Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

We give characterizations of the distributional derivatives D ^1,1, D ^1,0, D ^0,1 of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

Optimal Cubature Formulae and Recovery of Fast-Oscillating Functions from an Interpolational Class

The method of limit functions is used to construct optimal-by-accuracy and optimal-by-order (with constant not exceeding two) cubature formulae for the integration of fast oscillatory functions given by their values at a finite number of fixed nodes in a square region. The construction is based on explicit forms of the majorant and minorant in the given interpolational class C _1,L,N ² and the solution of the problem of optimal-by-accuracy recovery of functions from this class. It is shown that an appropriate choice of the grid in this interpolational class leads to a substantial reduction in a priori information required for the application of the proposed approach.This revised version was published online in October 2005 with corrections to the Cover Date.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

The auto‐correlation equation on the finite interval

Applying the Fourier cosine transformation, the quadratic auto‐correlation equation on the finite interval [0,T] of the positive real half‐axis ?₊ is reduced to a problem for the modulus of the finite complex Fourier transform of the solution. From the solutions of this problem L²‐solutions of the auto‐correlation equation are obtained in closed form. Moreover, as in the case of the equation on ?₊ a Lavrent'ev regularization procedure for the auto‐correlation equation is suggested. Copyright © 2003 John Wiley & Sons, Ltd.     

Translation representations for automorphic solutions of the wave equation in non-euclidean spaces. I

This paper deals with the spectral theory of the Laplace-Beltrami operator Δ acting on automorphic functions in n-dimensional hyperbolic space Hⁿ. We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [-(1/2(n - 1))², 0], and none less than (1/2(n -1))². Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (-∞, -(1/2(n - 1))²). Our approach uses the non-Euclidean wave equation introduced by Faddeev and Pavlov, Energy E_F is defined as (u_t, u_t)-(u, Lu), where the bracket is the L₂ scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L₂(R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations. u_tt-Lu = 0, L = Δ + (1/2(n - 1))².     

A Second Main Theorem on ℙ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> for difference operator

The main result of this article is an extension of the Second Main Theorem, of Halburd and Korhonen, for meromorphic functions of finite order. Their result replaces the counting function of the ramification divisor N _ramf(r) in the classical Second Main Theorem by the counting function of a finite difference divisor N _pair(r). In this article, the Second Main Theorem of Halburd and Korhonen is extended to the case of holomorphic maps into ℙ ⁿ of finite order.     

A note about stabilization in Aℝ(𝔻)

It is shown that for A_?(??) functions f₁ and f₂ with and f₁ being positive on real zeros of f₂ then there exists A_?(??) functions g₂ and g₁, g₁^–1 with and This result is connected to the computation of the stable rank of the algebra A_?(??) and to Control Theory (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Separation of Eigenvalues of the Wave Equation for the Unit Ball in RN

It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If W_l denotes the lth positive root for some fixed order x, the minimum gap property assures that {exp(±iw_lt|l = 1, 2,...} form a Riesz basis in L²(0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.     

A remark on theL ∞ bounds of the Ritz operator associated with a finite element approximation

Summary For the elliptic operatorA of second order, its finite element approximationA _h is considered by piecewise linear trial functions. AnL bound on the Ritz operator is shown by Stampacchia's method, which implies a discrete elliptic-Sobolev inequality forA _h.     

Waiting-Time Behavior in a Nonlinear Diffusion Equation

We study the nonlinear diffusion equation u_t*=(uⁿu_x)_x, which occurs in the study of a number of problems. Using singular-perturbation techniques, we construct approximate solutions of this equation in the limit of small n. These approximate solutions reveal simply the consequences of this variable diffusion coefficient, such as the finite propagation speed of interfaces and waiting-time behavior (when interfaces wait a finite time before beginning to move), and allow us to extend previous results for this equation.     

Solving hyperelastic problems using mixed LSFEM

The focus of this contribution is the solution of hyperelastic problems using the least-squares finite element method (LSFEM). In particular a mixed least-squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L₂-norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W^1,p with p > 2, standard polynomials are used. Shape functions belonging to a Raviart-Thomas space are applied for the stress interpolation. These vector-valued functions ensure a conforming discretization of the Sobolev space H(div, Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sommerfeld Half-plane Problems with Higher Order Boundary Conditions

Sommerfeld-type diffraction problems for a half-plane with arbitrary n-th order generalized impedance boundary conditions arc examined in a Sobolev space setting. The corresponding boundary-transmission problems for the two dimensional Helmholtz equation are shown to be well-posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener-Hopf type in [L₂⁺ (IR)]². Formulas for the solutions as well as the so-called edge conditions arc obtained for any n, by explicit canonical generalized factorization of the presymbols of the associated Wiener-Hopf operators.     

Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of (1) where F is a function nondecreasing in each parameter, and λ_k(Ω) is the k^th Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λ_N. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler-Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

The closure operator in a multivalued logic based on functional equations

An operator of FE-closure is introduced on the set of functions of a multivalued logic based on the systems of functional equations. It is proved that, for every k ≥ 2, the FE-closure operator generates a finite classification on the set P _k of functions of k-valued logic. The least class in this classification is shown to be the class H _k of all homogeneous functions. Also a series of corollaries are obtained concerning the finite FE-generating sets in the FE-closed classes.     

On the existence of convex decompositions of partially separable functions

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsf _i whose Hessians have nontrivial nullspacesN _i , Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionf _i is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf _i such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureN _i ¹ have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachN _i is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDL^T factorization without fill-in.     

Extended balance of linear momentum from higher gradient stress field theory

We present the equation of linear momentum considering higher gradients for stress and body force. Both are approximated via Taylor series expansion within a finite Cauchy cube of dimensions L_c. Whereas linear terms of the series expansion result to the classical balance of linear momentum, terms up to third order yield an extended balance equation. The extension includes an internal length scale L²_c caused by surface integrals on the cube. The approach makes use of Cauchy's theorem and standard Newtonian mechanics but constitutive assumptions are not applied. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak Galerkin finite element methods for a fourth order parabolic equation

This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L² and an H² type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results.     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.