Equations Of Infinite Order Research Articles

On the Generalized Cauchy Problem for One Class of Differential Equations of Infinite Order

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.

Нелокальна за часом задача для еволюційних сингулярних рівнянь нескінченного порядку

Нелокальна за часом задача для еволюційних сингулярних рівнянь нескінченного порядку

On Fredholm Solvability of the Dirichet Problem for Linear Differential Equations of Infinite Order

We propose a new approach to investigation of solvability of the Dirichlet problem for differential equations of infinite order. Namely, by using the embedding theorems obtained by the author in previous papers for the energy spaces, corresponding to operators of infinite order, the initial differential operator of infinite order is expressed as a sum of the main and subordinate operators of infinite order. The conditions, under which the above Dirichlet problems are soluble, are established by using the main term of the corresponding differential operator.

Intensification of nonlinearity by dispersed events in media with structure and generation of various dominant frequencies in P- and S-waves

A model of a continuum with a structure described by infinite order equations of motion is proposed. In case that a wave is very long as compared with the size of the structure, equations are reduced to the fourth-order equations. A closed equation of motion, including nonlinear, dispersed and wave members, is derived. It is shown that solutions in the form of soliton waves exist only in media where wave velocity grows with pressure. In the media, where soliton waves do not exist, quasi-stationary solutions with multiple frequencies prevail. It is found that the nonlinear effect of multiple frequencies is unexpectedly high even for small deformation as dispersion violently intensifies nonlinear events. Moreover, in the domain of small deformation, there exist solutions for longitudinal and transversal waves with the same length and different frequencies. The solutions for the same length waves with different frequencies most often occur in seismology and seismic explorations.

Nonlocal Problem Multipoint in Time for a Class of Partial Differential Equations of Infinite Order

We establish the correct solvability of a nonlocal problem multipoint in time for the evolutionary equation with differential operator of infinite order.

Linear and nonlinear abstract differential equations of high order

Abstract The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established. In application, the separability and spectral properties of nonlocal boundary value problem for the system of degenerate differential equations of infinite order is derived.

Electromagnetic Wave Scattering by Parallel Dielectric/Metamaterial Coated Elliptic Cylinders

An exact solution to the boundary value problem of multiple scattering by M parallel two-layer elliptic cylinders, based on the separation of variable technique and the addition theorem for Mathieu function is presented. It is expressed in terms of a system of simultaneous linear equations of infinite order. Numerical results for the special case of two parallel two-layer cylinders are presented and discussed for several geometrical cases and material parameters

Solvability of convolution equations in the Beurling spaces of ultradifferentiable functions of mean type on an interval

We establish a solvability criterion for convolution equations in the Beurling classes of ultradifferentiable functions of mean type on an interval. Under study is also the question of degeneration of convolution equations into infinite-order equations with constant coefficients.

Analytic Continuation of Taylor-Dirichlet Series and Non-Vanishing Solutions of a Differential Equation of Infinite Order

In the spirit of the Fabry-Polya Gap Theorems on the singularities of power series, we investigate similar phenomena for Taylor-Dirichlet series $$\sum\limits_{n = 1}^\infty {\left( {\sum\limits_{j = 0}^{\mu _n - 1} {c_n j^{z^j } } } \right)e^{\lambda _n z} }$$ associated to the positive real multiplicity-sequence Λ = {λn, μn}n=1∞. Assume that Λ has positive finite density d, counting multiplicities, and that Λ belongs to a certain class that we denote by U(d, 0). Let $$F(z) = \prod\limits_{n = 1}^\infty {\left( {1 - \frac{{z^2 }} {{\lambda _n ^2 }}} \right)^{\mu _n } } and g\left( {z,w} \right) = \frac{{\exp (wz)}} {{F(w)}}.$$ Then the series $$\sum\limits_{n = 1}^\infty {\left( {\sum\limits_{j = 0}^{\mu _n - 1} {c_{n,j} z^j } } \right)e^{\lambda _n z} = \sum {\operatorname{Re} s g} } \left( {z,\lambda _n } \right),$$ defines an analytic function in the open left half-plane ℂ_. This function cannot be extended analytically across any open interval lying on the imaginary axis having length greater than 2πd. Nevertheless, this series can be extended analytically as an even function to the open right half-plane ℂ+ across the open segment (—iπd, iπd). Applications are given to a differential equation of infinite order. Let D = d/dx and L(z) = zF(z). The equation L(D)f(x) = 0 with the boundary condition limx→+ -∞ f(x) = 0 has non-vanishing solutions. Extensions are given with D = d/dz.

Generation of nonlinear oscillations at weak perturbations and generalization of cracks at fracture

The paper studies properties of a continuum with structure. The characteristic size of the structure governs the fact that difference relations do not automatically transform into differential ones. It is impossible to distinguish an infinitesimal volume of a body to which we could apply the major conservation laws, because the minimum representative volume of the body should contain at least a few elementary microstructures. This gives motion equations of infinite order. Solutions for such equations include, along with sound waves, perturbations propagating with abnormally low velocities not bounded below. It is shown that in such media weak perturbations can increase or decrease without limit. Weak dispersion of structure sizes reduces the intensity of such increase and therefore stabilizes the medium, whereas unlimited growth of dispersion destabilizes it. The nonlinearity of the stress-strain curve is shown to cause a reduction of the specific surface of a cracked medium, i.e. the medium undergoes fracture with the formation of a system of a small number of large cracks as though it ignores a large number of small cracks.

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

A distributed control problem for n × n parabolic coupled systems involving operators with infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii-Milyutin theorem, the necessary and sufficient conditions of optimality are derived for the Dirichlet problem. Yet, the problem considered here is more general than the problems in El-Saify & Bahaa (2002, Optimal control for n × n hyperbolic systems involving operators of infinite order. Math. Slovaca, 52, 409-424), El-Zahaby (2002, Optimal control of systems governed by infinite order operators. Proceeding (Abstracts) of the International Conference of Mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28-31 December 2002. J. Egypt. Math. Soc. (submitted)), Gali & El-Saify (1983, Control of system governed by infinite order equation of hyperbolic type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 99-103), Gali et al. (1983, Distributed control of a system governed by Dirichlet and Neumann problems for elliptic equations of infinite order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 83-87) and Kotarski et al. (200b, Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order. Int. J. Pure Appl. Math., 1, 241-254).

Numerical studies of time-independent and time-dependent scattering by several elliptical cylinders

A numerical solution to the problem of time-dependent scattering by an array of elliptical cylinders with parallel axes is presented. The solution is an exact one, based on the separation-of-variables technique in the elliptical coordinate system, the addition theorem for Mathieu functions, and numerical integration. Time-independent solutions are described by a system of linear equations of infinite order which are truncated for numerical computations. Time-dependent solutions are obtained by numerical integration involving a large number of these solutions. First results of a software package generating these solutions are presented: wave propagation around three impenetrable elliptical scatterers. As far as we know, this method described has never been used for time-dependent multiple scattering.

Resonant Scattering From a Slit Coupled Cylindrically Eccentric Impedance Cavity: the Case of H-Polarized Plane Wave Incidence

The resonant scattering characteristics of a slit coupled cylindrically eccentric impedance cavity are examined using direct integration equation technique, and the incident wave is assumed to be a H-polarized plane wave. Such a slit coupled eccentric impedance cavity is assumed to be infinite in length and of infinitely thin wall surrounded by several impedance screens. By using the Galerkin's method combined with translational addition theorem for cylindrical wave functions, a system of linear equations of infinite order for determining the induced current distributions are derived in an analytical form. Therefore, numerical results are presented to show the effects of cavity geometrical and impedance parameters on the resonant scattering characteristics of some typical structures in the far zone.

Scattering of a plane wave by an infinite grating of circular dielectric cylinders at oblique incidence: E-polarization

The boundary value solution to the problem of scattering by an infinite grating of circular dielectric cylinders is treated for an obliquely incident vertically polarized plane electromagnetic wave. The generalized representations of the scattered electric and magnetic fields as well as the internal fields within the dielectric circular cylinders are acquired in terms of special functions and series. The solution is an exact one and based on the separation-of-variables technique and the addition theorem for cylindrical waves. The scattering coefficients for the electric and magnetic fields are expressed in terms of two systems of simultaneous linear equations of infinite order in coupled form. It has been demonstrated that the solutions to the infinitely long dielectric cylinder at oblique incidence, and to the infinite grating of circular dielectric cylinders at non-oblique incidence can be obtained from the most generalized solution presented here.

Multiple scattering of an E-polarisation plane wave from multiple cylindrical strips

Multiple scattering of an E-polarisation plane wave from multiple cylindrical strips is investigated using the technique of the direct integral equation. The induced surface currents on multiple perfectly conducting strips are expanded in the form of a series of Chebyshev polynomials of the first kind. By using Galerkin's procedure, all the unknown expanding coefficients are determined by a set of matrix equations of infinite order, and then numerical results are presented to show the characteristics of the copolarised far-zone field changing with the angular width and number of strips as well as their relative orientations.

The three-dimensional problem for an anisotropic plate of arbitrary thickness

By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external forces.

Exact effective-mass theory for heterostructures.

An exact effective-mass differential equation is derived for electrons in heterostructures. This equation is exactly equivalent to the Schr\"odinger equation, and is obtained by applying a k-space transformation of variables to the Burt envelope-function theory in which the Brillouin zone is mapped onto the infinite real axis. The mapping eliminates all nonlocal effects and long-range Gibbs oscillations in the Burt theory, producing an infinite-order differential equation in which interface effects are strongly localized to the immediate vicinity of the interface. A general procedure is given for obtaining finite-order boundary conditions from the infinite-order equation; the second-order theory reduces to the BenDaniel-Duke model with a \ensuremath{\delta}-function potential at the interface. The derivation is presented for a simple one-dimensional crystal but can easily be generalized for more complex situations.

Multiple scattering by parallel conducting elliptic cylinders: TE case

A solution to the boundary value problem of transverse electric (TE) multiple scattering by M parallel perfectly conducting elliptic cylinders is presented. The solution is an exact one and is based on the separation of variables technique and the addition theorem for Mathieu functions. It is expressed in terms of a system of simultaneous linear equations of infinite order, which is then truncated for numerical computations. Representative numerical results for the forwards and backscattered fields by two cylinders are then generated, for some selected sizes and orientations parameters, and presented.

The Cauchy problem for a system of two partial differential equations of infinite order

Starting from the generalized scheme of separation of variables, we propose a new effective method of constructing the solution of the Cauchy problem for a system of two partial differential equations, in general of infinite order with respect to the spatial variable. We consider the example of the Cauchy problem for the system of Lame equations in the case of a two-dimensional strain.

Boundary value problem for nonlinear parabolic equations of infinite order in Sobolev-Orlicz spaces

Boundary value problem for nonlinear parabolic equations of infinite order in Sobolev-Orlicz spaces