scholarly journals Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 924
Author(s):  
Marzieh Raei ◽  
Salvatore Cuomo ◽  
Giovanni Colecchia ◽  
Gerardo Severino
Keyword(s):  
Diffusion Coefficients ◽  
Reaction Diffusion ◽  
Point Method ◽  
Linear System Of Equations ◽  
Diffusion Dynamics ◽  
Closest Point Method ◽  
Non Linear ◽  
Variable Diffusion ◽  
And Performance ◽  
Non Linear System

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

scholarly journals Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1179 ◽  
Author(s):  
Kamel Al-Khaled ◽  
Ashwaq Hazaimeh
Keyword(s):  
Variational Iteration Method ◽  
The Other ◽  
Sinc Method ◽  
Linear System Of Equations ◽  
Non Linear ◽  
The Laplace Transform ◽  
Sturm Liouville ◽  
Linear Differential ◽  
Non Linear System ◽  
Adomian’S Polynomials

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

scholarly journals On Linear and Non-Linear Approximation In the Theory of Convective Heat Transfer

2021 ◽  
pp. 44-51
Author(s):  
M. Y. Davidzon
Keyword(s):  
Heat Transfer ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Stokes Equations ◽  
Linear Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear System Of Equations ◽  
Non Linear ◽  
Non Linear System

A system of linear equations that is currently widely used to describe convective heat transfer does not seem to be able to explain some experimental facts. One of the reasons for this may lie in using Newton’s and Fourier’s linear laws when deriving energy and Navier-Stokes equations. Replacing linear equations with nonlinear ones, as well as using an expression for surface heat flux density that is based on laws of physics instead of expressions called ‘cooling laws,’ would allow to solve a wider range of problems, and also would better agree with the experimental data. The use of proposed non-linear system of equations would also permit engineers in chemical, textile, defense, power, and other industries to design more economical and smaller-sized heat exchange devices.

scholarly journals A Fuzzy MLP Approach for Identification of Nonlinear Systems

2019 ◽  
Vol 65 (1) ◽  
pp. 44-53 ◽  
Author(s):  
A R Marakhimov ◽  
K K Khudaybergenov
Keyword(s):  
Linear Systems ◽  
Learning Algorithm ◽  
Back Propagation ◽  
Fuzzy Model ◽  
Performance Comparison ◽  
Neuro Fuzzy ◽  
Non Linear ◽  
Non Linear Systems ◽  
And Performance ◽  
Non Linear System

In case of decision making problems, identification of non-linear systems is an important issue. Identification of non-linear systems using a multilayer perceptron (MLP) trained with back propagation becomes much complex with an increase in number of input data, number of layers, number of nodes, and number of iterations in computation. In this paper, an attempt has been made to use fuzzy MLP and its learning algorithm for identification of non-linear system. The fuzzy MLP and its training algorithm which allows to accelerate a process of training, which exceeds in comparing with classical MLP is proposed. Results show a sharp reduction in search for optimal parameters of a neuro fuzzy model as compared to the classical MLP. A training performance comparison has been carried out between MLP and the proposed fuzzy-MLP model. The time and space complexities of the algorithms have been analyzed. It is observed, that number of epochs has sharply reduced and performance increased compared with classical MLP.

scholarly journals Existence and two-scale convergence of the generalised Poisson–Nernst–Planck problem with non-linear interface conditions

2020 ◽  
pp. 1-28
Author(s):  
V. A. KOVTUNENKO ◽  
A. V. ZUBKOVA
Keyword(s):  
Phase Interface ◽  
Interface Condition ◽  
Two Phase ◽  
Linear System Of Equations ◽  
Cross Diffusion ◽  
Non Linear ◽  
Convergence Method ◽  
Two Phases ◽  
Electric Flux ◽  
Non Linear System

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

scholarly journals Iterative least squares method for global positioning system

2011 ◽  
Vol 9 ◽  
pp. 203-208 ◽  
Author(s):  
Y. He ◽  
A. Bilgic
Keyword(s):  
Linear System ◽  
Least Squares ◽  
Global Positioning System ◽  
Least Squares Method ◽  
Positioning System ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Global Positioning ◽  
Non Linear ◽  
Non Linear System

Abstract. The efficient implementation of positioning algorithms is investigated for Global Positioning System (GPS). In order to do the positioning, the pseudoranges between the receiver and the satellites are required. The most commonly used algorithm for position computation from pseudoranges is non-linear Least Squares (LS) method. Linearization is done to convert the non-linear system of equations into an iterative procedure, which requires the solution of a linear system of equations in each iteration, i.e. linear LS method is applied iteratively. CORDIC-based approximate rotations are used while computing the QR decomposition for solving the LS problem in each iteration. By choosing accuracy of the approximation, e.g. with a chosen number of optimal CORDIC angles per rotation, the LS computation can be simplified. The accuracy of the positioning results is compared for various numbers of required iterations and various approximation accuracies using real GPS data. The results show that very coarse approximations are sufficient for reasonable positioning accuracy. Therefore, the presented method reduces the computational complexity significantly and is highly suited for hardware implementation.

A Multi-Agent Evolution Algorithm Used for Input Shaping of a Repetitive Non-Linear Dynamic System

2018 ◽  
Author(s):  
Niels C. Bender ◽  
Henrik C. Pedersen ◽  
Michael M. Bech ◽  
Torben O. Andersen
Keyword(s):  
Differential Evolution Algorithm ◽  
Electrical Energy ◽  
Performance Criteria ◽  
Fast Switching ◽  
Optimal Decisions ◽  
Non Linear ◽  
Multi Agent ◽  
Evolution Algorithm ◽  
And Performance ◽  
Non Linear System

This paper explores the challenges regarding designing a heuristic control algorithm for a dynamic non-linear system with multiple inputs and outputs. The presented algorithm aims to shape the voltage input (both magnitude and timing) applied to fast switching valves in a Digital Displacement® unit. This consists of multiple sub-systems, where optimal decisions must be made, based on the system design and performance criteria. In this regard good performance are defined as: low electrical energy required for switching, accurate switching timing and low plunger velocity near the seat. The proposed algorithm examines the design-space in a user-defined manner combined with stochastic decision making. The randomness of the algorithm is based on the standard deviation between located elite designs. This reveals several feasible input sequences to achieve the goal, and the optimums are benchmarked with a differential evolution algorithm. The techniques are demonstrated by simulation and the results compared showing similar performance of the optimums.

scholarly journals A COMBINATORIAL APPROACH TO DECIDE INITIAL ROOT VALUE FOR THE SOLUTION OF NON-LINEAR SYSTEM OF EQUATIONS

2019 ◽  
Vol 1228 ◽  
pp. 012063
Author(s):  
Gananathan Ramaiyan ◽  
S.M Sulaiman. ◽  
R Irene Hepzibah.
Keyword(s):  
Linear System ◽  
System Of Equations ◽  
Combinatorial Approach ◽  
Linear System Of Equations ◽  
Non Linear ◽  
Non Linear System ◽  
Initial Root
Sign in / Sign up

Export Citation Format

Share Document