Numerical solutions of coupled systems of nonlinear elliptic equations

Boglaev, Igor

doi:10.1002/num.20648

Cited by 6 publications

(4 citation statements)

References 7 publications

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“…Centroidal mean selected in developing this proposed scheme which is centroidal mean (M). This combination of Euler (E) and mean (M) produces proposed scheme known as Centroidal Polygon (CP) (E+M) [10], [11].…”

Section: Methodsmentioning

confidence: 99%

“…This analysis will use four different schemes: The Polygon scheme, Harmonic-Polygon scheme, Cube-Polygon scheme, and Centroidal Polygon scheme. The researcher will evaluate the linear equation to ensure that the equation is computable before continuing the non-linear equation [11]. The researcher will construct the mathematical program method as follows before beginning to test the schemes into the non-linear first order ODE: The Centroidal Polygon scheme will be compared to previous schemes using the SCILAB 6.0 program.…”

Section: Fig 1 -The Centroidal-polygon Scheme Formmentioning

confidence: 99%

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Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

Zulkifli

Samsudin

et al. 2022

IJIE

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The Euler method is one of the oldest methods to solve differential equation problems. The Euler method produces the simplest solution. However, although is not computationally expensive, the Eulermethod is lack of accuracy. To improve the Euler method, the researcher proposed a new scheme for better accuracy. The Euler method equation and the mean method were combined to enhance this method. As the improvement basis, the researcher used the Centroidal mean and the midpoint method or Polygon to improve the Euler method. The combination of the Euler and Centroidal mean is known as Centroidal Polygon (CP) scheme. The CP scheme was used to solve first-order non-linear Ordinary Differential Equations (ODE). The researcher used SCILAB 6.0 software to solve the equation and the CP scheme was tested in three different step sizes (0.1,0.01, and 0.001). Aside from that, the researcher had compared the CP scheme with previous schemes such as ZulZamri's Polygon (P) scheme, Nurhafizah's Harmoni-Polygon (HP) scheme, and Nooraida's Cube-Polygon (CuP) scheme to ensure that the CP scheme is more accurate than previous research. When the maximum error is calculated by subtracting the scheme and exact solution, the results show that the CP scheme delivers the highest accuracy results in the shortest amount of time. The new enhanced, modified Euler method is useful for other researchers to achieve good accuracy at low computational cost as an alternative to the more computationally expensive methods.

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Section: Methodsmentioning

confidence: 99%

Section: Fig 1 -The Centroidal-polygon Scheme Formmentioning

confidence: 99%

Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

Zulkifli

Samsudin

et al. 2022

IJIE

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“…The aim of this paper is to obtain an existence and uniqueness of the solutions on any bounded Lipschitz domain, in addition to carry out a new meshless (WEB-S) numerical method for the approximate solutions of the system. On an arbitrary domain finite element approximation of these equations is a highly sought after approach to obtain their numerical solution;In particular WEB-FEM combines the computational advantages of B-Splines and standard mesh-based finite elements.Further it attains the degree and smoothness to be chosen flexibly without substantially increasing the size of problem.Off late spectral Galerkin method [2], Lattice-Boltzmann schemes [3] etc have been used for numerical approximations.In [4] Boglaev have used method of upper and lower solutions , and construct monotone sequence for difference scheme to approximate the solution of coupled system and Xiu et.al [5] used stochastic Galerkin and stochastic collocation method in conjunction with the gPC expansions. In view of the computational advantages WEBS-FEA is one of highly desired approach to solve the coupled elliptic system.In the current literature no work is reported on WEBS-FEA of the general coupled elliptic problem.…”

Section: Introductionmentioning

confidence: 99%

Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

Chakraborty

Kumar

2018

Int J Adv Eng Sci Appl Math

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In this study we establish the existence and uniqueness of the solution of a coupled system of general elliptic equations with anisotropic diffusion , non-uniform advection and variably influencing reaction terms on Lipschitz continuous domain Ω ⊂ R m (m≥1) with a Dirichlet boundary. Later we consider the finite element (FE) approximation of the coupled equations in a meshless framework based on weighted extended B-Spine functions (WEBS).The a priori error estimates corresponding to the finite element analysis are derived to establish the convergence of the corresponding FE scheme and the numerical methodology has been tested on few examples.

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“…The application of the method of upper and lower solutions to coupled systems has more complexity. For solving coupled systems of nonlinear elliptic equations, monotone iterative methods based on the method of upper and lower solutions have been developed in [9] for continuous problems and in [4], [5], [6], [17] for discrete problems.…”

Section: Introductionmentioning

confidence: 99%

Block monotone iterative methods for solving coupled systems of nonlinear elliptic problems

Al-Sultani¹

2018

Preprint

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This paper investigates numerical methods for solving coupled system of nonlinear elliptic problems. We utilize block monotone iterative methods based on Jacobi and Gauss-Seidel methods to solve difference schemes which approximate the coupled system of nonlinear elliptic problems, where reaction functions are quasimonotone nondecreasing. In the view of upper and lower solutions method, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence of solutions to problems with quasimonotone nondecreasing reaction functions. Construction of initial upper and lower solutions is presented. The sequences of solutions generated by the block Gauss-Seidel method converge not slower than by the block Jacobi method.

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Numerical solutions of coupled systems of nonlinear elliptic equations

Cited by 6 publications

References 7 publications

Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

Block monotone iterative methods for solving coupled systems of nonlinear elliptic problems

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