A family of methods for the wave equation in one- and two-space dimensions

A family of finite-difference methods is developed for the numerical solution of the simple wave equation. Local truncation errors are calculated for each member of the family and each is analyzed for stability. The concepts of A₀ stability and L₀ stability, well used in the literature on other types of partial differential equation, are discussed in relation to second-order hyperbolic equations. The numerical methods arc extended to cover two-dimensional wave equations and the methods developed in this article are tested on three problems from the literature.     

The Navier-Stokes equations: Asymptotic solutions describing tangential discontinuities

Asymptotic solutions whose weak limit has a jump discontinuity on a two-dimensional surface are studied for the Navier-Stokes equations with small viscosity. The equations describing the leading term of the asymptotics and the correction terms are obtained and analyzed. A recursive procedure for constructing a formal asymptotic series satisfying the corresponding Cauchy problem is described. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 938–949, June, 2000.     

Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions

Materials which are heated by the passage of electricity are usually modeled by a nonlinear coupled system of two partial differential equations. The current equation is elliptic, while the temperature equation is parabolic. These equations are coupled one to another through the conductivities and the Joule effect. A computationally attractive discretization method is analyzed and shown to yield optimal error estimates in H¹. © 1993 John Wiley & Sons, Inc.     

S-Bounded distributions,application to partial differential equations

The S-boundness at infinity of a distribution is defined to give some informations on the behaviour of a large class of distributions at infinity. The subspace A′ ? D′ of S-bounded distributions has been characterized and the properties of elements of A′ have been analyzed especially those interesting for partial differential equations. At the end, some propositions, concerning the S-boundness of solutions of linear partial differential equations have been proved.     

A class of explicit multistep exponential integrators for semilinear problems

A class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided.     

Nonconforming <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed FEM for Sobolev equations on anisotropic meshes

A nonconforming H^1-Calerkin mixed finite element method is analyzed for Sobolev equations on anisotropic meshes. The error estimates are obtained without using Ritz-Volterra projection.     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Stochastic evolution equations with fractional Brownian motion

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail. Mathematics Subject Classification (2000): 60H15, 60G15     

Method of perturbations in vector electromagnetic problems of diffraction by a wedge

A vector electromagnetic problem of diffraction by a wedge-shaped region is reduced to a system of coupled functional equations by using Sommerfeld integrals. This system of functional equations is solved by the perturbation method, and the convergence of the related series is analyzed. The system of functional equations is further reduced to linear equations with contracting operators, and the solution is represented in the form of a Neumann series. Reduction to a system with compact operators is also considered. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 138–156. Translated by M. A. Lyalinov.     

Orientational phase transition in solid C60: Bifurcation approach

A simple model for the angle-dependent interaction betweenC ₆₀ molecules in the face-centered cubic lattice is proposed. The bifurcations of the solutions of the nonlinear integral equations for orientational distribution functions in the mean-field approximation are analyzed, and the orientational phase transition in solidC ₆₀ is described. The quantitative results for the orientational phase transition agree with the experimental data. Translated from Teoreticheskaya i matematicheskaya Fizika, Vol. 121, No. 3, pp. 479–491, December, 1999.     

Unconditionally stable explicit methods for parabolic equations

Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA ₀-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.     

On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.     

Geometry of Solutions of the N=2 SYM Theory in Harmonic Superspace

The classical equations of motion of the D=4, N=2 supersymmetric Yang–Mills (SYM) theory for Minkowski and Euclidean spaces are analyzed in harmonic superspace. We study dual superfield representations of equations and subsidiary conditions corresponding to classical SYM solutions with different symmetries. In particular, alternative superfield constructions of self-dual and static solutions are described in the framework of the harmonic approach.     

Multidimensional Inverse Scattering for First-Order Systems

A method for solving the inverse problem for a class of multidimensional first-order systems is given. The analysis yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data. The results are used to solve the multidimensional N-wave resonant interaction equations.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

Stability of solutions of a system of linear differential equations with periodic stochastic coefficients

Systems of equations for first and second moments are investigated and transformed. Stability of solutions of a first-order linear differential equations is analyzed. Stability of solutions of the stochastic Mathieu equation is investigated and the boundaries of the instability region are determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 119–126, 1986.     

A new technique for linear static state estimation based on weighted least absolute value approximations

This paper presents a new technique for solving the problem of linear static state estimation, based on weighted least absolute value (WLAV). A set ofm optimality equations is obtained, wherem=number of measurements, based on minimizing a WLAV performance index involvingn unknown state variables,m>n. These equations are solved using the left pseudo-inverse transformation, least-square sense, to obtain approximately the residual of each measurement.Ifk is the rank of the matrixH,k=n, we choose among the optimality equations a number of equations equal to the rankk and having the smallest residuals. The solution of thesen equations inn unknowns yields the best WLAV estimation. A numerical example is reported; the results for this example are obtained by using both WLS and WLAV techniques. It is shown that the best WLAV approximation is superior to the best WLS approximation when estimating the true form of data containing some inaccurate observations.This work was supported by the Natural Science and Engineering Research Council of Canada, Grant No. A4146.     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Exact and numerical solutions of a micropolar fluid in a vertical annulus

In this article, we have analyzed the effects of heat transfer on a peristaltic flow of a micropolar fluid in a vertical annulus. The governing equations of two‐dimensional micropolar fluid are simplified by using the assumptions of long wavelength and neglecting the wave number. A close form solutions are obtained for velocity field υ_x and microrotation component υ_θ. Further, the numerical solutions of the simplified equation of υ_θ are computed and the results are compared with the exact solution. The influence of pertinent parameters are analyzed through graphs. Trapping phenomena is also discussed for different parameters. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010