Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows us to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is not interpreted as the triviality of the right-hand side of an equation. It is understood in the sense that the left-hand side of an equation is a homogeneous function with respect to operators appearing in that equation. Solutions are represented as functional vector-valued series which are uniformly convergent and generated by solutions to a kth order ordinary differential equation, by the roots of the characteristic polynomial and by elements of a locally convex space. We find sufficient conditions for the continuous dependence of the solution on a generating set. We also solve the Cauchy problem for the considered equations and specify conditions for the existence and the uniqueness of the solution. Moreover, under certain hypotheses we find the general solution to the considered equations. It is a function which yields any particular solution. The investigation is realized by means of characteristics of operators such as the order and the type of an operator, as well as operator characteristics of vectors, namely, the operator order and the operator type of a vector relative to an operator. We also use a convergence of operator series with respect to an equicontinuous bornology.     

Periodic solutions of Quasilinear Hyperbolic integro-differential equations of second order

We study a periodic boundary-value problem for a quasilinear integro-differential equation with the d’Alembert operator on the left-hand side and a nonlinear integral operator on the right-hand side. We establish conditions under which the uniqueness theorems are true.     

An analog of the method of continuation of solution with respect to the parameter for nonlinear operator equations

A process of second order is constructed for the solution of nonlinear operator equations which is an analog of the method of continuation of solution with respect to the parameter. For each value of the parameter the Newton-Kantorovich iteration formula is applied only once in all. The quadratic convergence of the process is ensured by the specification of the parameter by a special formula. The process under consideration enables us to avoid the singular points of the derivative of the nonlinear operator on the left-hand side of the operator equation.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 601–606, April, 1978.     

On a contact problem for a viscoelastic von Karman plate and its semidiscretization

We deal with the system describing moderately large deflections of thin viscoelastic plates with an inner obstacle. In the case of a long memory the system consists of an integro-differential 4th order variational inequality for the deflection and an equation with a biharmonic left-hand side and an integro-differential right-hand side for the Airy stress function. The existence of a solution in a special case of the Dirichlet-Prony series is verified by transforming the problem into a sequence of stationary variational inequalities of von Karman type. We derive conditions for applying the Banach fixed point theorem enabling us to solve the biharmonic variational inequalities for each time step.     

Periodic solutions of second-order quasilinear hyperbolic equations

We study a periodic boundary-value problem for a quasilinear equation with the d'Alembert operator on the left-hand side and a nonlinear operator on the right-hand side and establish conditions under which the solution of the indicated problem is unique.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1370–1375, October, 1995.     

Vector-Valued Functions Generated by the Operator of Finite Order and Their Application to Solving Operator Equations in Locally Convex Spaces

This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics, the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.     

Ordinary p-Biharmonic Problems

The aim of this paper is to recall known results on boundary value problems for the quasilinear fourth order differential equation in one dimension (1) where p>1. The operator at the left-hand side is often called a p-biharmonic operator which reduces to u⁽⁴⁾ for p = 2. It is a fourth order analogue of the well-known p-Laplacian. We discuss spectral properties of the corresponding eigenvalue problems, and existence and global bifurcation of solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalization of the Lagrange Method to the Case of Second-Order Linear Differential Equations with Constant Operator Coefficients in Locally Convex Spaces

The well-known Lagrange method for linear inhomogeneous differential equations is generalized to the case of second-order equations with constant operator coefficients in locally convex spaces. The solutions are expressed in terms of uniformly convergent functional vector-valued series generated by a pair of elements of a locally convex space. Sufficient conditions for the continuous dependence of solutions on the generating pair are obtained. The solution of the Cauchy problem for the equations under consideration is also obtained and conditions for its existence and uniqueness are given. In addition, under certain conditions, the so-called general solution of the equations (a function of most general form from which any particular solution can be derived) is obtained. The study is carried out using the characteristics (order and type) of an operator and of a sequence of operators. Also, the convergence of operator series with respect to equicontinuous bornology is used.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

Divided powers and composition in integro-differential algebras

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.     

On the regularization of projection methods for solving III-posed problems

Summary In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.     

Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.     

Dynamical Systems Gradient Method for Solving Ill-Conditioned Linear Algebraic Systems

A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An a priori and a posteriori stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decomposition of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization.     

Further bounds for hardy type differences

In this paper we define a functional as a difference between the right-hand side and left-hand side of the refined Boas type inequality using the notation of superquadratic and subquadratic functions and study its properties, such as exponential and logarithmic convexity. We also, state and prove improvements and reverses of new weighted Boas type inequalities. As a special case of our result we obtain improvements and reverses of the Hardy inequality and its dual inequality. We introduce new Cauchy type mean and prove monotonicity property of this mean.     

Spectral multipliers for the Ornstein-Uhlenbeck semigroup

The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent. The four authors have received support from the European Commission via the TMR network “Harmonic Analysis”. The first and last authors were also partially supported by the Spanish DGICYT, under grant PB97-0030.     

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a “complicated” Vekua equation from positive formal powers and a generating sequence of a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.     

FPGA Realization of Fractional Order Neuron

In this paper fractional Hindmarsh Rose (HR) neuron, which mimics several behaviors of a real biological neuron is implemented on field programmable gate array (FPGA). The results show several differences in the dynamic characteristics of integer and fractional order Hindmarsh Rose neuron models. The integer order model shows only one type of firing characteristics when the parameters of model remains same. The fractional order model depicts several dynamical behaviors even for the same parameters as the order of the fractional operator is varied. The firing frequency increases when the order of the fractional operator decreases. The fractional order is therefore key in determining the firing characteristics of biological neurons. To implement this neuron model first the digital realization of different fractional operator approximations are obtained, then the fractional integrator is used to obtain the low power and low cost hardware realization of fractional HR neuron. The fractional neuron model has been implemented on a low voltage and low power circuit and then compared with its integer counter part. The hardware is used to demonstrate the different dynamical behaviors of fractional HR neuron for different type of approximations obtained for fractional operator in this paper. A coupled network of fractional order HR neurons is also implemented. The results also show that synchronization between neurons increases as long as coupling factor keeps on increasing.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Logarithmic Terms in Trace Expansions of Atiyah–Patodi–Singer Problems

For Dirac-type operator D on a manifold X with a spectral boundarycondition (defined by a pseudodifferential projection), the associated heatoperator trace has an expansion in integer and half-integer powers and log-powersof t; the interest in the expansion coefficients goes back to the work of Atiyah,Patodi and Singer. In the product case considered by APS, it is known that allthe log-coefficients vanish when dim X is odd, whereas the log-coefficients atinteger powers vanish when dim X is even. We investigate here whether this partialvanishing of logarithms holds more generally. One type of result, shown forgeneral D with well-posed boundary conditions, is that a perturbation of Dby a tangential differential operator vanishing to order k on the boundaryleaves the first k log-power terms invariant (and the nonlocal power termsof the same degree are only locally perturbed). Another type of result is thatfor perturbations of the APS product case by tangential operators commuting withthe tangential part of D, all the logarithmic terms vanish when dim X is odd(whereas they can all be expected to be nonzero when dim X is even). The treatmentis based on earlier joint work with R. Seeley and a recent systematic parameter-dependentpseudodifferential boundary operator calculus, applied to the resolvent.     

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.