Asymptotic solutions of the linear shallow-water equations with localized initial data

A new asymptotic method for solving Cauchy problems with localized initial data (perturbation) for the linearized shallow-water equation is suggested. The solution is decomposed into two parts: waves and vortices. Metamorphosis of the profile takes place for the wave part: it is localized in the neighborhood of the initial point and later on it is localized in the neighborhood of the front (1-D curve on the plane). Initially, the front is a smooth curve, but as the time increases turning (focal) and self-intersection points might appear. The vortical part is localized in the neighborhood of a point moving along the trajectory of the basic velocity vector field. Both parts are described by very simple formulae taking into account the form of the initial perturbation. These formulae are expressed by means of elementary functions for some special choice of the initial data. Due to the profile-metamorphosis phenomenon the method leading to the final formulae is not elementary. It makes use of semiclassical approximations and ray expansions. It is based on the generalization of the construction known as the Maslov canonical operator and boundary-layer expansions.     

Errors arising from irregular boundaries in ADI solutions of the shallow-water equations

Truncation errors introduced by operator splittingtechniques, such as the ADI scheme, are discussed in the contextof the two-dimensional shallow-water equations. Numericalexamples of flow along a uniform channel, with and withoutfriction, illustrate the practical significance of these errors when the boundary is not aligned with the computational gridaxes. With irregular boundaries the maximum time step aceptable ismore severely restricted than has been realized hitherto. The problem, which is shown to stem from the treatment of the boundary conditons, is inherent in any splitting scheme and is difficult to circumvent.     

Note on two-dimensional instability in shallow-water flows

Summary Linear temporal stability of a two-dimensional flow of shallow water is investigated in the framework of the Boussinesq equations, in which both the bottom friction and internal lateral friction are included. If internal friction is strong, it does not enter into the stability criterion explicity. However, it provides the large wavenumber cut-off that is necessary to meet the long-wave assumption of the shallow-water approximation. As internal friction diminishes, both the temporal growth rate and wavenumber of the most amplified disturbance increase. For weak internal friction, the flow can be unstable even if the long-wave disturbances are damped.     

Approximate angle factors for two-dimensional problems

We have approximated the special Ki_n(x) functions used in formulas for radiation characteristics. We have found the approximate intermediate functions for an isotropic incident flow by means of which [2] many angle factors have been expressed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.16, No. 3, pp. 499–503, March, 1969.     

Dispersive nonlinear shallow-water equations: some preliminary numerical results

The dispersive nonlinear shallow-water equations of Antuono et al. (Stud Appl Math 122:1–28, 2008) are solved by means of an explicit arbitrary high-order accurate finite-volume scheme for nonlinear hyperbolic systems with stiff source terms. Tests against typical benchmark solutions are used to illustrate the robustness and accuracy of the solver while typical solutions for the propagation of solitary waves on a slope highlight the solution value in reproducing nearshore flows.     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

Riemann invariants-like solutions for a class of rate-type materials

Summary An approach is developed to generate exact solutions to a hyperbolic model describing a class of rate-type materials. The leading idea of the analysis which is carried on herein is to require the model in point to be consistent with a pair of additional equations defining Riemann invariants-like quantities along the characteristic curves. Hence classes of model constitutive laws allowing a reduction procedure to hold are characterized.     

Approximate solutions to a nonlinear diffusion equation

Approximate similarity solutions to the porous-medium equation, c ₁ = · (c ^m c), are obtained in one and two dimensions. The problems considered arise in the modelling of dopant diffusion in semiconductors, the two-dimentional problems corresponding to diffusion under a mask edge.     

Approximate analytical solutions of a nonlinear diffusion equation

Approximate analytical solutions of a nonlinear diffusion equation $\frac{{\partial c}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {D(c)\frac{{\partial c}}{{\partial x}}} \right)$ are obtained in the practically important case of constant boundary conditions corresponding to the diffusion in a homogeneously doped half-space at a zero surface concentration for D(c) = ac, ac ², and a√c (a > 0). The error of approximation for these D(c) dependences in the concentration interval (1–2) × 10^?3 < c < (0.92–0.99) does not exceed 1–2%.     

Approximate determination of conductivity in two-dimensional (plane) figures

Justification and examples are provided for the utilization of a method to evaluate the conductivity in two-dimensional figures of complex shape. Two methods are proposed for the derivation of approximate conductivity formulas.     

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.     

Diagnostic equations for two-dimensional vortical flows

Summary A straightforward derivation shows that in two-dimensional vortical flows without the effect of unsteady irrotational straining there is a set of three diagnostic equations (i.e. not involving time derivatives) associated with the prognostic one for vorticity transport. These equations provide — apart from the primary diagnostic significance — a specific (topological, dynamical, geometrical) meaning treated in detail.     

General solutions of the coupled diffusion equations

New finite integral transform and the corresponding infinite series are introduced, which brings the solution of the coupled diffusion equations within the realm of integral transform theory. The formulation of this transform leads to an eigenvalue problem which is not of the conventional Sturm-Liuoville type and therefore a special integral condition was derived which serves as an orthogonality relation. The solution obtained can be applied when studying diffusion in a tubular reactor, heat transfer by a turbulenty flowing fluid-solids mixture in a pipe.     

Approximate solution of the equations of heat-exchanger dynamics on a digital computer

A modification of the method of lines is proposed for the approximate solution of the system of partial differential equations describing the thermal dynamics of a multistage forced-circulation heat exchanger (HE).Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.21, No.6, pp. 1053–1059, December, 1971.     

Approximate stress intensity factors compounded from known solutions

A versatile method for determining approximate stress intensity factors is presented. The accuracy of the method is assessed using three widely differing configurations for which alternative solutions are available. Finally an approximate solution is then obtained to a configuration for which no alternative solution exists.     

Approximate equations for the flexure of thin,incomplete, piezoelectric bimorphs

Summary In this paper the linear, three-dimensional, piezoelectric equations for a body in equilibrium are reduced to approximate, two-dimensional ones, treating the flexure of thin bimorphs, partly coated by electrodes (incomplete bimorphs). For that purpose two-dimensional equations are derived for piezoelectric plates and for bimorphs with completely coated faces. An assumption about the charge distribution on the inner electrode is given, stating that the charge vanishes on those parts where the outer faces are free of electrodes. This assumption allows the application of the mentioned, approximate equations for plates and bimorphs to the parts of incomplete bimorphs. By stating edge and continuity conditions, the approximate theory ls completed. The solution for a circular, incomplete, piezoceramic bimorph, loaded by a singular force in the centre, is given and compared with experimental results.     

梯度功能压电复合材料球壳的热弹性近似解

针对梯度功能压电复合材料球壳热弹性响应提出了简单而精确的近似解。借鉴纤维复合材料层合结构的研究方法，将梯度功能压电复合材料球壳沿径向分为若干层，各层视为均匀材料，从而导出力-电-温度多场耦合近似解。只要层数足够大，解将收敛于精确解。该方法的另一个优点在于解的方法对材料性能的变化方式（函数）没有要求，具有普适性。这对其它非均匀材料结构的分析具有推广、应用价值。     

Approximate solution of singular integro-differential equations in elastic contact problems

A method for the numerical solution of singular integro-differential equations is proposed. The approximate solution is sought in the form of the sum of a power series with unknown coefficients multiplied by a special term which controls the appropriate solution behaviour near and at the edges of the interval. The coefficients are to be determined from a system of linear algebraic equations. The method is applied to the solution of a contact problem of a disk inserted in an infinite elastic plane. Exact analytical solution is obtained for the particular case when the disk is of the same material as the plane. Comparison is made between the exact and the approximate solutions as well as with the solutions previously available in literature. The stability and the accuracy of the present method is investigated under variation of the parameters involved. The applicability of the method to the case when the boundary conditions for the unknown function are nonzero is discussed along with an illustrative example. A FORTRAN subroutine for the numerical solution of singular integro-differential equations is also provided.