An efficient 4-step block method for solution of first order initial value problems via shifted chebyshev polynomial

In this paper, we develop a four-step block method for solution of first order initial value problems of ordinary differential equations. The collocation and interpolation approach is adopted to obtain a continuous scheme for the derived method via Shifted Chebyshev Polynomials, truncated after sufficient terms. The properties of the proposed scheme such as order, zero-stability, consistency and convergence are also investigated. The derived scheme is implemented to obtain numerical solutions of some test problems, the result shows that the new scheme competes favorably with exact solution and some existing methods.

Integrating Oscillatory General Second-Order Initial Value Problems Using a Block Hybrid Method of Order 11

In some cases, high-order methods are known to provide greater accuracy with larger step-sizes than lower order methods. Hence, in this paper, we present a Block Hybrid Method (BHM) of order 11 for directly solving systems of general second-order initial value problems (IVPs), including Hamiltonian systems and partial differential equations (PDEs), which arise in multiple areas of science and engineering. The BHM is formulated from a continuous scheme based on a hybrid method of a linear multistep type with several off-grid points and then implemented in a block-by-block manner. The properties of the BHM are discussed and the performance of the method is demonstrated on some numerical examples. In particular, the superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is established numerically.

Asymptotic Stability of Quaternion-based Attitude Control System with Saturation Function

In the design of attitude control, rotational motion of the spacecraft is usually considered as a rotation of rigid body. Rotation matrix parameterization using quaternion can represent globally attitude of a rigid body rotational motions. However, the representation is not unique hence implies difficulties on the stability guarantee. This paper presents asymptotically stable analysis of a continuous scheme of quaternion-based control system that has saturation function. Simulations run show that the designed system applicable for a zero initial angular velocity case and a non-zero initial angular velocity case due to utilization of deadzone function as an element of the defined constraint in the stability analysis.

New Algorithm for First order Stiff Initial Value Problems

In this paper, we consider the development and implementation of algorithms for the solution of stiff first order initial value problems. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme that is evaluated at selected grid points to give discrete methods. The stability properties of the method is verified and numerical experiments show that the new method is efficient in handling stiff problems.

New algorithm for problems with exponential solutions

In this paper, we consider the development of algorithms for the solution of first order initial value problems whose solution exhibits exponential behaviour. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme. The discrete methods recovered from the continuous scheme are implemented in block form. The stability properties of the method is verified and numerical experiments show that our method is efficient in handling these problems

Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the dif-fusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

Estimating life expectancy of demented and institutionalized subjects from interval-censored observations of a multi-state model

We consider the problem of estimating life expectancy of demented and institutionalized subjects from interval-censored observations. A mixed discrete-continuous scheme of observation is a classical pattern in epidemiology because very often clinical status is assessed at discrete visit times while times of death or other events can be exactly observed. In this work, we jointly modelled dementia, institutionalization and death from data of a cohort study. Due to discrete time observations, it may happen that a subject developed dementia or was institutionalized between the last visit and the death. Consequently, there is an uncertainty about the precise number of diseased or institutionalized subjects. Moreover, the time of onset of dementia is interval censored. We use a penalized likelihood approach for estimating the transition intensities of the multi-state model. With these estimators, incidence and life expectancy can be computed easily. This approach deals with incomplete data due to the presence of left truncation and interval censoring. It can be generalized to take explanatory variables into account. We illustrate this approach by applying this model to the analysis of a large cohort study on cerebral ageing.