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Recent trends in formal and analytic solutions of diff. equations : Virtual Conference Formal and Analytic Solutions of Diff. Equations, June 28-July 2, 2021, University of Alcalá, Alcalá de Henares, Spain
This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.
eBook
On some aspects of oscillation theory and geometry
The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations
on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation we prove some new results in
both directions, ranging from oscillation and nonoscillation conditions for ODE’s that improve on classical criteria, to estimates in
the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications.
To keep our investigation basically self-contained we also collect some, more or less known, material which often appears in the
literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.
eBook
Geometric Optics for Surface Waves in Nonlinear Elasticity
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as \"the amplitude equation\", is an integrodifferential equation of nonlocal Burgers type. The authors begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^{\\varepsilon} $ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength $\\varepsilon $, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^{\\varepsilon}$ on a time interval independent of $\\varepsilon $. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.
eBook
Recent Trends in Formal and Analytic Solutions of Diff. Equations
This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcalá, Alcalá de Henares, Spain.The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, q-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions.The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.
eBook
Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations
We represent general solution to a homogeneous linear difference equation of second order in terms of a specially chosen solution to the equation and apply it to get a representation of general solution to the bilinear difference equation in terms of a solution to an associate difference equation of second order, considerably generalizing some recent results in an elegant way. We also present the corresponding representations for some systems of bilinear difference equations. Many historical notes not so known to wide audience are also presented, and we offer an answer to an open question regarding the attribution of the bilinear difference equation.
Journal Article