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 On the Singular Perturbations for Fractional Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Abdon Atangana
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Variational Iteration Method ◽  
Regular Perturbation ◽  
Homotopy Decomposition ◽  
New Development ◽  
The Laplace Transform ◽  
Nonhomogeneous Differential Equation ◽  
Fractional Differential ◽  
Linear Differential

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

Frequency Error Estimation in Nonlinear Systems Using Beats

1991 ◽  
Author(s):  
Richard Rodrigues Jettappa ◽  
Seshadri Sharma
Keyword(s):  
Nonlinear Systems ◽  
Linear System ◽  
Error Estimation ◽  
Scientific Community ◽  
Theory And Practice ◽  
The Other ◽  
Frequency Error ◽  
Non Linear ◽  
Linear Behaviour ◽  
Non Linear System

Abstract Real systems are inherently non-linear. True linear behaviour is rare. On the other hand, most of our models are linear. It is relatively easy to solve for a linear system compared to the corresponding non-linear system. This dichotomy between theory and practice is all pervasive. It is highly desirable that an analyst should be able to check the closeness of the predicted behavior with the actual behaviour of the system. In this paper we outline the method of beats towards this end. Even though the phenomenon of beats is well known to scientific community, its use in error analysis is unknown. This is the essence of the present paper.

scholarly journals Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1432
Author(s):  
Mutti-Ur Rehman ◽  
Jehad Alzabut ◽  
Arfan Hyder
Keyword(s):  
Dynamical Systems ◽  
Differential Inclusions ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Quadratic Stability ◽  
Linear Differential Inclusions ◽  
Non Linear ◽  
Linear Differential ◽  
The Stability ◽  
Non Linear System

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

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 Die ruimtelike vorm van 'n onelastiese, buigbare, geankerde kabel

1992 ◽  
Vol 11 (3) ◽  
pp. 94-101 ◽  
Author(s):  
T. P. Dreyer
Keyword(s):  
Unit Length ◽  
Linear Equations ◽  
Numerical Procedure ◽  
Continuous Model ◽  
The Other ◽  
End Point ◽  
Non Linear ◽  
Linear Differential ◽  
The One ◽  
External Forces

Consider an inelastic, perfectly flexible cable with given external forces acting on the total length of the cable. The one end-point is fixed in the origin and the other end-point is anchored at a given point (a;b;c) in space. The resulting configuration of the cable in space can be modelled by a system of non-linear differential equations. In this article it is shown that this continuous model of the cable can always be solved in terms of an integral. In the special case of a constant (i.e. independent of the position on the cable) external force per unit length the solution is given explicitly in terms of three constants that describe the tension at the origin. These three constants are determined by the boundary values a, b and c at the other end-point, and must be calculated in general by a numerical procedure from the three resulting simultaneous non-linear equations. A few applications of this method are shown.

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.

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