Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusion-wave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An imp...

This paper examines the stability and convergence of discretisa-tions of initial value p.d.e.'s using spatial discretisation followed by time integration with an explicit one-step method. A Cauchy integral representation is used to bound the growth in the discrete solution. New results are obtained regarding suucient conditions for both algebraic and strong stability. Suucient conditions are al...

This chapter is about numerical methods for a particular type of equation expressed as a matrix equality. The Lyapunov equation is the most common problem in the class of problems called matrix equations. Other examples of matrix equations: Sylvester equation, Stein equation, Riccati equation. Definition 4.0.1. Consider two square matrices A, W ∈ Rn×n. The problem to find a square matrix X ∈ Rn...

We derive a manifestly gauge invariant low energy blocked action for Yang-Mills theory using operator cutoff regularization, a prescription which renders the theory finite with a regulating smearing function constructed for the proper-time integration. By embedding the momentum cutoff scales in the smearing function, operator cutoff formalism allows for a direct application of Wilson-Kadanoff r...

Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem [5]. These schemes have attractive stability properties and they a...

This paper investigates the expected benefits of the implicit time integration scheme in the solution of jointed rock problems. Discontinuous Deformation Analyses (DDA) exploits the unconditional stability of the implicit time integration scheme, and allows the time step size of the solution to be much larger than the critical time step size dictated by the stability requirement of the explicit...

Article history: Received 31 August 2012 Accepted 26 January 2013 Available online 8 February 2013

A general class of convergent methods for the numerical solution of ordinary differential equations is employed to obtain a class of convergent generalized reducible quadrature methods for Volterra integral equations of the second kind and Volterra integro-differential equations.

This paper contains error estimates for covolume discretizations of Maxwell’s equations in three space dimensions. Several estimates are proved. First, an estimate for a semi-discrete scheme is given. Second, the estimate is extended to cover the classical interlaced time marching technique. Third, some of our unstructured mesh results are specialized to rectangular meshes, both uniform and non...