Integration Of Differential Systems Research Articles

Lp theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces

We study the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in a separable Banach space X. We derive conditions in terms of the Hausdorff measure of noncompactness under which mild solutions exist in L p (0, b; X). For illustration, a partial integral differential system is worked out.

Analytic normalization of analytic integrable systems and the embedding flows

This paper provides the normal forms of analytic integrable differential systems and diffeomorphisms via analytic normalizations. Furthermore, we consider the existence of embedding flows of an analytic integrable diffeomorphism.

The characteristic Chern type classes and integrability of multidimensional differential systems on Riemannian manifolds

AbstractThe differential‐geometric properties of generalized de Rham‐Hodge complexes naturally related with integrable multidimensional differential systems of M. Gromov type are analyzed. The geometric structure of Chern type characteristic classes are studied, special differential invariants of the Chern type are constructed. The integrability of multi‐dimensional nonlinear differential systems on Riemannian manifolds is discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a class of quartic surfaces and an associated integrable differential system

Given a unicursal quartic in a plane and four bi-tangential straight lines, there exist a pair of conics four times tangential to it. Choosing an arbitrary point K0 outside the plane, the lines and either one of the conics determine four planes and a cone, and a quartic surface can be found that stays tangential to them and presents 15 conic point singularities, including K0. We introduce a one-parameter family of symmetric tensors which are polynomial functions of degree 7 of the Cartesian coordinates. Each of them determines two families of curves on the surface, which are the null curves of the associated metric and are the integral curves of a differential system which turns out to possess the Painlevé property and to be integrable by quadratures. Differential systems of that form have been found (Gaffet 2006 J. Phys. A: Math. Gen. 39 99) to represent the rotating motion with precession of gas clouds expanding into a vacuum.

Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters

In this work, we develop a systematic and self-consistent approach to homogenize arbitrary nonmagnetic periodic metamaterials. The proposed method does not rely on the solution of an eigenvalue problem and can fully characterize the effects of frequency dispersion, magnetoelectric coupling, and spatial dispersion, even in frequency band gaps or when the materials are lossy. We formulate a homogenization problem to characterize a generic microstructured artificial material, and demonstrate that it is equivalent to an integral-differential system. We prove that this complex system can be reduced to a standard integral equation and solved using standard methods. To illustrate the application of the proposed method, we homogenize several important metamaterial configurations involving split-ring resonators and metallic wires.

Generalized de Rham-Hodge complexes, the related characteristic Chern classes, and some applications to integrable multidimensional differential systems on Riemannian manifolds

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed.

THE BIANCHI–DARBOUX TRANSFORM OF L-ISOTHERMIC SURFACES

We study an analogue of the classical Bäcklund transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi–Darboux transformation. First we show how to construct the Bianchi–Darboux transforms of an L-isothermic surface by solving an integrable linear differential system, then we establish a permutability theorem for iterated Bianchi–Darboux transforms.

Diagonally implicit multistage integration methods for pseudospectral solutions of the wave equation

Diagonally implicit multistage integration methods are employed for the numerical integration in time of first order hyperbolic systems arising from Chebyshev pseudospectral discretizations of the spatial derivatives in the wave equation. These methods have stage order q equal to the order p. The stage values can be utilized to recover approximations to the solution u of sufficiently high accuracy. The phenomenon of order reduction, which is present in the integration of differential systems by numerical methods of low stage order, such as explicit Runge–Kutta methods, is avoided.

A class of singularly perturbed reaction diffusion integral differential system

In this paper a class of singularly perturbed initial boundary value problems for the reaction diffusion integral differential, system are considered. Using the iteration, method and the differential inequalities, the existence, uniqueness and its asymptotic behavior of solution for the initial boundary value problems are studied.

New generating relations for the multivariate biorthogonal Wilson polynomials

The application of the elementary manipulation of series together with hypergeometric formulation is used to deduce a number of new multidimensional generating relations for the biorthogonal Wilson polynomials of several variables. These polynomials occur in connection with integrable difference systems.

Algebraic Methods for Creep Analysis of Continuous Composite Beams

This paper proposes a simplified time-dependent analysis for continuous steel-concrete composite beams with flexible shear connection, subjected to static actions, settlements of the supports, shrinkage, and prestressing of the concrete slab. By approximating the integral terms with particular quadrature rules (algebraic methods), the integral-differential system, which governs the problem, reduces to a differential system. This provides approximate solutions of the viscoelastic problem directly at the final time, without having to consider the real stress history. Some of the algebraic methods known in the literature (e.g., effective modulus, mean stress, and age adjusted effective modulus methods) are applied to continuous composite beams subjected to the four different actions examined separately. Subsequently, the numerical results obtained are compared with the results of the accurate numerical analysis (general method), and the validity of each simplified method is discussed. Finally, some informati...

Multivariable continuous Hahn and Wilson polynomials related to integrable difference systems

Multivariable generalizations of the continuous Hahn and Wilson polynomials are introduced as eigenfunctions of rational Ruijsenaars type difference systems with an external field.

A numerical study of thermal effects on nonlinear bubble oscillations

An integral-differential system was derived by Miksis and Ting describing the nonlinear oscillations of a gas bubble including thermal and viscous effects. Here, an efficient numerical scheme is presented to solve this system for long times relative to the forcing period. Examples are presented and compared with the Rayleigh–Plesset equation, which has no thermal damping. Numerical results show that the initial free oscillations are damped out much faster when thermal damping is included. Also shown is the fact that it is possible for a bubble to have its mean radius grow slowly with time. In this case for large time the bubble will approach a new constant mean radius, which is larger than the initial equilibrium radius. For the special case of small amplitude forcing, asymptotic periodic solutions for very large time are constructed and a method to systematically derive the higher-order terms is demonstrated.

Local structure of a completely integrable differential system with singularities

Local structure of a completely integrable differential system with singularities

On infinite prolongations of differential systems

From any exterior differential system with independent variables, it will be shown how to construct, on an infinite dimensional space, an equivalent completely integrable differential system. Local solutions are discussed in the real or complex analytic case. The calculus of exterior differential systems will be used freely [1; 2]. Prolongations are clearly described in [3]. All functions and forms are assumed infinitely differentiable unless otherwise restricted.