On the Variable Two-Step IMEX BDF Method for Parabolic Integro-differential Equations with Nonsmooth Initial Data Arising in Finance

2019 ◽  
Vol 57 (3) ◽  
pp. 1289-1317 ◽  
Author(s):  
Wansheng Wang ◽  
Yingzi Chen ◽  
Hua Fang
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Nonsmooth Initial Data ◽  
Bdf Method

scholarly journals Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

2018 ◽  
Vol 18 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Yan Yang ◽  
Yubin Yan ◽  
Neville J. Ford
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Homogeneous Problem ◽  
Fractional Diffusion ◽  
Nonsmooth Data ◽  
Time Stepping ◽  
Time Discretisation ◽  
Nonsmooth Initial Data ◽  
Densely Defined ◽  
Diffusion Problems

AbstractWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an {O(k)} convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k denotes the time step size. In this paper, we approximate the Riemann–Liouville fractional derivative by Diethelm’s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate {O(k)} with nonsmooth initial data for the homogeneous problem when A is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Model of the functioning of the optoelectronic information system for detecting space debris

2021 ◽  
Author(s):  
V.V. Pyatkov ◽  
I.V. Chebotar ◽  
R.A. Gudaev ◽  
S.V. Kulikov ◽  
R.R. Fattahov
Keyword(s):  
Information System ◽  
Initial Data ◽  
Differential Equations ◽  
Space Debris ◽  
Optoelectronic Devices ◽  
Structural Diagram ◽  
Electronic Information ◽  
Near Earth Space ◽  
Information Tools ◽  
Detection And Tracking

To determine the characteristics of optoelectronic devices, as a rule, models are used that do not take into account the peculiarities of the functioning of information tools, the conditions of visibility and observability, which does not allow to reliably assess their capabilities to obtain coordinate and non-coordinate information. Goal of the work is to investigate the model of an optical-electronic information tool in order to evaluate the characteristics and determine the possibility of obtaining coordinate and non-coordinate information in various conditions. A block model of a system for monitoring space debris in near-earth space by means of optical-electronic information means is considered. A structural diagram of the model's constituent parts is proposed. It is shown that the position of an object in the composition of space debris is determined based on the solution of the above differential equations. The interrelation of the influence of various conditions on the capabilities of optoelectronic information facilities, on the processes of detection and tracking is described. The equations and relationships underlying the model of operation of the optoelectronic information facility are described. The results of modeling are presented, which allow planning the rational placement of optoelectronic information facilities. The presented model makes it possible to obtain initial data for planning the rational placement of optoelectronic information facilities and to substantiate the requirements for their technical characteristics.

scholarly journals FORMATION OF SINGULARITY AND SMOOTH WAVE PROPAGATION FOR THE NON-ISENTROPIC COMPRESSIBLE EULER EQUATIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 671-690 ◽  
Author(s):  
GENG CHEN
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Initial Data ◽  
Differential Equations ◽  
Euler Equations ◽  
Hyperbolic Systems ◽  
Ideal Gas ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Shock Wave Formation

We define the notion of compressive and rarefactive waves and derive the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize Lax's singularity (shock wave) formation results (established in 1964 for hyperbolic systems with two variables) to the 3 × 3 compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.

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