Preconditioned rotated iterative methods in the solution of elliptic partial differential equation

A second-order finite difference scheme derived from rotated discretisation formula is employed in conjunction with a preconditioner to obtain highly accurate and fast numerical solution of the two-dimensional elliptic partial differential equation. The use of a ‘splitting’ preconditioning strategy will be shown to improve the spectral properties of the matrix of the linear system resulting from this discretisation by minimising the eigenvalue spectrum of the transformed matrix. The application of this technique to several acceleration iterative methods, such as Simultaneous displacement, Richardson's and Chebyshev accelerated methods, are presented and discussed.     

Dynamics of a fifth-order iterative method

In this paper, we study the dynamical behaviour of a two-point iterative method with order of convergence five to solve nonlinear equations in the complex plane. In fact, we complement the dynamical study started in previous works with a more systematic analysis for polynomials with at most two different roots and different multiplicities. In addition, we characterize some polynomials of degree greater or equal to 4, such that the related methods are not generally convergent. We also analyse the degrees of the rational functions associated with two-point methods when they are applied to polynomials of degree n, showing their dependence on n ² and how this fact considerably complicates the dynamical study.     

Acoustic resonant response of smooth or rough 3D small objects

This paper presents a new method in order to appreciate the roughness of elliptical or spherical small bodies by using ultrasound. This thing is possible by knowing about 2-4 acoustic resonance frequencies. The maximum diameter and medium roughness of the bodies are evaluated. Small bodies such as tumors can be detected, when other medical devices like ultrasound or tomography computerized cannot evaluate the surface shape because of limited spatial resolution. The method is intended to be used in oncological medicine for early non-invasive diagnostic for cancer, because tumor tissue has outer surface more or less rough, depending on whether or not it is malignant.     

Optimal control with constrained total variance for Markov jump linear systems with multiplicative noises

We consider in this paper the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises (MJLS-mn for short). Our objective is to present an optimal policy for the problem of maximising the system's total expected output over a finite-time horizon while restricting the weighted sum of its variance to a pre-specified upper-bound value. We obtain explicit conditions for the existence of an optimal control law for this problem as well as an algorithm for obtaining it, extending previous results in the literature. The paper is concluded by applying our results to a portfolio selection problem subject to regime switching.     

Gronwall inequality for hybrid systems

The Gronwall inequality, a well-known and useful result both for continuous-time and discrete-time signals, is extended to hybrid signals, namely those that combine continuous time and discrete time. An application of the result to establishing a bounded energy bounded state property for hybrid systems with inputs is provided.     

The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.     

Almost periodic solutions for impulsive fractional differential equations

In this paper, we consider the existence of almost periodic solutions for impulsive fractional evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, probability density functions, fixed point theorems and the techniques based on fractional calculus. An example is also discussed to illustrate the theory. Some known results are improved and generalized.     

Analysis on the stability of numerical schemes for a class of stochastic partial differential systems

The high-order accurate compact finite difference scheme which belongs to the finite difference methods is constructed to solve the system of partial differential equations with random noise. The error analysis and stability analysis are given and then the numerical simulation is executed. The simulation results verify the theoretical analysis results and have the faster computation speed and higher accuracy.     

Lattice Boltzmann models for two-dimensional coupled Burgers’ equations

In this paper, two lattice Boltzmann models for two-dimensional coupled Burgers’ equations are proposed through treating the part or all of convection items as the source term, where the spatial gradient can be calculated by the distribution function. The models can exactly recover the Burgers’ equations without any assumptions. Some numerical tests are also performed to validate the present models. It is found that the proposed models are more accurate and efficient in solving two-dimensional coupled Burgers’ equations.     

A simple solution method for the blade element momentum equations with guaranteed convergence

The blade element momentum (BEM) equations, though conceptually simple, can be challenging to solve reliably and efficiently with high precision. These requirements are particularly important for efficient rotor blade optimization that utilizes gradient‐based algorithms. Many solution approaches exist for numerically converging the axial and tangential induction factors. These methods all generally suffer from a lack of robustness in some regions of the rotor blade design space, or require significantly increased complexity to promote convergence. The approach described here allows for the BEM equations to be parameterized by one variable—the local inflow angle. This reduction is mathematically equivalent, but greatly simplifies the solution approach. Namely, it allows for the use of one‐dimensional root‐finding algorithms for which very robust and efficient algorithms exist. This paper also discusses an appropriate arrangement of the equation and corresponding bounds for the one‐dimensional search—intervals that bracket the solution and over which the function is well behaved. The result is a methodology for solving the BEM equations with guaranteed convergence and at a superlinear rate.Copyright © 2013 John Wiley & Sons, Ltd.