Erhaltungssätze und schwache lösungen in der gasdynamik

The fundamental laws of Gasdynamics can be formulated very naturally as conservation laws in the form of integral relations. This formulation includes not only continuously differentiable processes but also the very important discontinuous shocks. On the other side one has the tool of weak solutions of the differential equations of Gasdynamics due to P. D. Lax and several other authors. While the conservation laws of integral type are determined by Physics in an unique way the differential equations of Gasdynamics, even if written in divergence form, are not. Hence the question arises which form of the differential equations in the weak sense is the “correct” interpretation of the physical conservation laws. This paper tries to give an answer by investigating the connections between the two formulations. At first the integral equations of Gasdynamics are written in space-time divergence form. Thus, independently from Gasdynamics, one has Haar's lemma stating that for each weak solution of a partial differential equation (in divergence form) a corresponding integral equation of conservation law type is valid for almost every family member, the family consisting of some simple domains like spheres or squares. Moreover the converse of Haar's lemma is also true. In this paper Haar's lemma is extended to a more general class of domains. This yields that both formulations of conservation laws are essentially equivalent. Additionally a divergence definition due to C. Müller is considered. As is shown by a simple example C. Müller's divergence concept leads to a more general class of solutions, not all of them being solutions of the corresponding conservation laws.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Homoclinic solutions for nonlinear general fourth‐order differential equations

This work provides sufficient conditions for the existence of homoclinic solutions of fourth‐order nonlinear ordinary differential equations. Using Green's functions, we formulate a new modified integral equation that is equivalent to the original nonlinear equation. In an adequate function space, the corresponding nonlinear integral operator is compact, and it is proved an existence result by Schauder's fixed point theorem. Copyright © 2017 John Wiley & Sons, Ltd.     

广义函数x+λ和x-μ的乘积

For any complcx λ≠integral, and any natural number n, the product is calcuated and Hadmard''s finite part of obtained. The product is also obtained when λ+μ is a complex number of any other possible kind, and μ is a complex number ≠ integral. The products are also calculated. Any special example of these kinds of products has not yet been calculated.     

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's‐Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.     

Majorizing Sequences for Nonlinear Fredholm–Hammerstein Integral Equations

We use the method of majorizing sequences to study the applicability of Newton's method to solve nonlinear Fredholm–Hammerstein integral equations. For this, we use center convergence conditions on points different from the starting point of Newton's method, which is the point usually used by other authors until now when center conditions are required. In addition, the theoretical significance of the method is used to draw conclusions about the existence and uniqueness of solutions and about the region in which they are located. As a result, we modify the domain of starting points for Newton's method.     

Solution of Volterra's population model via block‐pulse functions and Lagrange‐interpolating polynomials

A numerical method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro‐differential equation where the integral term represents the effects of toxin. The approach is based on hybrid function approximations. The properties of hybrid functions that consist of block‐pulse and Lagrange‐interpolating polynomials are presented. The associated operational matrices of integration and product are then utilized to reduce the solution of Volterra's model to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. Applications are demonstrated through an illustrative example. Copyright © 2008 John Wiley & Sons, Ltd.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Wavelet compression of integral operators arising from boudary-domain integral method

The boundary-domain integral method uses Green's functions to write integral representations of partial differential equations. Since Green's functions are non-local, the systems of linear equations arising from the discretization of integral representations are fully populated. Several algorithms have been proposed, which yield a data-sparse approximation of these systems, such as the fast multipole method, adaptive cross approximation and among others, wavelet compression. In the framework of solving the Navier-Stokes equations in velocity-vorticity form one may utilize the boundary-domain integral method for the solution of the kinematics equation to calculate the boundary vorticity values. Since the kinematics equation is a Poisson type equation, usually its integral representation is written with the Green's function for the Laplace operator. In this work, we introduce a false time into the equation and parabolize its nature. Thus, a time-dependent Green's function may be used. This provides a new parameter, the time step, which can be set to control the Green's function. The time-dependent Green's function is a global function, but by carefully choosing the time step, its behaviour is almost local. This makes it a good candidate for wavelet compression, yielding much better compression ratios at a given accuracy than when using the Green's function for the Laplace operator. However, as false time is introduced, several time steps must be solved in order to reach a converged solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Intégration de multifonctions par rapport à une mesure paramétrée

We investigate some properties of the integral, with respect to a parametrized probability, of a multifunction whose values are compact (nonconvex) subsets of a Banach space. We obtain an approximation of the closure of the Aumann's integral of a multifunction with values in a separable Banach space by the Aumann's integral of multifunctions with values in finite dimensional spaces.     

A first Integral of Navier­Stokes equations for two-dimensional flows

As wellknown, Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity. Even in case of non-vanishing vorticity, a first integral from Euler's equations is obtained by using the so called Clebsch transformation [1] for inviscid flows. In contrast to this, a generalisation of this procedure towards viscous flows has not been established so far. In the present paper a first integral of Navier-Stokes equations is constructed in the case of two-dimensional flow by making use of an alternative representation of the fields in terms of complex coordinates and introducing a potential representation for the pressure. The associated boundary conditions are also considered. The first integral is a suitable tool for the development of new analytical methods and numerical codes in fluid dynamics. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interior Lp-estimates for functions with integral representations

For fnctions having either boundary integral representations or domain integral representations the present paper proves apriori estimate of first order derivatives in the L_pnorm. An overview of integral representations and their application to partial differential equations can be found in H. Begehr's and R. P.Gilbert's two volumes [2]. Interior esitmates in the Hijlder norm for very general (elliptic) systems of partial differential equations can be found in A. Douglis' and L. Nirenberg's paper [7], while estimates in the L_p-norm are contained, for instance, in the papers I ofthe threeauthors S. Agmon, A. Douglis and L. Nirenberg. In these papers [1,7] one can find further references.     

Towards a quaternionic function theory linked with the Lamé's wave functions

Over the past few years, considerable attention has been given to the role played by the Lamé's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lamé's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non‐commutative framework. We show that the theory of the LQWFs is determined by the Moisil‐Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lamé's wave equation, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel‐Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski‐Plemelj formulae, the ‐hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs. Copyright © 2015 John Wiley & Sons, Ltd.     

A Cauchy's Integral Formula for Functions with Values in a Universal Clifford Algebra and its Applications

In this paper, we obtain Cauchy's integral formula on certain distinguished boundary for functions with values in a universal Clifford algebra, which is similar to the classical Cauchy's integral formula on the distinguished boundary of polycylinder for several complex variables. By using it, both the mean value theorem and the maximum modulus theorem are given.     

A Novel Time‐Domain BIEM for Wave Propagation Analysis in Anisotropic Solids With Cracks

Analysis of elastic wave propagation in anisotropic solids with cracks is of particular interest to quantitative non‐destructive testing and fracture mechanics. For this purpose, a novel time‐domain boundary integral equation method (BIEM) is presented in this paper. A finite crack in an unbounded elastic solid of general anisotropy subjected to transient elastic wave loading is considered. Two‐dimensional plane strain or plane stress condition is assumed. The initial‐boundary value problem is formulated as a set of hypersingular time‐domain traction boundary integral equations (BIEs) with the crack‐opening‐displacements (CODs) as unknown quantities. A time‐stepping scheme is developed for solving the hypersingular time‐domain BIEs. The scheme uses the convolution quadrature formula of Lubich [1] for temporal convolution and a Galerkin method for spatial discretization of the BIEs. An important feature of the present time‐domain BIEM is that it uses the Laplace‐domain instead of the more complicated time‐domain Green's functions. Fourier integral representations of Laplace‐domain Green's functions are applied. No special technique is needed in the present time‐domain BIEM for evaluating hypersingular integrals.     

Weak Solutions of Fluid–Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gårding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.     

Shifted rectangular quadrature rule approximations to Dawson's integral F(x)

This paper investigates the relationship between some rapidly convergent series of exponential functions for computing Dawson's integral. These series are the result of approximating a certain improper integral by a shifted rectangular quadrature rule. Dawson's original method and a more recent expansion due to Rybicki are shown to be special cases of our quadrature approach. An error bound is derived to compare the accuracy of the resulting approximations.     

A Differentiation Theory for Itô's Calculus

Abstract

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.     

A new analytical technique to solve Volterra's integral equations

The purpose of the present paper is to show that the well‐known homotopy analysis method for solving ordinary and partial differential equations can be applied to solve linear and nonlinear integral equations of Volterra's type with high accuracy as well. Comparison of the present method with Adomian decomposition method (ADM), a well‐known method to solve integral equations, reveals that the ADM is only especial case of the present method. Furthermore, some illustrating examples such as linear, nonlinear and singular integral equations of Volterra's type are given to show high efficiency with reliable accuracy of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.