An exponentially convergent algorithm for nonlinear differential equations in Banach spaces

An exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in Banach space is proposed. The algorithm is based on an equivalent Volterra integral equation including the operator exponential generated by the operator coefficient. The operator exponential is represented by a Dunford-Cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient, and then the integrals involved are approximated using the Chebyshev interpolation and an appropriate Sinc quadrature. Numerical examples are given which confirm theoretical results.

     

Be careful with the Exp-function method – Additional remarks

It is shown that the solution produced by the Exp-function method may not hold for all initial conditions. Riccati and Maccari nonlinear differential equations are used to illustrate that fact. Conditions of existence for the produced solution in the space of initial conditions and in the space of system’s parameters are derived using the operator method based on the generalized operator of differentiation. The concept of the expansion of an ordinary differential equation is introduced and it is shown that the algebraic–analytical solution of Maccari equation can be produced by solving Riccati equation.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.     

A fluctuations limit for scaled age distributions and weighted Sobolev spaces

It is shown that under the central-limit scaling, the fluctuations of the space—time renormalized age distributions of particles (whose development is controlled by critical linear birth and death processes) around the law-of-large-numbers limit converge in a Hilbert space (containing the class of signed Radon measures with finite moment generating functionals) to a continuous Gaussian process satisfying a Langevin equation. So far, the space of rapidly decreasing functions has been considered to be the natural state space for the kind of limit theorem considered here. However, the space of rapidly decreasing functions is not suitable in the present context and we are led to define an appropriate family of Sobolev spaces. In fact, we construct a scale of Hilbert spaces based on the eigenfunctions expansions of an elliptic operator defined on a weightedL ²-space.This research was partially supported by an NSERC of Canada grant.     

Reflections on the almost Mathieu operator

An explicit derivation of a tridiagonal matrix form for the almost Mathieu operator (Harper's equation) is obtained via conjugation with a reflection operator, valid for all rational values of the rotation parameter. The difference between even and odd values of the denominator is highlighted. This tridiagonal form is useful for numerical eigenvalue computations; some Matlab code is included.This research was supported in part by an NSERC individual research grant.     

Abstract Differential Equation with a Spectral Parameter in the Boundary Conditions

In the present paper, we first establish a coerciveness estimate with respect to both the space variable and the spectral parameter of a boundary-value problem for an operator differential equation with nonlocal boundary conditions containing the spectral parameter. Then, using the coerciveness estimate, we prove that the operator generated by this problem is a Fredholm, and, moreover is an isomorphism between appropriate spaces. Finally, the results obtained are applied to a class of non local problems for a quasi-elliptic partial differential equation.     

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL ² space. It is assumed that the Markov semigroup corresponding to the state process isC ₀ on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.     

On contact equivalence of Abel cubic differential equations of second order

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some nonlinear differential operator. This operator is defined by a cubic on the Cartan distribution on the space of 1-jets. Based on this observation, we construct a contact-invariant {e}-structure associated with a non-degenerate Abel equation and obtain contact classification of such equations.     

Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.     

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.     

The Fourth-order Bessel–type Differential Equation

The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane.

There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation.

When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coefficients. Embedded in this form of the equation is a spectral parameter; this combination leads to the generation of self-adjoint operators in a weighted Hilbert function space. In the second-order case one of these associated operators has an eigenfunction expansion that leads to the Hankel integral transform.

This article is devoted to a study of the spectral theory of the Bessel-type differential equation of order four; considered on the positive real axis this equation has singularities at both end-points. In the associated Hilbert function space these singular end-points are classified, the minimal and maximal operators are defined and all associated self-adjoint operators are determined, including the Friedrichs self-adjoint operator. The spectral properties of these self-adjoint operators are given in explicit form.

From the properties of the domain of the maximal operator, in the associated Hilbert function space, it is possible to obtain a virial theorem for the fourth-order Bessel-type differential equation.

There are two solutions of this fourth-order equation that can be expressed in terms of classical Bessel functions of order zero and order one. However it appears that additional, independent solutions essentially involve new special functions not yet defined. The spectral properties of the self-adjoint operators suggest that there is an eigenfunction expansion similar to the Hankel transform, but details await a further study of the solutions of the differential equation.     

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.     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.     

Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials

Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials. The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's equation that is based on approximating with harmonic polynomials locally. Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

具有脉冲的分数阶Bagley-Torvik 模型边值问题

将具有脉冲的分数阶Bagley-Torvik微分方程边值问题巧妙地转化为积分方程,定义加权Banach空间及全连续算子,运用不动点定理获得该边值问题解的存在性定理.举例说明了定理的应用.最后提出有趣的研究问题.     

Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

The divergence theorem and the Laplacian in Minkowski space

Let (X, B) be a Minkowski space (finite-dimensional Banach space) with unit ball B. Using a Minkowski definition of unit normal to a hypersurface, a Minkowski analogue of Euclidean divergence is defined. We show that the divergence theorem holds. Using the Minkowski divergence, a Minkowski Laplacian is defined. We prove that this Laplacian is a second-order, constant-coefficient, elliptic, differential operator. Furthermore, the symbol of this Laplacian is computed and used to associate a natural Euclidean structure with (X, B).Supported, in part, by NSERC Operating Grant #4066.     

An eigenvalue problem related to cell growth

In this paper, the eigenvalues of the operator corresponding to the partial differential equation which describes the evolution of a population reproducing by simple fission are investigated. This is done by transforming the eigenvalue problem to an integral equation. The theory concerning positive operators on a Banach space appears to be very useful.