Existence and uniqueness of classical solution to the initial-boundary value problem for the unsteady general planar Broadwell model with four velocities

First-order partial derivatives of a mathematical model are an essential part of evaluating the measurement uncertainty of a liquid flow standard system according to the Guide to the expression of uncertainty in measurement (GUM). Although the GUM provides a straight-forward method to evaluate the measurement uncertainty of volume flow rate, the first-order partial derivatives can be complicated. The mathematical model of volume flow rate in a liquid flow standard system has a cross-correlation between liquid density and buoyancy correction factor. This cross-correlation can make derivation of the first-order partial derivatives difficult. Monte Carlo simulation can be used as an alternative method to circumvent the difficulty in partial derivation. However, the Monte Carlo simulation requires large computational resources for a correct simulation because it considers the completeness issue whether an ideal or a real operator conducts an experiment to evaluate the measurement uncertainty. Thus, the Monte Carlo simulation needs a large number of samples to ensure that the uncertainty evaluation is as close to the GUM as possible. Unscented transform can alleviate this problem because unscented transform can be regarded as a Monte Carlo simulation with an infinite number of samples. This idea means that unscented transform considers the uncertainty evaluation with respect to the ideal operator. Thus, unscented transform can evaluate the measurement uncertainty the same as the uncertainty that the GUM provides.

Undergraduates enrolled in two sections of an elementary differential equations course completed a sequence of tasks to identify the surface plots of selected functions of two variables and the first-order partial derivatives. Average scores on the series of tasks exceeded chance-level expected values and showed an increase over time. Participants' suggestions for improving performance on the identification tasks included increasing student feedback and interaction among students, and a review of the graphics literature pointed to the potential for improvement offered by adaptive surface plotting, graphical design variations, and attention to contour diagrams.

Here we derive multivariate weighted fractional representation formulae involving ordinary partial derivatives of first order. Then we present related multivariate weighted fractional Ostrowski type inequalities with respect to uniform norm.

A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension

In this paper, we study the approximation of d-dimensional $$\varrho $$ -weighted integrals over unbounded domains $${\mathbb {R}}_+^d$$ or $${\mathbb {R}}^d$$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $$L_p$$ norm of mixed partial derivatives of first order, where $$p\in [1,+\infty ].$$ The main results give sufficient conditions on the change of variables $$\nu $$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $$\varrho $$ and p. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

Here we derive a multivariate fractional representation formula involving ordinary partial derivatives of first order. Then we prove a related multivariate fractional Ostrowski type inequality with respect to uniform norm.

Our goal is to obtain an estimate of the W q,1-norm of the solution of Eq. (1) on an interior domain Ω′ ⊂ Ω via the norms of the functions e, f and the L1-norm of the solution on the greater domain Ω. This estimate is a refinement of the result stated by Morrey in his book [1, p. 156] not quite correctly: with Ω′ = Ω, which is leads to an incorrect result, e.g., for the Laplace equation on the disk. The proof of Morrey’s estimate with a study of the dependence of the constant on the coefficients was given in [2], where the same mistake as in Morrey’s book appears. Arguments from [2] are valid for solutions with zero boundary condition. We shall assume somewhat stronger conditions on the coefficients of the equation than those in [1] and [2]. By W q,1(Ω) we denote the Sobolev space of functions belonging to Lq(Ω) together with their generalized partial derivatives of first order. This space is endowed with the standard norm ‖f‖W q,1(Ω) := ‖f‖q,1 := ‖f‖Lq(Ω) + ‖∇f‖Lq(Ω), E-mail: shaposh.st@ru.net.

The paper deals with the description of a method and the accompanying software, the package LABSUP, for representing C 1 interpolating surfaces. The application to the lagoon of Venice's bed is also proposed. The surfaces are built over the Delaunay triangulation and the polynomial patches used for the representation can be chosen among the Q 18 element, the Clough-Tocher or the Powell–Sabin finite elements or simply using global Bezier methods. The first three patches require the knowledge of the gradients at the nodes, or at least a suitable estimation of them. Therefore, interesting in itself is the derivative estimation process based on the minimization of the energy functional associated with the interpolant. For the representation of the lagoon of Venice's bed we only used the reduced Clough-Tocher finite element, due to the high number of points involved for which one needs to compute the Delaunay triangulation, and simply the partial derivatives of first order. A brief description of the software modules together with some graphical results of parts of the lagoon of Venice's bed are also presented.

The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study alocal method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called acell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines theglobal Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed.

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

An interval uncertainty analysis method for structural response bounds using feedforward neural network differentiation

In $n \geq 1$ spatial dimensions, we study the Cauchy problem for a quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology) of regular initial data that are close to the data of a simple plane wave, we give a sharp, constructive proof of shock formation in which the transport variable remains bounded but its first-order Cartesian coordinate partial derivatives blow up in finite time. Moreover, we prove that the singularity does not propagate into the symmetric hyperbolic variables: they and their first-order Cartesian coordinate partial derivatives remain bounded, even though they interact with the transport variable all the way up to its singularity. The formation of the singularity is tied to the finite-time degeneration, relative to the Cartesian coordinates, of a system of geometric coordinates adapted to the characteristics of the transport operator. Two crucial features of the proof are that relative to the geometric coordinates, all solution variables remain smooth, and that the finite-time degeneration coincides with the intersection of the characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to wave equations, the main new features of the present work are: i) we develop a theory of nonlinear geometric optics for transport operators, which is compatible with the coupling and which allows us to implement a quasilinear geometric vectorfield method, even though the regularity properties of the corresponding eikonal function are less favorable compared to the wave equation case and ii) we allow for a full quasilinear coupling, i.e., the principal coefficients in all equations are allowed to depend on all solution variables.

In this paper, we propose a method to determine the state function and control function of the dynamical system represented by a system of equations with first-order partial derivatives under boundary and the first-order partial derivatives conditions of state function where are real matrices with corresponding sizes. The basis of the theory is a method to prove the orderable cascade decomposition to transform the original system into the equivalent system in the type . In the final step, we obtain the state function satisfying the conditions and substituting this in the previous step. Continuing this process, we can find out the state function and controllable function of the original dynamical system.

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.