Scielo RSS <![CDATA[Computational & Applied Mathematics]]> vol. 31 num. 2 lang. en <![CDATA[SciELO Logo]]> <![CDATA[<b>Exact travelling wave solutions for some nonlinear (<i>N</i>+1)-dimensional evolution equations</b>]]> In this paper, we implement the tanh-coth function method to construct the travelling wave solutions for (N + 1)-dimensional nonlinear evolution equations. Four models, namely the (N + 1)-dimensional generalized Boussinesq equation, (N + 1)-dimensional sine-cosine-Gordon equation, (N + 1)-double sinh-Gordon equation and (N + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. These equations play a very important role in mathematical physics and engineering sciences. The implemented algorithm is quite efficient and is practically well suited for these problems. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. Mathematical subject classification: 35K58, 35C06, 35A25. <![CDATA[<b>Solutions to the recurrence relation <i>u<sub>n</sub></i><sub>+1</sub> = <i>v</i><sub>n+1</sub> + <i>u<sub>n</sub></i> </b><b>⊗</b><b> <i>v<sub>n</sub></i> in terms of Bell polynomials</b>]]> Motivated by time series analysis, we consider the problem of solving the recurrence relation u n+1 = v n+1 + u n ⊗ v n for n ≠ 0 and u n, given the sequence v n. A solution is given as a Bell polynomial. When v n can be written as a weighted sum of nth powers, then the solution u n also takes this form. Mathematical subject classification: 33E99. <![CDATA[<b>A special class of continuous general linear methods</b>]]> We consider the construction of a class of numerical methods based on the general matrix inverse [14] which provides continuous interpolant for dense approximations (output). Their stability properties are similar to those for Runge-Kutta methods. These methods provide a unifying scope for many families of traditional methods. They are self-starting, to change stepsize during integration is not difficult when using them. We exploited these properties by first obtaining the direct block methods associated with the continuous schemes and then converting the block methods into uniformly A-stable high order general linear methods that are acceptable for solving stiff initial value problems. However, we will limit our formulation only for the step numbers k = 2, 3, 4. From our preliminary experiments we present some numerical results of some initial value problems in ordinary differential equations illustrating various features of the new class of methods. Mathematical subject classification: 65L05. <![CDATA[<b>The sodium pump controls the frequency of action-potential-induced calcium oscillations</b>]]> Calcium plays a significant role in a number of cellular processes, like muscle contraction, gene expression, synaptic plasticity, signal transduction, but the significance of calcium oscillations (CaOs) is not yet completely understood in most of the cell types. It is a widely accepted fact that CaOs are a frequency encoded signal that allows a cell to use calcium as a second messenger while avoiding its toxic effects. These intracellular CaOs are primarily driven by some agonist-dependent pathways or fluctuations in membrane potential. The present mathematical model is of the latter type. The model incorporates expression for all major intracellular ionic species and membrane proteins. Especially, it integrates the coupling effect of sodium pump and Na+ / Ca2+ exchanger over CaOs. By varying sodium pump current, it is found that, sodium pump is a key player in modulating intracellular CaOs. The model predicts that the sodium pump can play a decisive role in regulating intercellular cell signaling process. The present study forms the basis for sodium pump controlled intercellular signaling process and requires further experimental verification. Mathematical subject classification: 34M10, 92C20. <![CDATA[<b>Piecewise constant bounds for the solution of nonlinear Volterra-Fredholm integral equations</b>]]> In this paper, we compute piecewise constant bounds on the solution of mixed nonlinear Volterra-Fredholm integral equations. The enclosures are in the form of intervals which are guaranteed to contain the exact solution considering all round-off and truncation errors, so the width of interval solutions allows us to control the error estimation. An iterative algorithm to improve the accuracy of initial enclosures is given and its convergence are also investigated. Our numerical experiments show that the precision of interval solutions are reasonable in comparison to the classical methods and the obtained conditions and initial enclosure of the proposed algorithm are not restrictive. Mathematical subject classification: 65G20, 45G10, 65G40. <![CDATA[<b>Integrating Ridge-type regularization in fuzzy nonlinear regression</b>]]> In this paper, we deal with the ridge-type estimator for fuzzy nonlinear regression models using fuzzy numbers and Gaussian basis functions. Shrinkage regularization methods are used in linear and nonlinear regression models to yield consistent estimators. Here, we propose a weighted ridge penalty on a fuzzy nonlinear regression model, then select the number of basis functions and smoothing parameter. In order to select tuning parameters in the regularization method, we use the Hausdorff distance for fuzzy numbers which was first suggested by Dubois and Prade [8]. The cross-validation procedure for selecting the optimal value of the smoothing parameter and the number of basis functions are fuzzified to fit the presented model. The simulation results show that our fuzzy nonlinear modelling performs well in various situations. Mathematical subject classification: Primary: 62J86; Secondary: 62J07. <![CDATA[<b>An alternating LHSS preconditioner for saddle point problems</b>]]> In this paper, we present a new alternating local Hermitian and skew-Hermitian splitting preconditioner for solving saddle point problems. The spectral property of the preconditioned matrices is studies in detail. Theoretical results show all eigenvalues of the preconditioned matrices will generate two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter tends to zero from positive. Numerical experiments are given to validate the performances of the preconditioner. Mathematical suject classification: Primary: 65F10; Secondary: 65F50. <![CDATA[<b>Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices</b>]]> An n × n real matrix P is said to be a generalized reflection matrix if P T = P and P² = I (where P T is the transpose of P). A matrix A ∈ Rn×n is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = P A P (A = - P A P). The reflexive and anti-reflexive matrices have wide applications in many fields. In this article, two iterative algorithms are proposed to solve the coupled matrix equations { A1 XB1 + C1X T D1 = M1. A2 XB2 + C2X T D2 = M2. over reflexive and anti-reflexive matrices, respectively. We prove that the first (second) algorithm converges to the reflexive (anti-reflexive) solution of the coupled matrix equations for any initial reflexive (anti-reflexive) matrix. Finally two numerical examples are used to illustrate the efficiency of the proposed algorithms. Mathematical subject classification: 15A06, 15A24, 65F15, 65F20. <![CDATA[<b>Wavelet Galerkin method for solving singular integral equations</b>]]> An effective technique upon linear B-spline wavelets has been developed for solving weakly singular Fredholm integral equations. Properties of these wavelets and some operational matrices are first presented. These properties are then used to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. Mathematical subject classification: 45A05, 32A55, 34A25, 65T60. <![CDATA[<b>A filled function method for nonlinear systems of equalities and inequalities</b>]]> In this paper a filled function method is suggested for solving nonlinear systems of equalities and inequalities. Firstly, the original problem is reformulated into an equivalent constrained global optimization problem. Subsequently, a new filled function with one parameter is constructed based on the special characteristics of the reformulated optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed filled function for solving nonlinear systems of equalities and inequalities is presented. The objective function value can be reduced by half in each iteration of our filled function algorithm. The implementation of the algorithm on several test problems is reported with numerical results. Mathematical subject classification: 65K05, 90C30. <![CDATA[<b>A new double trust regions SQP method without a penalty function or a filter</b>]]> A new trust-region SQP method for equality constrained optimization is considered. This method avoids using a penalty function or a filter, and yet can be globally convergent to first-order critical points under some reasonable assumptions. Each SQP step is composed of a normal step and a tangential step for which different trust regions are applied in the spirit of Gould and Toint [Math. Program., 122 (2010), pp. 155-196]. Numerical results demonstrate that this new approach is potentially useful. Mathematical subject classification: 65K05, 90C30, 90C55.