Finite-element approximation for singularly perturbed nonlinear elliptic boundary-value problems

Numerical solution of a two-dimensional nonlinear singularly perturbed elliptic partial differential equation ∈ Δu = f(x, u), 0 < x, y < 1, with Dirichlet boundary condition is discussed here. The modified Newton method of third-order convergence is employed to linearize the nonlinear problem in place of the standard Newton method. The finite-element method is used to find the solution of the nonlinear differential equation. Numerical results are provided to demonstrate the usefulness of the method.     

Numerical evaluation of Goursat’s infinite integral

The infinite integral ò₀^￥x dx/(1+x⁶sin²x)\int_0^{\infty}x\,dx/(1+x^6\sin^2x) converges but is hard to evaluate because the integrand f(x) = x/(1 + x ⁶sin² x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.     

The period function of a delay differential equation and an application

We consider the delay differential equation [(x)\dot](t) = - mx(t) + f(x(t - t))\dot x(t) = - \mu x(t) + f(x(t - \tau )), where μ, τ are positive parameters and f is a strictly monotone, nonlinear C ¹-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.     

Iterative methods based on the signum function approach for solving nonlinear equations

For finding a root of an equation f(x) = 0 on an interval (a, b), we develop an iterative method using the signum function and the trapezoidal rule for numerical integrations based on the recent work (Yun, Appl Math Comput 198:691–699, 2008). This method, so-called signum iteration method, depends only on the signum function sgn(f(x)){\rm{sgn}}\left(f(x)\right) independently of the behavior of f(x), and the error bound of the kth approximation is (b − a)/(2N ^k ), where N is the number of integration points for the trapezoidal rule in each iteration. In addition we suggest hybrid methods which combine the signum iteration method with usual methods such as Newton, Ostrowski and secant methods. In particular the hybrid method combined with the signum iteration and the secant method is a predictor-corrector type method (Noor and Ahmad, Appl Math Comput 180:167–172, 2006). The proposed methods result in the rapidly convergent approximations, without worry about choosing a proper initial guess. By some numerical examples we show the superiority of the presented methods over the existing iterative methods.     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β ₁ ∈ (0,1) and β ₂ ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.     

Convergence of a non-monotone scheme for Hamilton–Jacobi–Bellman equations with discontinous initial data

We prove the convergence of a non-monotonous scheme for a one-dimensional first order Hamilton–Jacobi–Bellman equation of the form v _t + max _α (f(x, α)v _x ) = 0, v(0, x) = v ₀(x). The scheme is related to the HJB-UltraBee scheme suggested in Bokanowski and Zidani (J Sci Comput 30(1):1–33, 2007). We show for general discontinuous initial data a first-order convergence of the scheme, in L ¹-norm, towards the viscosity solution. We also illustrate the non-diffusive behavior of the scheme on several numerical examples.     

Linear independence measures for infinite products

Let f be an entire transcendental function with rational coefficients in its power series about the origin. Further, let f satisfy a functional equation f(qz)= (z−c)f(z)+Q(z) with and some particular c∈ℚ. Then the linear independence of 1,f(α), f(−α) over ℚ for non-zero α∈ℚ is proved, and a linear independence measure for these numbers is given. Clearly, for Q= 0 the function f can be written as an infinite product. Received: 19 September 2000 / Revised version: 14 March 2001     

Elliptic equation with singular potential

We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R ⁿ , n ≥ 3:

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x″+ p ( t ) x′+ q ( t ) f ( x ) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 694–700, May, 2008.     

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]