Parametric splines on a hyperbolic paraboloid

A hyperbolic paraboloid over a tetrahedron, constructed in B–B algebraic reduced form with its barycentric coordinate system, can be conveniently represented by two parameters. An arc on the surface, obtained by determining a type of function relation about the two parameters, has multiformity and consistent endpoint properties. We analyze the equivalence and boundedness of an arc’s curvature, and give a process of the proof. These arcs can be connected into an approximate G²$G^2$-continuity space curve for fitting to a sequence of points with their advantages, and the curves, connected by this type arcs, are quite different from other algebraic and parametric splines.     

Inequalities on generalized trigonometric and hyperbolic functions

In this paper, we prove some inequalities involving the generalized trigonometric and hyperbolic functions.     

Spline collocation method for linear singular hyperbolic systems

Some classes of singular systems of partial differential equations with variable matrix coefficients and internal hyperbolic structure are considered. The spline collocation method is used to numerically solve such systems. Sufficient conditions for the convergence of the numerical procedure are obtained. Numerical results are presented.     

Spline methods for the solution of hyperbolic equation with variable coefficients

In this study, we developed the methods based on nonpolynomial cubic spline for numerical solution of second‐order nonhomogeneous hyperbolic partial differential equation. Using nonpolynomial cubic spline in space and finite difference in time directions, we obtained the implicit three level methods of O(k² + h²) and O(k² + h⁴). The proposed methods are applicable to the problems having singularity at x = 0, too. Stability analysis of the presented methods have been carried out. The presented methods are applied to the nonhomogeneous examples of different types. Numerical comparison with Mohanty's method (Mohanty, Appl Math Comput, 165 (2005), 229–236) shows the superiority of our presented schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Inequalities for the generalized trigonometric and hyperbolic functions

The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional p$p$-Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinovi?–Adamovi?’s inequality, Lazarevi?’s inequality, Huygens-type inequalities, Wilker-type inequalities, and Cusa–Huygens-type inequalities, are generalized to the case of generalized functions.     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Coefficient estimation in a first-order nonlinear hyperbolic Cauchy problem

This paper deals with the estimation and approximation of coefficient function in a first-order, nonlinear, hyperbolic Cauchy problem. The estimation is accomplished by minimizing a functional which measures the error between a finite set of given observations and the corresponding values of the solution generated by the coefficient function. A class of admissible coefficient functions is defined, and it is proved that minimizing coefficient function always exists within this class. We also develop an approximation by a sequence of solutions of associated finite-dimensional minimization problems.     

A reduced Hsieh-Clough-Tocher element with splitting based on an arbitrary interior point

We present formulas for a reduced Hsieh-Clough-Tocher (rHCT) element with splitting based on an arbitrary interior point. These formulas use local barycentric coordinates in each of the subtriangles and are not significantly more complicated than formulas for an rHCT element with splitting based on the centroid.     

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

The formation of Turing vegetation patterns in flat arid environments is investigated in the framework of a generalized version of the hyperbolic Klausmeier model. The extensions here considered involve, on the one hand, the strength of the rate at which rainfall water enters into the soil and, on the other hand, the functional dependence of the inertial times on vegetation biomass and soil water. The study aims at elucidating how the inclusion of these generalized quantities affects the onset of bifurcation of supercritical Turing patterns as well as the transient dynamics observed from an uniformly vegetated state towards a patterned state. To achieve these goals, linear and multiple-scales weakly nonlinear stability analysis are addressed, this latter being inspected in both large and small spatial domains. Analytical results are then corroborated by numerical simulations, which also serve to describe more deeply the spatio-temporal evolution of the emerging patterns as well as to characterize the different timescales involved in vegetation dynamics.     

基于基样条局部逼近散乱数据拟合中的Shepard方法

本文针对散乱数据拟合的shepard方法,提出了一种局部逼近的新方法.该方法以局部三次基样条函数作为Shepard公式中的权函数,新的权函数具有良好的衰减性和二阶连续性,从而改进了传统方法的不足之处,使实际应用效果更好.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trigonometric and hyperbolic type solutions to a generalized Drinfel’d-Sokolov equation

A class of trigonometric and hyperbolic type solutions to the generalized Drinfel’d-Sokolov (GDS) equations

     

Minimization of convex functions on the convex hull of a point set

A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R ⁿ is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal to m. The method is based on a new procedure for finding the projection of the gradient of the objective function onto a simplicial cone in R ^m , which is the tangent cone at the current point to the simplex defined by the usual constraints on barycentric coordinates. It is shown that this projection can be computed in O(m log m) operations. For strictly convex quadratic functions, the basic method can be refined to a noniterative method terminating with the optimal solution.     

一类三角域上的三次样条插值(英文)

In this paper, we consider spaces of cubic C^1-spline on a class of triangulations. By using the inductive algorithm, the posed Lagrange interpolation sets are constructed for cubic spline space. It is shown that the class of triangulations considered in this paper are nonsingular for S1/3 spaces. Moreover, the dimensions of those spaces exactly equal to L. L. Schuraaker＇s low bounds of the dimensions. At the end of this paper, we present an approach to construct triangulations from any scattered planar points, which ensures that the obtained triangulations for S1/3 space are nonsingular.     

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.     

Differentiability of a class of semi-linear hyperbolic systems with respect to a parameter

If a parameter is contained in a hyperbolic semi-linear equation, then the solution to the equation will depend on the parameter. It is proved by using the theory of semi-groups of bounded operators that the solution continuously depends on the parameter and is Fréchet continuously differentiable, under certain mild assumptions.     

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J₁(w)= ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w)– (x)||₀² can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Notes on a generalized Fresnel class

In this paper we establish a theorem for the generalized Fresnel class F _A1,A2 ensuring that various functions are in F _A1,A2. We also prove a translation theorem for the analytic Feynman integral of functions in F _A1,A2.This research was supported in part by the Basic Science Research Program, Ministry of Education.