D03PFF integrates a system of linear or nonlinear convection-diffusion equations in one space dimension, with optional source terms. The system must be posed in conservative form. Convection terms are discretized using a sophisticated upwind scheme involving a user-supplied numerical flux function based on the solution of a Riemann problem at each mesh point. The method of lines is employed to ...

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spec...

In this paper we introduce a general type of differential equations with piecewise constant argument (EPCAG), and consider the problem of backward continuation of solutions. We establish the existence of global integral manifolds of quasilinear EPCAG, consisting of solutions back continued to −∞, while the solutions starting outside the invariant sets may not be back continued. The smoothness o...

In this paper, second order non-linear ordinary differential equations of Lane-Emden type are solved using the boundary value technique. A class of second derivative backward differentiation formula is derived from some continuous multistep schemes using the multistep collocation technique. The technique transforms the numerical methods to a system of non-linear equations represented as a tridi...

Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they c...

Given a function f ∈ C[0, 1] and some q ∈ (0, 1), we look at the approximation for the Hadamard finite-part integral = ∫ 1 0 x−q−1f(x)dx based on a piecewise linear interpolant for f at n equispaced nodes (i.e., the product trapezoidal rule). The main purpose of this paper is to give sufficient conditions for the sequence of approximations to converge against the correct value of the integral i...

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give th...

A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tang’s maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a Markov process to that of a non-Markov process, and establish a generalized Itô-Kunita-Wentzell’s formula allowing the test function to be a rando...

We consider the numerical solution of differential Riccati equations. We review the existing methods and investigate whether they are suitable for large-scale problems arising in LQR and LQG design for semi-discretized partial differential equations. Based on this review, we suggest an efficient matrix-valued implementation of the BDF for differential Riccati equations.

Statutory records of underground utility apparatus (such as pipes and cables) are notoriously inaccurate, so street surveys are usually undertaken before road excavation takes place to minimize the extent and duration of excavation and for health and safety reasons. This involves the use of sensors such as Ground Penetrating Radar (GPR). The GPR scans are then manually interpreted and combined ...