We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomia...

A modified regional self-interaction correction (mRSIC) method is proposed for obtaining accurate core-excitation energies in time-dependent density functional theory (TDDFT) calculations. The mRSIC method is an improvement of the RSIC method (Tsuneda et al. J Comput Chem 2003, 24, 1592). It takes into account the energy contributions from 2s and higher atomic orbitals that the RSIC method negl...

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multiorder fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is ve...

This paper proposes a pseudo-haptic feedback method conveying simulated soft surface stiffness information through a visual interface. The method exploits a combination of two feedback techniques, namely visual feedback of soft surface deformation and control of the indenter avatar speed, to convey stiffness information of a simulated surface of a soft object in virtual environments. The propos...

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as a set of non-overlapping cubic regions, plus a set of maps to identify the faces of adjoining regions. The differential structure on these manifolds is fixe...

In the past few years we have been involved in the development and implementation of parallel finite element and pseudo-spectral methods for porous media flow. In this paper we present a brief review of our methods and show several examples of highly nonlinear simulations in two and three-dimensions, on different current high performance computers. New issues on finite element techniques and ps...

An orthogonal system of rational functions is introduced. Some results on rational approximations based on various orthogonal projections and interpolations are established. These results form the mathematical foundation of the related spectral method and pseudospectral method for solving differential equations on the half line. The error estimates of the rational spectral method and rational p...