A response to Tisdell on ‘Critical perspectives of the “new” difference equation solution method’ of Rivera-Figueroa and Rivera-Rebolledo

ABSTRACT

Recently, Christopher C. Tisdell questions some claims and results that appear in our article [Rivera-Figueroa A, Rivera-Rebolledo JM. A new method to solve the second-order linear difference equations with constant coefficients. Int J Math Educ Sci Technol. 2016;47(4):636–649.] regarding the novelty and simplicity of a method to solve second-order linear difference equations. He argues that our method was not new because the method already appeared in the mathematical literature, neither was simpler than existing ones. After carefully analysing and comparing the references quoted by Tisdell, we confirm that our method differs from that quoted by Tisdell. Moreover, our approach is simpler because it avoids the uniqueness theorem and the method of variation of parameters, in this context, it offers some clear didactic and epistemological advantages.     

Computational thinking and constructionism: creating difference equations in spreadsheets

The invention of the computer has led to the establishment of a new research paradigm, computation, which has recently become more and more popular in scientific exploration. However, computation is not well represented in high school and university curricula in science and engineering, although it applies to a wide range of disciplines beyond computer science and software engineering. In light of the increasing need to provide students with computational education, this paper presents a novel way to develop computational thinking among students. The proposed approach is based on the implementation of Papert's theory of constructionism in electronic spreadsheets. In this approach, students build their knowledge while constructing the difference equation that describes a physical (or engineering) phenomenon, based on specific cases investigated in the spreadsheet. The method does not require the students to write code or perform complex calculations in the spreadsheet and makes it possible to teach advanced subjects at a relatively early stage. The method is demonstrated through contents taken from the secondary and tertiary curricula in mechanics and electromagnetism.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

线性方程组的正解

讨论了线性方程组正解的若干性质,给出了线性方程组有正解的一个充要条件,以及由此得到的求正解的一般方法,还介绍了正解问题的若干应用.     

Kantorovich's Majorization and Functional Equations

Abstract

We consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization. The main challenge in this context is to determine the convergence domain of the corresponding majorant generator. As it turns out, dealing with this task leads to solution of functional equations of a certain kind. After considering several examples, we formulate two generic models and develop an approach to their solution.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

The numerical inversion of the Laplace transform in reproducing kernel Hilbert spaces for ill-posed problems

In this paper we have converted the Laplace transform into an integral equation of the first kind of convolution type, which is an ill-posed problem, and used a statistical regularization method to solve it. The method is applied to three examples. It gives a good approximation to the true solution and compares well with the method given by Rodriguez.     

On the eigenvalue problem of a class of linear partial difference equations

In this paper, the eigenvalue problem of a class of linear partial difference equations is studied. The results concern the existence of eigenvalues, their character (real, positive), as well as the behavior of its eigenfunctions (positivity, oscillation). Moreover a theorem is given concerning the existence of a unique solution of an associated non-homogeneous partial difference equation. The results generalize previously known results for ordinary linear difference equations. The method used is a functional-analytic one, which transforms the eigenvalue problem for the difference equation into the equivalent problem of the eigenvalues of an operator defined on an abstract separable Hilbert space.     

A modified modulus method for symmetric positive‐definite linear complementarity problems

By reformulating the linear complementarity problem into a new equivalent fixed‐point equation, we deduce a modified modulus method, which is a generalization of the classical one. Convergence for this new method and the optima of the parameter involved are analyzed. Then, an inexact iteration process for this new method is presented, which adopts some kind of iterative methods for determining an approximate solution to each system of linear equations involved in the outer iteration. Global convergence for this inexact modulus method and two specific implementations for the inner iterations are discussed. Numerical results show that our new methods are more efficient than the classical one under suitable conditions. Copyright © 2008 John Wiley & Sons, Ltd.     

Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3 × 3 stencil

Hirota's bilinear method (‘direct method’) has been very effective for constructing soliton solutions to many integrable equations. The construction of one-soliton solution (1SS) and two-soliton solution (2SS) is possible even for non-integrable bilinear equations, but the existence of a generic three-soliton solution (3SS) imposes severe constraints and is in fact equivalent to integrability. This property has been used before in searching for integrable partial differential equations, and in this paper we apply it to two-dimensional (2D) partial difference equations defined on a 3 × 3 stencil. We also discuss how the obtained equations are related to projections and limits of the 3D master equations of Hirota and Miwa, and find that sometimes a singular limit is needed.     

On differential equations with interval coefficients

In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.     

An efficient algorithm for finding all solutions of separable systems of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of nonlinear equations with separable mappings. This algorithm is based on interval analysis, the dual simplex method, the contraction method, and a special technique which makes the algorithm not require large memory space and not require copying tableaus. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 2000 nonlinear equations in acceptable computation time. AMS subject classification (2000)  65H10, 65G10     

New third-order method for solving systems of nonlinear equations

In this paper, we present a new iterative method to solve systems of nonlinear equations. The main advantages of the method are: it has order three, it does not require the evaluation of any second or higher order Fréchet derivative and it permits that the Jacobian be singular at some points. Thus, the problem due to the fact that the Jacobian is numerically singular is solved. The third order convergence in both one dimension and for the multivariate case are given. The numerical results illustrate the efficiency of the method for systems of nonlinear equations.      

一种建立和求解递归关系的方法

本文利用有限差分算子和组合恒等式为工具，给出了线性递归关系中序列｛ａｎ｝求解的新方法，与原来特征多项式法比较，它有两点好处；其一是简化了计算的过程；其二是避免了建立递归关系时复杂的推导。     

Critical perspectives of the ‘new’ difference equation solution method of Rivera-Figueroa and Rivera-Rebolledo

Recently, claims of a ‘new and straightforward’ method of solution to second-order linear difference equations have appeared in the mathematics education literature from Rivera-Figueroa and Rivera-Rebolledo. The claim of novelty is based on an assumption that ‘since the equation is worked in its canonical form’, the method within this context must be new. In addition, the assertion of straightforwardness is based on the position that ‘the solution comes naturally’ through this method, rather than artificially. In this article, we subject these claims and assumptions to closer scrutiny, examination and analysis. We note that the method has been published before, and we present the method in a more succinct form. We also discuss how the method can be extended to solve difference equations with non-constant coefficients, illustrating this via a discussion of an example.     

An inexact parallel splitting augmented Lagrangian method for large system of linear equations

Parallel iterative methods are powerful in solving large systems of linear equations (LEs). The existing parallel computing research results focus mainly on sparse systems or others with particular structure. Most are based on parallel implementation of the classical relaxation methods such as Gauss-Seidel, SOR, and AOR methods which can be efficiently carried out on multiprocessor system. In this paper, we propose a novel parallel splitting operator method in which we divide the coefficient matrix into two or three parts. Then we convert the original problem (LEs) into a monotone (linear) variational inequality problem (VI) with separable structure. Finally, an inexact parallel splitting augmented Lagrangian method is proposed to solve the variational inequality problem (VI). To avoid dealing with the matrix inverse operator, we introduce proper inexact terms in subproblems such that the complexity of each iteration of the proposed method is O(n²). In addition, the proposed method does not require any special structure of system of LEs under consideration. Convergence of the proposed methods in dealing with two and three separable operators respectively, is proved. Numerical computations are provided to show the applicability and robustness of the proposed methods.     

Two-level iteration penalty and variational multiscale method for steady incompressible flows

In this paper, we study two-level iteration penalty and variational multiscale method for the approximation of steady Navier-Stokes equations at high Reynolds number. Comparing with classical penalty method, this new method does not require very small penalty parameter $\varepsilon$. Moreover, two-level mesh method can save a large amount of CPU time. The error estimates in $H^1$ norm for velocity and in $L^2$ norm for pressure are derived. Finally, two numerical experiments are shown to support the efficiency of this new method.     

The Immersed Interface Method for a Nonlinear Chemical Diffusion Equation with Local Sites of Reactions

A diffusion equation with nonlinear localized chemical reactions is considered in this paper. As a result of the reactions, although the equation is parabolic, the derivatives of the solution are discontinuous across the interfaces (local sites of reactions). A second-order accurate immersed interface method is constructed for the diffusion equation involving interfaces. The new method is more accurate than the standard approach and it does not require the interfaces to be grid points. Several experiments that confirm second-order accuracy are presented. The efficiency of the proposed algorithm is also demonstrated for solving blow up problems. The proposed technique could be extended for construction of efficient numerical algorithms on uniform grids for the present equations with moving interfaces [9] but more analysis is required.     

Direct and iterative methods for interval parametric algebraic systems producing parametric solutions

This paper deals with interval parametric linear systems with general dependencies. Motivated by the so‐called parameterized solution introduced by Kolev, we consider the enclosures of the solution set in a revised affine form. This form is advantageous to a classical interval solution because it enables us to obtain both outer and inner bounds for the parametric solution set and, thus, intervals containing the endpoints of the hull solution, among others. We propose two solution methods, a direct method called the generalized expansion method and an iterative method based on interval‐affine Krawczyk iterations. For the iterative method, we discuss its convergence and show the respective sufficient criterion. For both methods, we perform theoretical and numerical comparisons with some other approaches. The numerical experiments, including also interval parametric linear systems arising in practical problems of structural and electrical engineering, indicate the great usefulness of the proposed methodology and its superiority over most of the existing approaches to solving interval parametric linear systems.     

修正的LMS方法

ＢｉｔｍｅａｄＲ．Ｒ和ＡｎｄｅｒｓｏｎＤ．Ｏ在文献［１］中为任意线性方程组的求解提出了一种颇为有效的算法，称为ＬＭＳ方法．文献［２］详细地论述了算法的收敛性，指出收敛极限是方程组的最小二乘解．本文为使解线性方程组的ＬＭＳ算法具有更广泛、更方便的应用性．对文献［２］中的ＬＭＳ算法作了修正．理论和实践证明修正后的算法是成功的．