Asymptotic Methods in Equations of Mathematical Physics

By B. Vainberg. 2024

374 citations recorded by the source corpus.

Part 1 The stationary phase method: on asymptotic expansions the stationary phase method the stationary phase method, the multidimensional case the problem of waves on the surface of a liquid the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation the scattering problem the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order general partial differential equations of the first order the Hamilton-Jacobi equation example propagation of light waves in an inhomogenous medium characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula problems with rapidly oscillating initial data discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium the Lagrange maniford the precanonical operator the canonical operator, construction of a formal asymptotic solution a field in an isotropic medium with parabolic wave front more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces elliptic problems elliptic problems with a parameter inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial equations and systems of the type of the Helmholtz equation radiation conditions the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients equations with variable coefficients and problems in the interior of a bounded domain the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t introduction short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering proof of theorems 1 and 2. Part contents...

Subfields: Applied Mathematics, Mechanical Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/6311473c7c614ad29b5c0e6b94deec428219113f

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Citations in the corpus

Global Strichartz Estimates for Nontrapping Geometries: About an Article by H. Smith and C. Sogge (2002, background): “5 was proven for |λ| >> 1 in greater generalities by Lax and Phillips [6], Melrose and Sjöstrand [7, 8]), Vainberg[13], Tang and Zworski [11] (see also [3] for a self contained proof which joined with the results in [1] would relax the smoothness assumption required in these papers to a C2 assumptio”
Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media (2008, methodology): “Theorem 3. ([58, 59 ], see also [13] for a brief description of this result.)”