Showing papers in "Mathematical Methods in The Applied Sciences in 1984"
TL;DR: In this article, a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small is studied and convergence proof is carried out.
Abstract: This paper is devoted to the study of a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small. In the three studied limits (e 0 then ω 0), (ω 0 then e 0) and lately (e and ω 0 together) the three dimensional equation of elasticity are approached by the two dimensional general equations of a linear anisotropic plate, the stretching and bending being coupled. This study points out the importance of the ratio of the two small parameters, indeed the moduli occuring in the two dimensional equations are different in the three limits. In each case a convergence proof is carried out.
332 citations
TL;DR: The existence of weak LL-solutions of the initial value problem for Vlasov's equation is proved under rather general assumptions as mentioned in this paper, and the solutions exist on the entire time axis.
Abstract: The existence of weak LL-solutions of the initial value problem for Vlasov's equation is proved under rather general assumptions. In the three-dimensional case the solutions exist on the entire time axis.
125 citations
TL;DR: In this article, the boundary integral method is applied to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics, which are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order -1 which provide certain coercivity properties.
Abstract: Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L2. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.
94 citations
TL;DR: In this paper, the S-Matrix was constructed using the Rayleigh-Bloch wave expansion and the reduced Grating Propagator (GRP) for the Scattered Fields problem.
Abstract: 1. Physical Theory.- 1. The Physical Problem.- 2. The Mathematical Formulation.- 3. Solution of the Initial-Boundary Value Problem.- 4. The Reference Problem and Its Eigenfunctions.- 5. Rayleigh-Bloch Diffracted Plane Waves for Gratings.- 6. Rayleigh-Bloch Surface Waves for Gratings.- 7. Rayleigh-Bloch Wave Expansions.- 8. Wave and Scattering Operators for Gratings.- 9. Asymptotic Wave Functions for Gratings.- 10. The Scattering of Signals from Remote Sources.- 2. Mathematical Theory.- 1. Grating Domains and Grating Propagators.- 2. Rayleigh-Bloch Waves.- 3. The Reduced Grating Propagator Ap.- 4. Analytic Continuation of the Resolvent of Ap.- 5. Proofs of the Results of 4.- 6. The Eigenfunction Expansion for Ap.- 7. Proofs of the Results of 6.- 8. The Rayleigh-Bloch Wave Expansions for A.- 9. Proofs of the Results of 8.- 10. The Initial-Boundary Value Problems for the Scattered Fields.- 11. Construction of the Wave Operators for AP and Ao,p.- 12. Construction of the Wave Operators for A and Ao.- 13. Asymptotic Wave Functions and Energy Distributions.- 14. Construction and Structure of the S-Matrix.- 15. The Scattering of Signals by Diffraction Gratings.- References.
87 citations
TL;DR: In this paper, a modification of the Vlasov-Poisson equation, obtained by adding a diffusion term with respect to velocity, was studied, which describes a plasma in thermal equilibrium, in a mean field limit situation.
Abstract: We study a modification of the Vlasov-Poisson equation, obtained by adding a diffusion term with respect to velocity. It describes, from a physical point of view, a plasma in thermal equilibrium, in a mean field limit situation. We find that the already known results concerning existence and uniqueness of the solutions for the ordinary Vlasov equation (constructive results and counterexamples) translate to our case.
29 citations
TL;DR: In this article, the authors discuss the asymptotic behavior of acoustic and electromagnetic waves generated by given time-harmonic exterior forces with frequency ω, in the unbounded region between the parallel planes X3 = 0 and X3= 1, and show that the principle of limiting amplitude is violated if ω = πn(n = 1, 2, 3, 4, 5, 6, 7).
Abstract: We discuss the asymptotic behaviour of acoustic and electromagnetic waves, generated by given time-harmonic exterior forces with frequency ω, in the unbounded region between the parallel planes X3 = 0 and X3 = 1, and show that the principle of limiting amplitude is violated if ω = πn(n = 1, 2,…). For these values of the frequency, forces with compact support can be chosen such that the amplitudes of the waves increase with a logarithmic rate as t ∞.
25 citations
TL;DR: In this article, it was shown that existence of the exact solution implies existence of Faedo-Galerkin approximations to the solution of the evolution equation u' = Au + Nuu(0) = ϕ======
Abstract: We consider the evolution equation u' = Au + Nuu(0) = ϕ
u is a function on a real interval [0, T] with values in a Hilbert space H, A is a linear operator in H generating a continuous semigroup eAt and N is a nonlinear operator in H.
We show that
1
Existence of the exact solution implies existence of the Faedo-Galerkin Approximations.
2
Existence of the Faedo-Galerkin Approximations implies existence of the exact solution.
3
Uniform convergence of the Faedo-Galerkin Approximations to the exact solution.
The Paper consists of two parts. In the first five sections we require that A possesses a complete orthonormal system of eigenfunctions, in section 6 we drop this requirement.
20 citations
TL;DR: In this article, Sobolev considered un problem de valeur limite-initiale for un ecoulement de gaz visqueux, conducteur de la chaleur dans un domaine limite D. On the other hand, in this paper, we assume that the frontiere S de D consiste en de two surfaces disjointes S 1 et S 2 de classe C 2 and that le gaz penetre D par la surface S 1 and en sort par the surface S 2.
Abstract: On considere un probleme de valeur limite-initiale pour un ecoulement de gaz visqueux, conducteur de la chaleur dans un domaine limite D. On suppose que la frontiere S de D consiste en deux surfaces disjointes S 1 et S 2 de classe C 2 et que le gaz penetre D par la surface S 1 et en sort par la surface S 2 . Preuve de l'existence (localement dans le temps) d'une solution du probleme dans les espaces anisotropes de Sobolev
20 citations
TL;DR: In this article, the existence and unicity of a solution of the Maxwell-Norton quasi-static problem were investigated and the convergence of the discrete solutions to the initial problem was proved.
Abstract: In this paper, we deal with the existence and unicity of a solution of the Maxwell-Norton quasi-static problem. Some fundamental results concerning the functional spaces are first recalled. Then, by an implicit scheme, we obtain a discretized problem which is well-posed. By monotonicity techniques, we finally prove the convergence of the discrete solutions to the solution of the initial problem.
20 citations
18 citations
TL;DR: In this article, it was proved that the Carleman equations possess a solution on the time interval on which a smooth solution of the fluid-like equation exists, which gives a justification of the Chapman-Enskog procedure as an asymptotic expansion method.
Abstract: The Chapman-Enskog procedure is applied to the Carleman model of the Boltzmann equation. It has been proved that the Carleman equations possess a solution on the time interval on which a smooth solution of the fluid-like equation exists. The calculations have been performed up to the first order i.e., to the Navier-Stokes-like equation. It has been shown that in this case a difference between an exact solution and the Chapman-Enskog solution is of order ϵ2. Extension of the results to higher orders is also possible. This gives a justification of the Chapman-Enskog procedure as an asymptotic expansion method.
TL;DR: In this article, a qualitative description of the free boundary of an annular domain in the x-y plane is given, involving inflection points and local maxima and minima relative to specified directions.
Abstract: In the x-y plane, let Ω be an annular domain whose interior boundary Γ* is a known Jordan curve and whose exterior boundary Γ is a free boundary characterized by the condition that|ΔU| = 1 on Γ, where U is the capacity potential in Ω. We obtain a qualitative description of Γ involving such things as its inflection points and local maxima and minima relative to specified directions. We also extend these results to obtain qualitative properties of the free boundary when it is subjected to various geometric contraints. Our results generalize those in [1].
TL;DR: In this article, the existence and uniqueness of solution for a class of singularly perturbed problems for differential equations are discussed by using classical results of Poincare and Birkhoff.
Abstract: By using classical results of Poincare and Birkhoff we discuss the existence and uniqueness of solution for a class of singularly perturbed problems for differential equations. The Tau method formulation of Ortiz [6] is applied to the construction of approximate solutions of these problems. Sharp error bounds are deduced.
These error bounds are applied to the discussions of a model problem, a simple one-dimensional analogue of Navier-Stokes equation, which has been considered recently by several authors (see [2], [3], [8], [10]). Numerical results for this problem [8] show that the Tau method leads to more accurate approximations than specially designed finite difference or finite element schemes.
TL;DR: In this paper, the authors consider the initial-boundary value problem for the backward heat equation assuming that some error has been made in characterizing the geometry of the domain under consideration and show that solutions which belong to an appropriately defined constraint set depend continuously in L 2 on errors in the geometry.
Abstract: In this paper we are concerned with the development of criteria for stabilizing inherently unstable initial-boundary value problems under small errors in the geometry of the underlying domain. We consider in particular the initial-boundary-value problem for the backward heat equation assuming that some error has been made in characterizing the geometry of the domain under consideration. It is shown that solutions which belong to an appropriately defined constraint set depend continuously in L2 on errors in the geometry.
TL;DR: On demontre la validite de l'equation de Boltzmann, localement en temps, en montrant que sa solution est une limite des solutions de la hierarchie BBGKY as discussed by the authors.
Abstract: On demontre la validite de l'equation de Boltzmann, localement en temps, en montrant que sa solution est une limite des solutions de la hierarchie BBGKY
TL;DR: In this article, a free boundary problem for a flow around a circle is analyzed, and it is shown that bifurcations from the trivial flow actually take place in Golubitsky-Schaeffer theory, together with a formula concerning variations of domains.
Abstract: A free boundary problem for a flow around a circle is analyzed. We find and mathematically prove that bifurcations from the trivial flow actually take place. Golubitsky-Schaeffer theory, together with a formula concerning variations of domains, enables us to clarify the behaviors of the branches of nontrivial solutions.
TL;DR: In this article, the existence and uniqueness of the solution of the transfer of polarized light in a homogeneous semi-infinite or finite plane-parallel medium were proved for a general LL-space formulation, where 1 ≤ p < ∞.
Abstract: On neglecting reflection by the surface the existence and uniqueness are proved for the solution of the equation of transfer of polarized light in a homogeneous semi-infinite or finite plane-parallel medium. A general LL-space formulation, where 1 ≤ p < ∞, is adopted. The analysis concerns a vector-valued convolution equation, which is an equivalent form of the equation of radiative transfer and is solved with the help of Wiener-Hopf factorization, Fredholm index and cone preservation methods. The results are also proved for the equations obtained from the full equation of transfer by means of Fourier expansion and symmetry relations.
TL;DR: In this article, the authors considered the linearized problem for the ideal fluid flow induced by the horizontal motion of a fully immersed body, where the system of equations is made up of an elliptic problem (P) and an initial-value problem (R) which are coupled by a pseudo-differential operator T. They defined a regularized Cauchy problem using the Yosida approximation of T; they gave energy and wave resistance estimates and finally they obtained existence uniqueness and regularity of the weak solution of R by taking the limit when ϵ goes to zero.
Abstract: We consider the linearized problem for the ideal fluid flow induced by the horizontal motion of a fully immersed body. The system of equations is made up of an elliptic problem (P) and an initialvalue problem (R) which are coupled by a pseudo-differential operator T.
We define a regularized Cauchy problem (Rϵ) using the Yosida approximation of T; we give energy and wave resistance estimates and finally we obtain existence uniqueness and regularity of the weak solution of (R) by taking the limit when ϵ goes to zero.
TL;DR: In this article, a transmutation method for equations of the form x2 φ + x2(k2) - q(x)) φ = (v2 - (1/4)) π, with v as spectral variable, correspond to problems in quantum scattering theory at fixed energy k2 (here v ˜ l + (1 2) with l complex angular momentum).
Abstract: Transmutation methods are developed for equations of the form x2 φ“ + x2(k2” - q(x)) φ = (v2 - (1/4)) φ, with v as spectral variable, which correspond to problems in quantum scattering theory at fixed energy k2 (here v ˜ l + (1/2) with l complex angular momentum). Spectral formulas for transmutation kernels are constructed and the machinery of transmutation theory developed by the author for spectral variable k is shown to have a version here. General Kontrorovic-Lebedev theorems are also proved.
TL;DR: In this paper, a string fixed at both ends A and B, can oscillate in a plane in which there is a fixed point obstacle, placed in the middle of the line AB.
Abstract: A string fixed at both ends A and B, can oscillate in a plane in which there is a fixed point obstacle, placed in the middle of the line AB. The string is initially at rest with a prescribed shape, symmetric with respect to the normal mid-plane of the segment A B. Using results established before, we find new periodic motions.
TL;DR: In this article, the authors propose a methode de prolongement standard for calculing the branche solution, which is defined as a function of the matrices carree jacobienne d'ordre (n+1) associee.
Abstract: Soit F une application de R n+1 dans R n , x(t) une paramerisation, pour |t−t 0 |
TL;DR: In this paper, a generalized Wiener-Hopf-type integral equation is derived by solving the transmission problem for the upper and the lower half-plane involving a Neumann condition at y = 0.
Abstract: The plane transmission problem of the Helmholtz equation for quadrants is characterized by a one-dimensional singular integral equation, which refers to the Fourier transform of the normal derivative of the solution along the x-axis. It is derived by solving the transmission problem for the upper and the lower half-plane involving a Neumann condition at y = 0. This is done by a two-dimensional Laplace transform technique. The inverse Laplace transform with respect to the second cartesian coordinate and the restriction of this one to y = 0 then lead to the integral equation. Thereby the transmission conditions of the original problem at y = 0 have to be taken into account. The resulting integral equation is of generalized Wiener-Hopf-type. It is solved via the contraction theorem imposing restricting conditions on the wave numbers.
TL;DR: In this article, a linear equation for a particle steady-state transport process in a homogeneous slab of finite thickness with boundary conditions of general type is derived, which differs from the well-known integral equation for no-reentry boundary conditions because of the presence of a per-turbance linear operator which describes the effect of the reemission of the particles incident at the wall.
Abstract: A linear equation for a particle steady-state transport process in a homogeneous slab of finite thickness with boundary conditions of general type is derived. This equation differs from the well-known integral equation for no-reentry boundary conditions because of the presence of a per-turbance linear operator which describes the effect of the re-emission of the particles incident at the wall. The properties of the resulting operator are investigated. The dependence of the first positive eigenvalue on physical parameters is studies in detail. The results obtained are enough to discuss the existence of a critical strictly positive solution of the physical problems which motivated this research.
TL;DR: In this article, an integral equation method for the exterior Robin problem for the Helmholtz equation where the boundary condition is interpreted in the L 2-sense is presented, which is uniquely solvable for all wave numbers.
Abstract: In this paper we study an integral equation method for the exterior Robin problem for the Helmholtz equation where the boundary condition is interpreted in the L2-sense. In particular, we derive a composite integral equation from Green's theorem which is uniquely solvable for all wave numbers.
TL;DR: On traite des equations d'onde a non linearites scalaires and demontre la relation entre la theorie de la bifurcation d'un systeme associe d'equations integrales de Hammerstein and l'existence de solutions periodiques de l'equation d'where non lineaire as mentioned in this paper.
Abstract: On traite des equations d'onde a non linearites scalaires et on demontre la relation entre la theorie de la bifurcation d'un systeme associe d'equations integrales de Hammerstein et l'existence de solutions periodiques de l'equation d'onde non lineaire
TL;DR: In this paper, the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners was proved.
Abstract: The aim of this paper is to prove the existence and uniqueness of local solutions of some initial boundary value problems for the Euler equations of an incompressible fluid in a bounded domain Ω ⊂ R2 with corners. We consider two cases of a nonvanishing normal component of velocity on the boundary. In three-dimensional case such problems have been considered in papers [12], [13], [14]. Similar problems in domains without corners have been considered in [2]–[6], [11]. In this paper the relation between the maximal corner angle of the boundary and the smoothness of the solutions is shown. The paper consists of four sections. In section 1 two initial boundary value problems for the Euler equations are formulated. In section 2 the existence and uniqueness of solutions of the Laplace equation in twodimensional domain with corners for the Dirichlet and Neumann problems is proved in the Sobolev spaces. In sections 3 and 4 we prove the existence and uniqueness of solutions of problems formulated in section 1, using the method of successive approximations.
TL;DR: In this paper, an integral operator is defined, which allows to solve the equation by Neumann series for sufficiently large k > 0, and the kernel is constructed by a modification of the WKB-method.
Abstract: An integral operator is defined, which allows to solve the equation
by Neumann series for sufficiently large k > 0. The kernel is constructed by a modification of the WKB-method. This kernel is so simple that the operator can be used effectively for numerical calculations. Numerical results are discussed.
TL;DR: In this article, the authors discuss the problems of contact roulant in termes d'inegalite variationnelle avec a certain operateur pseudodifferentiel.
Abstract: Formulation du probleme de contact roulant en termes d'inegalite variationnelle avec certain operateur pseudodifferentiel. Discussion d'une technique d'elements finis pour sa resolution
TL;DR: In this paper, the delta function initial condition solution v*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function for a sufficiently rapidly increasing and unbounded positive sequence {yy}.
Abstract: The delta function initial condition solution v*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function for a sufficiently rapidly increasing and unbounded positive sequence {yy}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.
TL;DR: Analyse mathematique de l'equation de Fokker-Planck simplifiee de physique statistique des plasmas. Pour le probleme de Cauchy, une methode d'approximation constructive est introduite.
Abstract: Analyse mathematique de l'equation de Fokker-Planck simplifiee de physique statistique des plasmas. Pour le probleme de Cauchy, une methode d'approximation constructive est introduite. Ce procede fournit une serie (f n ) de densites approchees, qui converge vers la solution globale classique du probleme avec une vitesse de convergence lineaire