Most mathematical models contain uncertainties that may be originated from various sources such as initial and boundary conditions, geometry representation of the domain and input parameters. When these sources are expressed as random processes or random fields, partial differential equations describing the underlying models become stochastic partial differential equations (SPDEs). Stochastic m...

This paper illustrates the use of the differentiation matrix technique for solving differential equations in finance. The technique provides a compact and unified formulation for a variety of discretisation and time-stepping algorithms for solving problems in one and two dimensions. Using differentiation matrix models, we compare time-stepping algorithms for option pricing computations and pres...

Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ‖u‖L∞ ≤ cMk on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussia...

ÐThe unsteady ̄ow of blood in a straight, long, rigid pipe, driven by an oscillatory pressure gradient is studied. Three di€erent non±newtonian models for blood are considered and compared. One of them turns out to o€er the best ®t of experimental data, when the rheological parameters are suitably ®xed. Numerical results are obtained by a spectral collocation method in space and the implicit tr...

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for th...

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we transform the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equ...

Abstract This paper describes an experiment that attempts to compare a range of existing collocation extraction techniques as well as the implementation of a new technique based on tests for lexical substitutability. After a description of the experiment details, the techniques are discussed with particular emphasis on any adaptations that are required in order to evaluate it in the way propose...