A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

Atomic radial basis functions in numerical algorithms for solving boundary-value problems for the Laplace equation

A numerical method for solution of boundary-value problems of mathematical physics is described that is based on the use of radial atomic basis functions. Atomic functions are compactly supported solutions of functional-differential equations of special form. The convergence of this numerical method is investigated for the case of using an atomic function in solving the Dirichlet boundary-value problem for the Laplace equation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 165–178, July–August 2008.     

一种改进的RBF全局优化方法

径向基函数（Radial Basis Functions）由于具有良好的近似效果和运算简单的特点,被应用于全局优化中,成为解决黑箱函数全局优化问题的有效方法。然而现有的基于RBF的全局优化算法存在迭代过程中RBF模型重构效率低下,以及采样方法不合理导致函数估值次数过多等问题。在此提出几个改进思路：采用基于矩阵分块的增量RBF方法以减少模型重构时间提高效率;采用增量LHD采样方法以确保具有更好的空间填充性;采用算法重启策略以降低估值次数。通过实验验证改进方法的优势。     

New exact solutions of the (n+1)-dimensional sine-Gordon equation using double elliptic equation method

In this paper, the (n+1)-dimensional sine-Gordon equation is studied using double elliptic equation method. With the aid of Maple, more exact solutions expressed by Jacobi elliptic functions are obtained. When the modulus m of Jacobi elliptic function is driven to the limit 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained, respectively.     

On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

The current investigation describes a computational technique to solve one- and two-dimensional Fredholm integral equations of the second kind. The method estimates the solution using the discrete collocation method by combining locally supported radial basis functions (RBFs) constructed on a small set of nodes instead of all points over the analysed domain. In this work, we employ the Gauss–Legendre integration rule on the influence domains of shape functions to approximate the local integrals appearing in the method. In comparison with the globally supported RBFs for solving integral equations, the proposed method is stable and uses much less computer memory. The scheme does not require any cell structures, so it is meshless. We also obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is high. Illustrative examples clearly show the reliability and efficiency of the new method.     

Consideration effect of uncertainty in power system reliability indices using radial basis function network and fuzzy logic theory

Reliability assessment of composite power systems is a critical and important part of power investigations especially in the market-driven environments. Therefore, the reliability indices as criteria for the comparison of the reliability of the power systems should be evaluated precisely and carefully. Because of the nonlinear behavior of the systems as the effect of different parameters like weather conditions, load pattern changes and some others, reliability indices always contain much uncertainty. In this paper a neuro-fuzzy based method is proposed to reduce the degree of the uncertainty in the reliability indices and therefore to evaluate the reliability of the composite power systems precisely. Fuzzy logic theory makes it possible to make use of the human experts knowledge in the reliability evaluations. Also by the use of RBFNN and its powerful characteristic to learn any nonlinear mapping between two states it would be possible to evaluate the reliability indices for every short time interval needed so that reliability evaluation in real time would be achievable and feasible.In this paper the RBFNN is trained by the training patterns that are achieved by the use of fuzzy logic theory, then the results are examined on a standard Reliability Test System (RTS-96).     

A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ. A stability analysis is performed, showing that even for the explicit scheme that θ=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.     

Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions

Two numerical techniques are presented for solving the solution of Riccati differential equation. These methods use the cubic B-spline scaling functions and Chebyshev cardinal functions. The methods consist of expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the new techniques. The methods are easy to implement and produce very accurate results.     

一种径向基函数插值的ENO格式

将ENO格式和径向基函数插值相结合,提出了求解双曲型偏微分方程的径向基函数插值的ENO方法。该方法依据ENO思想建立自适应模板,在选定的模板上利用径向基函数进行逼近,能够很好地处理具有间断解的问题,消除间断点处数值振荡现象。以一维双曲型偏微分方程为例,对该方法进行了验证,并通过与多项式ENO格式比较,表明该方法更具有优势。     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor-corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.     

Application of the local radial basis function-based finite difference method for pricing American options

In this paper we discuss a local radial basis function-based finite difference (RBF-FD) scheme for numerical solution of multi-asset American option problems. The governing equation is discretized by the θ-method and the option price is approximated by the RBF-FD method. Numerical experiments are performed with the multiquadratic radial basis function for single and double asset problem and results obtained are compared with existing ones. We show numerically that the scheme is second-order accurate. Stability of the scheme is also discussed.     

Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Since neural networks have universal approximation capabilities, therefore it is possible to postulate them as solutions for given differential equations that define unsupervised errors. In this paper, we present a wide survey and classification of different Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network techniques, which are used for solving differential equations of various kinds. Our main purpose is to provide a synthesis of the published research works in this area and stimulate further research interest and effort in the identified topics. Here, we describe the crux of various research articles published by numerous researchers, mostly within the last 10 years to get a better knowledge about the present scenario.     

采样算子调整的径向基网络增量映射学习算法

为了提高增量映射学习(IPL)算法的效率,调整了径向基神经网络基函数的中心及方差,以达到调整采样算子的目的,同时,通过神经元函数相关性的计算,确定添加新神经元时,相关函数的阈值,为系统结构调整提供相应依据.新方法步骤相对简单,所以算法速度较快;仿真结果表明,由于系统参数得到调整,对于同一问题,改进IPL算法得到的径向基神经网络结构较一般算法得到的网络结构简单,输出结果也较为精确.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

WPT-Interp-RBF网络:多组分分光光度同时测定数据解析的方法

针对UV-Vis分光光度多组分同时测定数据处理中经常遇到的变量多、样本少问题,提出了用小波包变换-线性插值- RBF网络解析多组分分光光度同时测定数据的方法。该方法采用小波包变换对测定数据滤噪,用线性插值处理使训练集样本对待辩识空问形成较好的覆盖,从而使RBF网络能够提取到更多的特征信息。改善网络性能、提高预测准确性。对系列实验数据的解析应用证明,该方法获得的测定结果优于常用的解析变量多、样本少数据的其它方法。由于该方法预测误差小,易在Matlab上实现编程,因此具有广泛的推广应用前景。本文以解析与炼油厂催化裂化装置生产相关的Fe、Ni、V的同时测定数据为例,对该方法的应用做了详细介绍,并与PLS法、RBF网络、小波压缩数据-RBF网络、小波压缩数据-PLS法、线性插值-RBF网络、小波变换-线性插值-RBF网络、小波包变换-线性插值-PLS法等方法傲了应用比较。     

Improvement of RBF neural networks using Fuzzy-OSD algorithm in an online radar pulse classification system

In this paper a new methodology for training radial basis function (RBF) neural networks is introduced and examined. This novel approach, called Fuzzy-OSD, could be used in applications, which need real-time capabilities for retraining neural networks. The proposed method uses fuzzy clustering in order to improve the functionality of the Optimum Steepest Descent (OSD) learning algorithm. This improvement is due to initialization of RBF units more precisely using fuzzy C-Means clustering algorithm that results in producing better and the same network response in different retraining attempts. In addition, adjusting RBF units in the network with great accuracy will result in better performance in fewer train iterations, which is essential when fast retraining of the network is needed, especially in the real-time systems. We employed this new method in an online radar pulse classification system, which needs quick retraining of the network once new unseen emitters detected. Having compared result of applying the new algorithm and Three-Phase OSD method to benchmark problems from Proben1 database and also using them in our system, we achieved improvement in the results as presented in this paper.     

A symmetric eight-step predictor-corrector method for the numerical solution of the radial Schrödinger equation and related IVPs with oscillating solutions

In this paper we present a new optimized symmetric eight-step predictor-corrector method with phase-lag of order infinity (phase-fitted). The method is based on the symmetric multistep method of Quinlan–Tremaine, with eight steps and eighth algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation during the resonance problem with the use of the Woods–Saxon potential. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.     

A hybrid equivalent dipole moment and adaptive modified characteristic basis function method for electromagnetic scattering by multilayered dielectric bodies

In this article, the equivalent dipole moment (EDM) method is extended and applied to analyze the electromagnetic scattering of multilayered dielectric bodies. Besides the advantage of reducing the computation time, the proposed EDM method makes it unnecessary to treat the boundary condition on the surface or the interface of the multilayered dielectric body. However, the EDM method is incapable of saving memory. In this case, the adaptive modified characteristic basis function (AMCBF) method is introduced and combined with the EDM method. Numerical results are given to demonstrate the merits of this EDM‐AMCBF method. © 2009 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.