Initial value problem

About: Initial value problem is a research topic. Over the lifetime, 27818 publications have been published within this topic receiving 519226 citations. The topic is also known as: cauchy problem.

Fractional diffusion and wave equations

Abstract: Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.

Basic theory of fractional differential equations

Abstract: In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.

Uniformly high-order accurate nonoscillatory schemes

Abstract: We begin the construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper we construct a uniformly second-order approximation, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise-linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem and an average of this approximate solution over each cell.

A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts

Abstract: Knowledge of the probability distribution of initial conditions is central to almost all practical studies of predictability and to improvements in stochastic prediction of the atmosphere. Traditionally, data assimilation for atmospheric predictability or prediction experiments has attempted to find a single “best” estimate of the initial state. Additional information about the initial condition probability distribution is then obtained primarily through heuristic techniques that attempt to generate representative perturbations around the best estimate. However, a classical theory for generating an estimate of the complete probability distribution of an initial state given a set of observations exists. This nonlinear filtering theory can be applied to unify the data assimilation and ensemble generation problem and to produce superior estimates of the probability distribution of the initial state of the atmosphere (or ocean) on regional or global scales. A Monte Carlo implementation of the fully n...