Relative error propagation in the recursive solution of linear recurrence relations

This paper is concerned with the numerical solution of the general initial value problem for linear recurrence relations. An error analysis of direct recursion is given, based on relative rather than absolute error, and a theory of relative stability developed.Miller's algorithm for second order homogeneous relations is extended to more general cases, and the propagation of errors analysed in a similar manner. The practical significance of the theoretical results is indicated by applying them to particular classes of problem.     

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.     

A Gautschi-type method for oscillatory second-order differential equations

Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation. Received January 21, 1998 / Published online: June 29, 1999     

On the combination of kernel principal component analysis and neural networks for process indirect control

ABSTRACT

A new adaptive kernel principal component analysis (KPCA) for non-linear discrete system control is proposed. The proposed approach can be treated as a new proposition for data pre-processing techniques. Indeed, the input vector of neural network controller is pre-processed by the KPCA method. Then, the obtained reduced neural network controller is applied in the indirect adaptive control. The influence of the input data pre-processing on the accuracy of neural network controller results is discussed by using numerical examples of the cases of time-varying parameters of single-input single-output non-linear discrete system and multi-input multi-output system. It is concluded that, using the KPCA method, a significant reduction in the control error and the identification error is obtained. The lowest mean squared error and mean absolute error are shown that the KPCA neural network with the sigmoid kernel function is the best.     

A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs

In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations.     

Bernstein series solution of linear second‐order partial differential equations with mixed conditions

The purpose of this study is to present a new collocation method for numerical solution of linear PDEs under the most general conditions. The method is given with a priori error estimate. By using the residual correction procedure, the absolute error can be estimated. Also, one can specify the optimal truncation limit n, which gives better result in any norm ∥ ∥ . Finally, the effectiveness of the method is illustrated in some numerical experiments. Numerical results are consistent with the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Septic B-spline method of the Korteweg-de Vries–Burger’s equation

In this paper, a numerical solution for the Korteweg-de Vries–Burger’s equation (KdVB) by using the collocation method using the septic splines is proposed. Applying the Von-Neumann stability analysis technique we show that the method is unconditionally stable. By conducting a comparison between the absolute error for the obtained numerical results and the analytic solution of the equation we will test the accuracy of the proposed method.     

A predictor–corrector scheme for the improved Boussinesq equation

An implicit finite-difference method based on rational approximants of second order to the matrix-exponential term in a three-time level recurrence relation has been proposed for the numerical solution of the improved Boussinesq equation already known from the bibliography. The method, which is analyzed for local truncation error and stability, leads to the solution of a nonlinear system. To overcome this difficulty a predictor–corrector (P–C) scheme in which the predictor is also a second order implicit one is proposed. The efficiency of the proposed method is tested to various wave packets and the results arising from the experiments are compared with the relevant ones known in the bibliography.     

Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations

In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell-Whitehead-Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.     

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.     

Analytical and numerical solution of multi-term nonlinear differential equations of arbitrary orders

We are concerned here with a nonlinear multi-term fractional differential equation (FDE). The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM) applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Some numerical examples are given, their ADM solutions are compared with a numerical method solutions. This numerical method is introduced in Podlubny (Fractional Differential Equations, Chap. 8, Academic Press, San Diego, 1999).     

Richardson-extrapolated sequential splitting and its application

During numerical time integration, the accuracy of the numerical solution obtained with a given step size often proves unsatisfactory. In this case one usually reduces the step size and repeats the computation, while the results obtained for the coarser grid are not used. However, we can also combine the two solutions and obtain a better result. This idea is based on the Richardson extrapolation, a general technique for increasing the order of an approximation method. This technique also allows us to estimate the absolute error of the underlying method. In this paper we apply Richardson extrapolation to the sequential splitting, and investigate the performance of the resulting scheme on several test examples.     

Numerical approach for the Hunter Saxton equation arising in liquid crystal model through Cocktail party graphs Clique polynomial

In this study, we extracted the Clique polynomials from the Cocktail party graph (CPG) and generated the generalized operational matrix of integration through the Clique polynomials of CPG. Then developed an effective computational technique called Cocktail party graphs Clique polynomial collocation method (CCCM) to obtain an approximate numerical solution for the nonlinear liquid crystal model called the Hunter-Saxton equation (HSE). The operational matrix of CPG has been used to reduce HSE into an algebraic system of nonlinear equations that makes the solution quite superficial. These nonlinear algebraic equations have been solved by the Newton Raphson method. This projected that CCCM is considerably efficacious on the computational ground for higher accuracy and convergence of numerical solutions. The solution of the HSE is presented through figures and tables for different values of and . The accuracy and efficiency of the proposed technique are analyzed based on absolute errors. We also provided the convergence and error analysis of our method and verified the results through two examples to confirm the accuracy of the theoretical results.     

High-order exponential spline method for pricing European options

ABSTRACT

The key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.     

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.     

基于双尺度渐近分析的有限元算法

1．引言正如文山所说，由于复合材料和周期结构的材料系数ail（x）在局部区域内间断且跳跃性很大，加上区域内含有周期性洞穴或裂缝，且周期长度很小．一般而言，直接采用有限元方法进行数值模拟，其计算量大得惊人，甚至难以实现．文山针对这种特征，提出了一种可计算的双尺度渐近分析模式，本文在此基础上给出了相应的有限元算法，它包括：1．周期解在一个基本构造上的有限元计算；2．边界层的有限元计算．同时，给出了相应的误差分析．2．周期解的有限元计算首先考虑下列形式的边值问题；其中把，代E尸（on叫，iii（0关于E—（EI，ZZ…     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

Justification and numerical realization of the uniform method for finding point estimates of interval elicited scaling constants

The form of the utility function over multi-dimensional consequences depends on the point estimates of the scaling constants. Fuzzy rational decision makers elicit those in the form of uncertainty intervals. The paper proposes an analytical justification and a numerical realization of the uniform method that finds point estimates of interval scaling constants. The main assumption of the technique is that constants are uniformly distributed in their uncertainty intervals. The density of the constants’ sum is constructed using preliminarily chosen knots. A new numerical procedure to calculate the I type error p _value of a two-tail test for singularity of the constants’ sum is proposed. All numerical procedures are embodied into program functions. The application of the method is demonstrated in examples. The connection between precision and time for analysis is investigated. Comparison of the analytical uniform method and an earlier proposed simulation realization is also conducted.