Differential Equations In Space Research Articles

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

Numerical methods for nonlinear second-order hyperbolic partial differential equations. II – Rothe’s techniques for 1-D problems

Two families of Rothe’s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second-order hyperbolic equations with damping and nonlinear source terms are presented. The first family is based on time integration and results in a Volterra integro-differential equation which, upon approximating certain time integrals, can be written as an ordinary differential equation in space. Upon discretizing the spatial derivatives by means of finite difference formula, this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step, and its accuracy is shown to be lower than that of an explicit second-order accurate discretization in both space and time of second-order hyperbolic equations. The second family of methods is based on the discretization of the time derivatives and time linearization, and results in linear ordinary differential equations in space. Upon freezing the coefficients of these equations, one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space, and three-point finite difference equations which depend exponentially on the time step, spatial grid size, and the diffusion, relaxation, damping and reaction times. The finite difference equations are shown to result in non-diagonally dominant matrices unless the time step is smaller than the characteristic relaxation, diffusion, damping and reaction times. To avoid this problem, two Rothe’s techniques that do not account for the Jacobian of the reaction terms in the ordinary differential operator are developed, and it is shown that the Rothe’s techniques belonging to the second family are as accurate as the linearly implicit methods presented in Part I provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space. When the relaxation time is greater than the critical one, Rothe’s methods have been found to be less accurate than the finite difference techniques of Part I. The Rothe’s techniques belonging to the second family are generalized to systems of nonlinear second-order hyperbolic equations and mixed systems of parabolic and second-order hyperbolic partial differential equations.

Immersed boundary method: The existence of approximate solution of the two-dimensional heat equation

This paper deals with the heat equation in which the source term involves a Dirac function and describes the elastic reaction of the immersed boundary. We analyze the existence of the approximate solution in two-dimensional case with Dirac function approximated by differentiable function. We obtain the result via finite element method, the Banach Fixed-Point Theorem and a theorem in nonlinear ordinary differential equations in abstract space.

Mathematical analysis of thermal diffusion shock waves

Thermal diffusion, also known as the Ludwig-Soret effect, refers to the separation of mixtures in a temperature gradient. For a binary mixture the time dependence of the change in concentration of each species is governed by a nonlinear partial differential equation in space and time. Here, an exact solution of the Ludwig-Soret equation without mass diffusion for a sinusoidal temperature field is given. The solution shows that counterpropagating shock waves are produced which slow and eventually come to a halt. Expressions are found for the shock time for two limiting values of the starting density fraction. The effects of diffusion on the development of the concentration profile in time and space are found by numerical integration of the nonlinear differential equation.

Modelling and simulation of railway cable systems

Mathematical models and numerical methods for the computation of both static equilibria and dynamic oscillations of railroad catenaries are derived and analyzed. These cable systems form a complex network of string and beam elements and lead to coupled partial differential equations in space and time where constraints and corresponding Lagrange multipliers express the interaction between carrier, contact wire, and pantograph head. For computing static equlibria, three different algorithms are presented and compared, while the dynamic case is treated by a finite element method in space, combined with stabilized time integration of the resulting differential algebraic system. Simulation examples based on reference data from industry illustrate the potential of such computational tools.

Thermal Diffusion Shock Waves

The Ludwig-Soret effect or thermal diffusion, which refers to the separation of liquid mixtures in a temperature gradient, is governed by a nonlinear, partial differential equation in space and time. It is shown here that the solution to the nonlinear differential equation for a binary mixture predicts the existence of shock waves completely analogous to fluid shocks and obeys an expression for the shock velocity that is an exact analogue of the Rankine-Hugoniot relations. Direct measurements of the time dependent, spatial absorption profile of a suspension of nanometer sized particles subjected to a sinusoidal temperature field generated by a pair of continuous laser beams, as well as self-diffraction experiments, show motion of the particles in agreement with the predictions of nonlinear theory.

Differential equations in spaces of compact operators

We study a system of differential equations in C ( H ) , the space of all compact operators on a separable complex Hilbert space, H . The systems considered are infinite-dimensional generalizations of mathematical models of learning, implementable as artificial neural networks. In this new setting, in addition to the usual questions of existence and uniqueness of solutions, we discuss issues which are operator theoretic in nature. Under some restrictions on the initial condition, we explicitly solve the system and represent the solution in terms of the spectral representation of the initial condition. We also discuss the stability of those solutions, and describe the weak, strong, and uniform limit sets in terms of their respective spectral properties.

Numerical Treatment of Catenary Systems

AbstractSimulation is of great importance in the development of railway cable systems. Mathematical models and numerical methods for the computation of both static equilibria and dynamic oscillations of railroad catenaries are being derived. These cable systems form a complex network of string and beam elements and lead to coupled partial differential equations in space and time where constraints and corresponding Lagrange multipliers express the interaction between carrier, contact wire, and pantograph head. For computing static equilibria, two different algorithms are presented, while the dynamic case is treated by a finite element method in space, combined with stabilized time integration of the resulting differential algebraic system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Surfactant-driven motion and splitting of droplets on a substrate

A theoretical and computational model is presented to predict the motion of a small sessile liquid droplet, lying on a solid substrate including surfactant effects. The model, as formulated, consists of coupled partial differential equations in space and time, and several auxilliary relationships. The validity of the long-wave, or ‘lubrication’ approximation is assumed. It is shown that there are circumstances where surfactant injection or production will cause the droplet to split into two daughter droplets. It is conjectured that the results are relevant to basic mechanisms involved in biological cell division (cytokinesis). It is also demonstrated that motion of a droplet, analogous to the motility of a cell, can be produced by surfactant addition. Computed examples are given here, in both two and three space dimensions. Approximate energy requirements are also calculated for these processes. These are found to be suitably small.

Approximation of Abstract Differential Equations

This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces. Most of the finite difference, finite element, and projection methods can be considered from the point of view of general approximation schemes (see, e.g., [207,210,211] for such a representation). Results obtained for general approximation schemes make the formulation of concrete numerical methods easier and give an overview of methods which are suitable for different classes of problems. The qualitative theory of differential equations in Banach spaces is presented in many brilliant papers and books. We can refer to the bibliography [218], which contains about 3000 references. Unfortunately, no books or reviews on general approximation theory appear for differential equations in abstract spaces during last 20 years. Any information on the subject can be found in the original papers only. It seems that such a review is the first step towards describing a complete picture of discretization methods for abstract differential equations in Banach spaces. In Sec. 2 we describe the general approximation scheme, different types of convergence of operators, and the relation between the convergence and the approximation of spectra. Also, such a convergence analysis can be used if one considers elliptic problems, i.e., the problems which do not depend on time. Section 3 contains a complete picture of the theory of discretization of semigroups on Banach spaces. It summarizes Trotter–Kato and Lax–Richtmyer theorems from the general and common point of view and related problems. The approximation of ill-posed problems is considered in Sec. 4, which is based on the theory of approximation of local C-semigroups. Since the backward Cauchy problem is very important in applications and admits a stochastic noise, we also consider approximation using a stochastic regularization. Such an approach was never considered in the literature before to the best of our knowledge. In Sec. 5, we present discrete coercive inequalities for abstract parabolic equations in Cτn([0, T ];En), C τn([0, T ];En), L p τn([0, T ];En), and Bτn([0, T ];C (Ωh)) spaces.

Differential Equations in Banach Spaces II. Theory of Cosine Operator Functions

More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should be the first part of a large work devoted to abstract differential equations and methods for solving them. However, the troubles being in the Russian science during the whole this period have influenced also on the authors, and instead of two years supposed, the preparation of the second part has occupied considerably more time. During the last 10 years, the work in the field of differential equations in abstract spaces was very active (in foreign countries), and every year several books and a heavy number of papers devoted to this direction appear in the world (of course, the most of them are not available for the Russian reader). At the same time, only two books [33, 75] of such a type appeared being translated by the authors of the present survey and [20], which were edited by Yu. A. Daletskii. Therefore, the work whose second part is proposed to the reader will be undoubtedly useful for the Russian reader. Its style coincides with that of [16], i.e., the material is often presented without proofs, and the main attention is paid to the structure of presentation, although we present certain proofs from foreign sources that are almost inaccessible for Russian readers. From our viewpoint, this allows us to demonstrate clearly the philosophy, to describe the results obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey.

Nonlinear Volterra difference equations in space lp

We consider a class of vector nonlinear discrete‐time Volterra equations in space lp and derive estimates for the norms of solutions. These estimates give us explicit stability conditions, which allow us to avoid finding Lyapunov functionals.

Differential equations in spaces of Hilbert space valued distributions

A Gaussian measure is introduced on the space of Hilbert space valued tempered distributions. It is used to define a Hilbert space valued Q-Wiener process and a white noise process with a nuclear covariance operator Q. The proposed construction is used for solving operator-differential equations with additive noise with the operator coefficient generating an n-times integrated exponentially bounded semigroup.

Stochastic differential equations on product loop manifolds

We study infinite systems of stochastic differential equations in spaces of loops with values in compact Riemannian manifolds. We prove existence of solutions with deterministic initial conditions and continuity of the corresponding paths.

Stochastic Reduced Basis Methods

Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.

Differential equations in spaces of abstract stochastic distributions

We develop the theory of stochastic distributions with values in a separable Hilbert space, and apply this theory to the investigation of abstract stochastic evolution equations with additive noise.

Differential Equations in Spaces with Asymptotics on Manifolds with Cusp Singularities

Differential Equations in Spaces with Asymptotics on Manifolds with Cusp Singularities

Constitutive equations and reciprocity

The differential equations that must be satisfied for most fields (physical variables) in geophysics are determined primarily by conservation equations which relate the divergence of the flux of the field, the field's time rate of change, and its sources and sinks. These conservation (or equilibrium) equations do not provide sufficient constraints to determine the fluxes and fields even when boundary conditions (both in space and time) are specified. To constrain the fields completely, it is necessary to introduce the properties of the media; that is, the constitutive equations. Because the conservation equations can be determined without considering the properties of the media, these equations are valid for the most general media; that is, heterogeneous, anisotropic, time-varying, non-linear, etc. media in which many field variables can interact or be coupled. When the fields can be described by ‘self-adjoint’ differential equations in space–time, these media exhibit reciprocity; that is, upon interchange of ‘sources’ and ‘detectors’, the same result is obtained. We show that viscoelastic, elastodynamic problems relating to generalized Kelvin–Voigt and generalized Maxwell media satisfy the conditions for reciprocity. In addition, we show that the introduction of tensor ‘densities’ (which relate the inertial-force density to the particle-acceleration, particle-velocity and particle-displacement vectors in the inertial force's constitutive equation) do not invalidate the reciprocity conditions. The two constitutive equations (the stress/strain and the inertial-force ones) lead to dispersion and attenuation in the propagation of the fields even though none of the material constants in the constitutive or conservation equations is complex (i.e. with real and imaginary parts). Complex material properties cannot exist in nature for actual materials or media; nor can the material constants or properties be functions of frequency. However, ‘apparent’ or ‘equivalent’ properties may be complex and functions of frequency if time-harmonic fields are assumed to exist.

Parameter dependence of solutions of partial differential equations in spaces of real analytic functions

LetΩ⊆Rn\Omega \subseteq \mathbb {R}^nbe an open set and letA(Ω)\mathrm {A}(\Omega )denote the class of real analytic functions onΩ\Omega. It is proved that for every surjective linear partial differential operatorP(D,x):A(Ω)→A(Ω)P(D,x):\mathrm {A}(\Omega )\to \mathrm {A}(\Omega )and every family(fλ)⊆A(Ω)(f_ \lambda )\subseteq \mathrm {A}(\Omega )depending holomorphically onλ∈Cm\lambda \in \mathbb {C}^mthere is a solution family(uλ)⊆A(Ω)(u_ \lambda )\subseteq \mathrm {A}(\Omega )depending onλ\lambdain the same way such thatP(D,x)uλ=fλ,forλ∈Cm.\begin{equation*} P(D,x)u_ \lambda =f_\lambda , \qquad \mathrm {for } \lambda \in \mathbb {C}^m. \end{equation*}The result is a consequence of a characterization of Fréchet spacesEEsuch that the class of “weakly” real analyticEE-valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid ifCm\mathbb {C}^mis replaced by another set.

Wave–field reciprocity and optimization in remote sensing

A unified approach to local optimization techniques and wave–field reciprocity as applied to constructing solutions to remote sensing, imaging and inversion problems in acoustic, elastic and electromagnetic wave theory is presented. The starting point is a system of linear, first–order partial differential equations in space–time of the class to which the indicated wave phenomena gives rise. For this system, three types of remote sensing problems — the inverse–source, the inverse–scattering, and the inverse–transducted–wave–field problems — are formulated, and the construction of their solutions via local optimization techniques is discussed. Emphasis is placed on iterative algorithms that are based on a guaranteed decrease in the mismatch between modelled and observed data at each update of the medium. Subsequently, the wave–field reciprocity theorems of the time–convolution and the time–correlation types are derived and their occurrence in the optimization procedures is discussed. Also, attention is paid to approximate methods, in particular to the Rayleigh–Gans–Born approximation. Approximations of this sort provide the means to invoke the method of preconditioning in the process of inverting the operator equations. ‘Exotic media’ (for example, chiral media in electromagnetics) are included in the analysis.