Quadrature-iteration method and quadrature-splitting method for Volterra integral equations of the second kind

We construct and analyze for convergence a quadrature-iteration method for Volterra integral equations of the second kind and a quadrature-splitting method for linear equations. The iteration processes producing an approximate solution are accelerated, because the integral operator is approximated by a quadrature operator with an arbitrarily small residual operator.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 11–16, 1992;     

A contractive operator solution of an airfoil design inverse problem

This paper deals with a numerical method for an airfoil design which was presented in [3, 5]. This method is intended for design of an airfoil from a given velocity distribution along a mean camber line. The method is based on searching for a fixed point of a contractive operator. This operator combines an inexact inverse operator and equations describing the flow. A subsonic flow is assumed, the flow is described by a system of the Euler equations which is solved by an implicit finite volume method. The Newton method is applied to the solution of the nonlinear system. The resulting system of linear algebraic equations is solved by GMRES method, the Jacobian-free version is described. The inexact inverse operator consists of a middle curve function and a thickness function, both depending on the given velocity distribution. In addition to the velocity distribution the velocity in infinity is given. The angle of attack is determined so that the stagnation point is in a specific position. Successful numerical results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 443–452, April, 2008.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

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.     

Recursion and group structures of soliton equations

The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered.     

Numerical conformal mapping based on the generalised conjugation operator

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators.

     

The dressing method and separation of variables: The two-dimensional case

Separation of variables is very convenient for obtaining solutions of linear differential equations in explicit form. We use the dressing method to widen the class of such equations. As an example, we dress a two-dimensional linear differential operator, including an operator with constant coefficients.     

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r₀) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako , where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt , Mennicken and Naboko . In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27–75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations $G(u)=0$ , where $G:H \rightarrow K'$ , is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski’s method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newton’s type methods where derivatives are approximated by divided differences.     

Banach空间一类H-增生算子的混合拟变分包含的邻近算子方程

本文在Banach空间中引入一类H-增生算子的混合拟变分包含,并提出求该变分包含问题解的邻近点法.通过H-增生算子的预解算子技术,建立了混合拟变分包含问题与邻近算子方程的等价关系,由这个等价关系得到求解邻近算子方程的迭代算法,该算法收敛于上述混合拟变分包含问题的解.     

A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry

ABSTRACT

This paper presents a novel variational method for treating three-dimensional rotational Navier-Stokes equations in flow channel of turbomachines. The proposed method establishes a new semi-geodesic coordinate system on the central surface of blades. From the perspective of differential geometry, the system under concern is split into a set of membrane operator equations on two-dimensional manifolds and bending operator equations along hub circle. The third variable of the new coordinate system is approximated by the central difference scheme. We derive a new formulation of two-dimensional Navier-Stokes equations with three components on the manifolds in the variational sense. The well-posedness of the proposed variational formulation is rigorously justified.     

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.     

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.