A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. doi:10.1017/S1446181121000377

Modeling of Hygro-Mechanical Behaviour through Raffia vinifera Fiber during the Water Diffusion Phenomenon

In an industrial context where the use of friendly materials is encouraged, natural fibers of vegetable origin become more solicited for the reinforcement of composite materials. This work deals with the modeling of the hygro-mechanical behavior through raffia vinifera fiber during the diffusion phenomenon. The modeling of water diffusion through the raffia vinifera fiber is described by Fick’s second law and taking into account the swelling phenomenon of the fiber. The equation obtained is solved numerically by the finite difference method, and the evolution of the fiber radius as a function of time is obtained. By applying the Leibniz integration rule, a mathematical expression to predict the evolution of this radius as a function of time is proposed. It is observed numerically and analytically an increase of the dimensionless fiber radius with time up to a critical value after what one observes the saturation. This model allowed us to propose a mathematical model describing the absorption kinetics of the raffia vinifera fiber through its absorption ratio. By comparing the results of this model with the experimental results from the literature, one observes a good agreement. Moreover, the induced stresses in the fiber during the water diffusion can also be estimated with the proposed mathematical model expression of fiber. These stresses increase with time and can reach between 5 and 7 GPa. The results of this work can be used to predict the behavior of the raffia vinifera fiber inside a composite material during its development.

A MESHLESS LOCAL GALERKIN INTEGRAL EQUATION METHOD FOR SOLVING A TYPE OF DARBOUX PROBLEMS BASED ON RADIAL BASIS FUNCTIONS

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions.

Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative

The behavior of any complex dynamic system is a natural result of the interaction between the components of that system. Important examples of these systems are biological models that describe the characteristics of complex interactions between certain organisms in a biological environment. The study of these systems requires the use of precise and advanced computational methods in mathematics. In this paper, we discuss a prey–predator interaction model that includes two competitive predators and one prey with a generalized interaction functional. The primary presumption in the model construction is the competition between two predators on the only prey, which gives a strong implication of the real-world situation. We successfully establish the existence and stability of the equilibria. Further, we investigate the impact of the memory measured by fractional time derivative on the temporal behavior. We test the obtained mathematical results numerically by a proper numerical scheme built using the Caputo fractional-derivative operator and the trapezoidal product-integration rule.

Evaluation of weighted fusion for scalar images in multi-sensor network

The regular image fusion method based on scalar has the problem how to prioritize and proportionally enrich image details in multi-sensor network. Based on multiple sensors to fuse and manipulate patterns of computer vision is practical. A fusion (integration) rule, bit-depth conversion, and truncation (due to conflict of size) on the image information are studied. Through multi-sensor images, the fusion rule based on weighted priority is employed to restructure prescriptive details of a fused image. Investigational results confirm that the associated details between multiple images are possibly fused, the prescription is executed and finally, features are improved. Visualization for both spatial and frequency domains to support the image analysis is also presented.

Iterative Method for a System of Nonlinear Fredholm Integral Equations

In this paper, the iteration method is proposed to solve a class of system of Fredholm-type nonlinear integral equations. First, the existence and uniqueness of solution are theoretically proven by the fixed-point theorem. Second, the approximation solution method is given by using the appropriate integration rule. The error analysis for the approximated solution with the exact solution is discussed for infinity-norm, and the rates of convergence are obtained. Furthermore, an iteration algorithm is constructed, and the convergence of the proposed numerical method is rigorously derived. Finally, some numerical examples are given to illustrate the theoretical results.

A Valid Dynamical Control on the Reverse Osmosis System Using the CESTAC Method

The aim of this study is to present a novel method to find the optimal solution of the reverse osmosis (RO) system. We apply the Sinc integration rule with single exponential (SE) and double exponential (DE) decays to find the approximate solution of the RO. Moreover, we introduce the stochastic arithmetic (SA), the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library instead of the mathematical methods based on the floating point arithmetic (FPA). Applying this technique, we would be able to find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorems are proved to support the method analytically. Based on these theorems, we can apply a new stopping condition in the numerical procedure instead of the traditional absolute error. These theorems show that the number of common significant digits (NCSDs) of exact and approximate solutions are almost equal to the NCSDs of two successive approximations. The numerical results are obtained for both SE and DE Sinc integration rules based on the FPA and the SA. Moreover, the number of iterations for various ε are computed in the FPA. Clearly, the DE case is more accurate and faster than the SE for finding the optimal approximation, the optimal error and the optimal iteration of the RO system.

A Novel Algorithm to Evaluate Definite Integrals by the Gauss-Legendre Integration Rule Based on the Stochastic Arithmetic: Application in the Model of Osmosis System

Finding the optimal iteration of Gaussian quadrature rule is one of the important problems in the computational methods. In this study, we apply the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library to find the optimal iteration and optimal approximation of the Gauss-Legendre integration rule (G-LIR). A theorem is proved to show the validation of the presented method based on the concept of the common significant digits. Applying this method, an improper integral in the solution of the model of the osmosis system is evaluated and the optimal results are obtained. Moreover, the accuracy of method is demonstrated by evaluating other definite integrals. The results of examples illustrate the importance of using the stochastic arithmetic in discrete case in comparison with the common computer arithmetic.