On the completeness of the spherical vector wave functions

The set of the regular and radiating spherical vector wave functions (SVWF) is shown to be complete in L₂(S) where S is a Lyapunov surface. The completeness fails when k² is an eigenvalue of the Dirichlet problem for the Helmholtz equation (▽² + k²)F = 0 in the region D_i bounded by S. On the other hand, the set of radiating SVWF is shown to be complete for all values of k². It is also proved that any vector function, which is continuous in D_i + S and satisfies the Helmholtz equation in D_i., can be approximated uniformly in D_i and in the mean square sense on S by a sequence of linear combinations of the regular SVWF (assuming the set is complete). Similar results are obtained for the exterior problem with the set of radiating SVWF. These results are extended to the set composed of the regular and radiating SVWF on two nonintersecting Lyapunov surfaces, one of which encloses the other.     

Full low-frequency asymptotic expansion for second-order elliptic equations in two dimensions

The present paper contains the low-frequency expansions of solutions of a large class of exterior boundary value problems involving second-order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k² and k² (ln k + a)^?1, where k is the wave number and a is a constant. In other cases it has the form of a power series with respect to k², coefficients of which depend polynomially on In k. The procedure for determining the full low-frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.     

A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide

For the Helmholtz equation Δu + k ² u = 0 in a domain Ω with a cylindrical outlet Q ₊ = ω × ?₊ to infinity, we construct a fictitious scattering operator $\mathfrak{S}$ that is unitary in L ₂(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of $\mathfrak{S}$ corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λ_n ≤ k ² of the Dirichlet problem for the Laplace operator on the cross-section ω.     

Some representations of solutions of elliptic equations

The Dirichlet problem for the region of the plane inside closed smooth curve C for second-order elliptic equations is considered. It is shown that under certain circumstances the solution u can be written uniquely in the form u(P) = ∝_cF(P, Q) g(Q) ds_Q, where F(P, Q) is the fundamental solution of the elliptic equation, and g?L² if the boundary value function f is absolutely continuous with square integrable derivative (f?W); and u(P) = p(F(P, ·)) where p is a unique bounded linear functional on W if f?L². These representations are valid in the exterior of C also. As special cases with slight modifications, the exterior Dirichlet problems for the Helmholtz and Laplace equations are mentioned.It is shown also that if kernel F(P′, Q), with P′ and Q on C, has a complete set of eigenfunctions {ψ_k(P′)} then u(P) can be expanded in a series of their extensions {ψ_k(P)}, where ψ_k(P) = λ_k ∝_cF(P, Q) ψ_k(Q) ds_Q.     

On the average orders of the error term in the Dirichlet divisor problem

Let Δ(x) be the error term in the Dirichlet divisor problem. The purpose of this paper is to study the difference between two kinds of mean value formulas of Δ(x), that is, the mean value formulas and ∑_n?xΔ(n)^k with a natural number k. In particular we study the case k=2 and 3 in detail.     

Exterior dirichlet problem for the reduced wave equation: asymptotic analysis of low frequencies

The order of convergence for the low frequency asymptotics of the exterior Dirichlet problem for the Helmholtz equation with variable, possibly non-smooth coefficients is shown to be Ω², the square of the frequency, except of some singular cases. In these cases the asymptotics are characterized completely by some lower order terms in the spherical harmonics expansion of the solution to the static problem.     

On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Implicit difference schemes of O(k⁴ + k²h² + h⁴), where k0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.     

An Initial-Boundary Value Problem for the Korteweg–de Vries Equation with Dominant Surface Tension

We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix s(k) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for s(k,t) is found. Then, the time-dependent scattering matrix s(k,t) is expressed in terms of s(k)=s(k,0) and of solutions of the self-conjugate SP. Knowing s(k,t), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

A new frequency‐uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L²(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star‐shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star‐combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second‐kind integral operator for which convergence of the Galerkin method in L²(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star‐combined operator implies frequency‐explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high‐frequency case. The proof of coercivity of the star‐combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.     

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.     

On the mean value of the mixed exponential sums with Dirichlet characters and general gauss sum

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers m, n, k, q, with k ? 1 and q ? 3, and Dirichlet characters χ, χ? modulo q we define a mixed exponential sum $$C(m,n;k;\chi ;\overline \chi ;q) = \sum\limits_{a = 1}^q {\chi (a){G_k}(a,\overline \chi )e\left( {{{m{a^k} + n\overline {{a^k}} } \over q}} \right)} ,$$ , with Dirichlet character χ and general Gauss sum G _k (a, χ?) as coefficient, where Σ′ denotes the summation over all a such that (a, q) = 1, aā ≡ 1 mod q and e(y) = e^2πiy . We mean value of $$\sum\limits_m {\sum\limits_\chi {\sum\limits_{\overline \chi } {C{{\left| {m,n;k;\chi ;\overline \chi ;q} \right|}^4}} } } ,$$ , and give an exact computational formula for it.     

Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star‐shaped domain in ?^d, d ≥ 2. Conventional boundary‐based methods require a root search in eigenfrequency k, hence take O(N³) effort per eigenpair found, where N = O(k^d?1) is the number of unknowns required to discretize the boundary. Our method is O(N) faster, achieved by linearizing with respect to k the spectrum of a weighted interior Neumann‐to‐Dirichlet (NtD) operator for the Helmholtz equation. Approximations to the square roots k_j of all O(N) eigenvalues lying in [k ? ?, k], where ? = O(1), are found with O(N³) effort. We prove an error estimate with C independent of k. We present a higher‐order variant with eigenvalue error scaling empirically as O(?⁵) and eigenfunction error as O(?³), the former improving upon the “scaling method” of Vergini and Saraceno. For planar domains (d = 2), with an assumption of absence of spectral concentration, we also prove rigorous error bounds that are close to those numerically observed. For d = 2 we compute robustly the spectrum of the NtD operator via potential theory, Nyström discretization, and the Cayley transform. At high frequencies (400 wavelengths across), with eigenfrequency relative error 10^?10, we show that the method is 10³ times faster than standard ones based upon a root search. © 2014 Wiley Periodicals, Inc.     

On an exact control problem for a semilinear wave equation with an unknown source

The exact controllability of a semilinear wave equation, with Dirichlet boundary control on a part of the boundary and an unknown source, is shown. The nonlinear term has at most a linear growth, the initial and target spaces are L²(Ω)×H⁻¹(Ω).     

Analytic continuation of a class of dirichlet series

We consider Dirichlet series of the type Σ(logk)ⁿ(k)(logk)?^-s. We prove the existence of an analytic continuation to the cut plane and give exact information about the singularity. We use this to generalize results, which occur in Ramanujan’s second notebook.     

Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers

A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients in a square domain is considered. Dirichlet conditions are specified on two sides orthogonal to the flow, and Neumann conditions are set on the other two sides. The right-hand side and the boundary functions are assumed to be sufficiently smooth, which ensures the required smoothness of the desired solution in the domain, except for neighborhoods of the corner points. Only zero-order compatibility conditions are assumed to hold at the corner points. The problem is solved numerically by applying an inhomogeneous monotone difference scheme on a rectangular piecewise uniform Shishkin mesh. The inhomogeneity of the scheme lies in that the approximating difference equations are not identical at different grid nodes but depend on the perturbation parameter. Under the assumptions made, the numerical solution is proved to converge ?-uniformly to the exact solution in a discrete uniform metric at an O(N ^?3/2ln² N) rate, where N is the number of grid nodes in each coordinate direction.     

Rawlins' problem for half-plane diffraction: its generalized eigenfunctions with real wave numbers

This paper deals with a famous diffraction problem for a single half-plane Σ: x>0, y=0 as an obstacle and for some time-harmonic plane incident wave field. Rawlins in 1975 was the first to solve the mixed (Dirichlet/Neumann) boundary value problem for the scalar Helmholtz equation. He also was the first to solve the equivalent pair of coupled Wiener–Hopf equations explicitly by factoring their discontinuous 2×2 Fourier matrix symbol in 1980. Although for real wave numbers k the usual factorization procedure fails it will serve as the basis: Following the lines given by Ali Mehmeti in his habilitation thesis [1] for the (Dirichlet/Dirichlet) boundary value problem we combine the idea of integral path deforming along the branch cuts of the characteristic square root √(ξ²−k²) given in Meister's book [13] with the modern Wiener–Hopf method solution derived by Speck [24] explicitly in a H^1+ε, ε⩾0, Sobolev space setting. The symmetry of the intermediate spaces H^s, H^-s, ∣s∣<1 2, which is due to generalized factorization, plays a key role in deforming the Fourier integral paths in order to get Laplace transform representations of the generalized eigenfunctions of the problem. As a remarkable fact 0<ε<¼ must hold here. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A proof for the Burton and Miller integral equation approach for the Helmholtz equation

The A. J. Burton and G. F. Miller integral equation formulation for the exterior Neumann problem for the Helmholtz equation [Proc. Roy. Soc. London Ser. A323 (1971), 201–210] is one of the most important integral equation approaches in that area. However, the kind of space settings they are working with is not clear. Evidently, the Fredholm integral equation of the second kind which they deduced is not well defined on the usual C(S) or L²(S), where S is a closed bounded smooth surface. In this paper, appropriate space settings are found and a rigorous existence and uniqueness proof for their integral equation formulation is given.     

On the Morse critical groups for indefinite sublinear elliptic problems

We consider the Dirichlet problem for the equation −Δu=αu+m(x)u|u|^q−2+g(x,u), where q∈(1,2) and m changes sign. We prove that the Morse critical groups at zero of the energy functional of the problem are trivial. As a consequence, existence and bifurcation of nontrivial solutions of the problem are established.     

Explicit representations of the integral containing the error term in the divisor problem

We shall investigate several properties of the integral $$ \int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt} $$ with a natural number k, a non-negative integer j and a complex variable θ, where Δ _k (x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.