Numerical solutions of index-1 differential algebraic equations can be computed in polynomial time

The cost of solving an initial value problem for index-1 differential algebraic equations to accuracy ɛ is polynomial in ln(1/ɛ). This cost is obtained for an algorithm based on the Taylor series method for solving differential algebraic equations developed by Pryce. This result extends a recent result by Corless for solutions of ordinary differential equations. The results of the standard theory of information-based complexity give exponential cost for solving ordinary differential equations, being based on a different model.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing

A new, simple algorithm of order 2 is presented to approximate weakly stochastic differential equations. It is then applied to the problem of pricing Asian options under the Heston stochastic volatility model.

2000 Mathematics Subject Classification, 65C30, 65C05.     

An Implementation for the Algorithm of Janet bases of Linear Differential Ideals in the Maple System

In this paper, an algorithm for computing the Janet bases of linear differential equations is described, which is the differential analogue of the algorithm JanetBasis improved by Gerdt. An implementation of the algorithm in Maple is given. The implemented algorithm includes some subalgorithms: Janet division,Pommaret division, the judgement of involutive divisor and reducible, the judgement of conventional divisor and reducible, involutive normal form and conventional normal form, involutive autoreduction and conventional autoreduction, PJ-autoreduction and so on. As an application, the Janet Bases of the determining system of classical Lie symmetries of some partial differential equations are obtained using our package.     

Interaction of weak discontinuities and the hodograph method as applied to electric field fractionation of a two-component mixture

The hodograph method is used to construct a solution describing the interaction of weak discontinuities (rarefaction waves) for the problem of mass transfer by an electric field (zonal electrophoresis). Mathematically, the problem is reduced to the study of a system of two first-order quasilinear hyperbolic partial differential equations with data on characteristics (Goursat problem). The solution is constructed analytically in the form of implicit relations. An efficient numerical algorithm is described that reduces the system of quasilinear partial differential equations to ordinary differential equations. For the zonal electrophoresis equations, the Riemann problem with initial discontinuities specified at two different spatial points is completely solved.     

A Richardson-type iterative approach for identification of delamination boundaries

A direct problem of applied mathematical modelling is to determine the response of a system given the governing partial differential equations, the geometry of interest, the complete boundary and initial conditions, and material properties. When one or more of the conditions for the solution of the direct problem are unknown, an inverse problem can be formulated. One of the methods frequently used for the solution of inverse problems involves finding the values of the unknowns in a mathematical formulation such that the behavior calculated with the model matches the measured response to a degree evaluated in terms of the classical L ₂ norm. Considered in this sense, the inverse problem is equivalent to an ill-posed optimization problem for the estimation of parameters whose solution in the majority of cases is a real mathematical challenge. In this contribution, we report a novel approach that avoids the mathematical difficulties inspired by the ill-posed character of the model. Our method is devoted to the computation of inverse problems furnished by second-order elliptical systems of partial differential equations and falls in the same conceptual line with the method initiated by Kozlov et al. and further extended and algorithmized by Weikl et al. We construct and employ a weak version of the algorithm found by Weikl et al. Proofs for the convergence and regularity of this version are given for the case of a single layer. The computational realization of the algorithm (called briefly AICRA) is applied and numerical results are obtained. Comparison with experiments demonstrates a good significance and representativeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 209–230, 2006.     

An initial value theory for linear causal boundary value problems

A new formalism in the theory of linear boundary value problems involving causal functional differential equations is presented. The approach depends on the construction of a differentiable family of boundary problems into which the original boundary value problem is imbedded. The formalism then generates an initial value problem which is equivalent to the family of imbedded problems. An important aspect of the method is that the equations in the initial value algorithm are ordinary differential equations rather than functional differential equations, although nonlinear and of higher dimension. Applications of the theory to differential-delay and difference equations are given.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

Compact gradient tracking in shape optimization

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that, in contrast to the situation of gradient tracking on the whole domain, the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the Cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed by means of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient algorithm to solve the problem under consideration.     

A boundary-value problem in an unbounded strip

Using the generalized scheme of the method of separation of variables we propose a new approach to the study of a boundary-value problem in an unbounded strip for hyperbolic partial differential equations of even order with respect to time. In the space of entire functions of exponential type (with respect to the spatial variable) we single out a unique solution class for the problem and exhibit an algorithm for constructing a solution of it. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 16–21.     

Optimal impulsive control of compartment models,II: Algorithm

The optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. Under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of an ordinary finite-dimensional convex programming problem. An iterative algorithm is given for the situation in which the critical points cannot all be determineda priori.This work was supported in part by the National Science Foundation under Grant No. MPS-74-13332.     

Transformation of an axialsymmetric disk problem for the helmholtz equation into an ordinary differential equation

The problem of solving the three-dimensional Helmholtz equation in the exterior of a circular disk is considered where radially symmetric Dirichlet data on the disk are assumed to be prescribed. This problem for example arises in the scattering of plane (sound) waves at an infinite plane screen with a circular aperture if the direction of the incident wave is normal to the screen, as well as in the process of diffusion through a circular hole. By applying the factorization technique developed in [N. GORENFLO, M. WERNER,Solution of a finite convolution equation with a Hankel kernel by matrix factorization, SIAM J. Math. Anal., 28 (1997), pp. 434–451] to the disk problem an equivalent ordinary differential equation is derived, whose solution leads directly to the solution of the disk problem. This differential equation belongs to a class of ordinary differential equations which are of higher complexity than the standard ordinary differential equations of mathematical physics. The examination of this new class of differential equations therefore is motivated.     

LQG量测反馈最优控制的精细积分

对于线性二次型高斯(LQG)量测反馈最优控制问题,提出了精细积分解法。根据分离性原理,LQG控制问题可以分成为最优状态反馈控制问题以及最优状态估计问题,即:离线计算的两套黎卡提微分方程的求解以及状态向量的时变微分方程的在线积分解。该算法不仅适用于求解二点边值问题及其相应的黎卡提微分方程,也适用于求解状态估计的时变微分方程。精细积分高精度的特点,对控制和估计都是有利的。数值算例表明了算法的高精度及有效性。     

Completely integrable nonlinear dynamical systems of the Langmuir chains type

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Padé approximations, and on the Sturm-Liouville method for finite difference equations is used. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 588–602, October, 1997. Translated by A. M. Chebotarev     

Numerical analysis of the dispersion of electroelastic waves in a cylinder whose physical properties have a longitudinal axis of symmetry

We develop a numerical method of analyzing the dispersion of electroelastic waves in a hollow circular cylinder. The method is based on the reduction of the wave problem to a spectral problem for a system of ordinary differential equations. This system is solved numerically. The roots of the dispersion equation are determined by the method of bisection. The practical convergence of the algorithm is shown using a specific example. One figure. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 151–156.     

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.     

<Emphasis Type="Italic">r</Emphasis>-Tuple almost product structures

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampère equations and Jacobi equations.     

Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lamé equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lamé equations (a version of the inverse scattering method).     

The method of lines for nonlinear parabolic differential equations with mixed derivatives

Summary By the so-called longitudinal method of lines the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper, for a wide class of nonlinear parabolic differential equations the spatial derivatives occuring in the original problem are replaced by suitable differences such that monotonicity methods become applicable. A convergence theorem is proved. Special interest is devoted to the equationu _t=f(x,t,u,u _x,u _xx), if the matrix of first order derivatives off(x,t,z,p,r) with respect tor may be estimated by a suitable Minkowski matrix.