scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

scholarly journals APPROXIMATE SOLUTIONS OF SOME NONLINEAR PROBLEMS FOR THE MONGE-AMPERE EQUATIO

2020 ◽  
pp. 97-105
Author(s):  
N.B. Iskakova ◽  
◽  
А.S. Rysbek ◽  
N.S. Serik ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Initial Conditions ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Mathcad Software ◽  
The Right ◽  
Monge Ampere

Due to numerous applications in various fields of science, including gas dynamics, meteorology, differential geometry, and others, the Monge – ampere equation is one of the most intensively studied equations of nonlinear mathematical physics.In this report, we study a nonlinear boundary value problem for the inhomogeneous Monge-ampere equation, the right part of which contains power nonlinearities in derivatives and arbitrary nonlinearity from the desired function.Based on linearization, the studied boundary value problems are reduced to a system of ordinary first-order differential equations with initial conditions that depend on the parameter.Methods for constructing exact and approximate solutions of some boundary value problems for the Monge-ampere equation are proposed.Using the Mathcad software package, numerical implementation of methods for constructing approximate solutions of the obtained systems of ordinary differential equations with a parameter is performed.Three-dimensional graphs of exact and approximate solutions of the problems under consideration in the Grafikus service are constructed.

scholarly journals Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

2021 ◽  
Vol 2128 (1) ◽  
pp. 012035
Author(s):  
W. Abbas ◽  
Mohamed Fathy ◽  
M. Mostafa ◽  
A. M. A Hesham
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Nonlinear Boundary ◽  
Analytic Solution ◽  
Boundary Value ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Algebraic Set ◽  
The Galerkin Method

Abstract In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

scholarly journals Bezier Polynomials with Applications

2018 ◽  
Vol 66 (2) ◽  
pp. 157-162
Author(s):  
Nazrul Islam ◽  
Md Shafiqul Islam
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Boundary ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Tabular Form ◽  
Nonlinear Boundary Value Problems ◽  
Matrix Formulation ◽  
Linear And Nonlinear ◽  
The Galerkin Method ◽  
Linear And Nonlinear Systems

In this paper, we use the Galerkin technique for solving higher order linear and nonlinear boundary value problems (BVPs). The well-known Bezier polynomials are exploited as basis functions in the technique. To use the Bezier polynomials, we need to satisfy the corresponding homogeneous form of the boundary conditions and modification is thus needed. A rigorous matrix formulation is developed by the Galerkin method for linear and nonlinear systems and solved it using Bezier polynomials. The approximate solutions are compared to the exact solutions through tabular form. All problems are computed using the software MATHEMATICA. Dhaka Univ. J. Sci. 66(2): 157-162, 2018 (July)

Numerical Methods for Time-Dependent, Nonlinear Boundary Value Problems

1971 ◽  
Vol 11 (04) ◽  
pp. 374-388 ◽  
Author(s):  
W.E. Culham ◽  
Richard S. Varga
Keyword(s):  
Numerical Methods ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Analytic Solution ◽  
Higher Order ◽  
Spatial Variable ◽  
Galerkin Methods ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
The Galerkin Method

Abstract This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear boundary value problems. This class of problems includes the equation that describes the flow of a fully compressible fluid in a porous medium. The Galerkin method with several different piecewise polynomial subspaces and a non-Galerkin piecewise polynomial subspaces and a non-Galerkin method specifically employing cubic spline functions are used to approximate the solution of a nonlinear parabolic equation with one spatial variable. With a parabolic equation with one spatial variable. With a known analytic solution of the problem, the accuracies of these approximations are determined and compared with conventional finite-difference approximations. Specially, the various methods are compared on the basis of the amount of computer time necessary to achieve a given accuracy, as well as with respect to the order oil convergence and computer core storage required. These tests indicate that the higher-order Galerkin methods require the least amount of computer time for a given range of accuracy. Introduction The purpose of this paper is to outline in detail the application of the Galerkin method, employing piecewise polynomials, to solve nonlinear piecewise polynomials, to solve nonlinear boundary-value problems and compare the computational efficiency of the Galerkin method with more conventional numerical methods. Numerical methods compared with the Galerkin technique include a non-Galerkin method that utilizes cubic spline interpolation and the conventional finite-difference methods. Four conventional time approximations were also studied in conjunction with the above mentioned space discretization methods. In an earlier paper, Price and Varga showed theoretically that higher-order approximations to certain semilinear convection-diffusion equations were possible by means of Galerkin techniques, but complete numerical results for such approximations were not given. Also, in a paper that introduced the Galerkin method to the petroleum industry, Price et al. demonstrated that higher-order approximations were far superior numerically to the conventional methods used to solve certain linear convection-diffusion type problems. Jennings, Douglas and Dupont and Douglas et al. have considered the application of Galerkin methods to various nonlinear problems, but again complete numerical results, problems, but again complete numerical results, including comprehensive comparisons with existing numerical methods, were not given. Thus, in addition to presenting some new and computationally efficient Galerkin formulations for nonlinear problems and numerically demonstrating their problems and numerically demonstrating their higher order accuracies, it was also desirable to test these. methods to determine if they also exhibited the same superiority in regard to computational efficiency as was demonstrated for the Galerkin methods applied to linear problems. If so, then the Galerkin technique could prove to be an important advancement toward developing faster numerical models for field application. To test and compare each method of solution, a problem involving the nonlinear gas-flow equation problem involving the nonlinear gas-flow equation in one spatial variable with a specific volumetric source term was chosen, for which a closed-form or analytic solution was known. Using this particular problem and its analytic solution, it was possible problem and its analytic solution, it was possible to determine numerically the order of convergence of each method, to compare each method on the basis of computer time expended to obtain a given accuracy, and to compare each method with respect to computer core storage required. In addition, the experimental data were used to define "consistent quadrature" and "consistent interpolation" schemes for the Galerkin methods. Finally, it was possible to formulate conclusions regarding the computational efficiency of the four time approximations investigated. SPEJ P. 374

Haar wavelet based numerical solution of nonlinear differential equations arising in fluid dynamics

2016 ◽  
Vol 05 (02) ◽  
pp. 1650010 ◽  
Author(s):  
S. C. Shiralashetti ◽  
M. H. Kantli ◽  
A. B. Deshi
Keyword(s):  
Fluid Dynamics ◽  
Differential Equations ◽  
Error Analysis ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Elastohydrodynamic Lubrication ◽  
Haar Wavelet ◽  
Nonlinear Boundary Value Problems

In this paper, we obtained the Haar wavelet-based numerical solution of the nonlinear differential equations arising in fluid dynamics, i.e., electrohydrodynamic flow, elastohydrodynamic lubrication and nonlinear boundary value problems. Error analysis is observed, it shows that the Haar wavelet-based results give better accuracy than the existing methods, which is justified through illustrative examples.

scholarly journals Solution of nonlinear problems by a new analytical technique using Daftardar-Gejji and Jafari polynomials

2019 ◽  
Vol 11 (12) ◽  
pp. 168781401989696 ◽  
Author(s):  
Liaqat Ali ◽  
Saeed Islam ◽  
Taza Gul ◽  
Iraj Sadegh Amiri
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Term ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Nonlinear Boundary Value Problems ◽  
Analytical Technique ◽  
Homotopy Perturbation

This article shows the solution of nonlinear differential equations by a new analytical technique called modified optimal homotopy perturbation method. Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the equation. Four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders are solved by modified optimal homotopy perturbation method as well as optimal homotopy perturbation method. The achieved consequences are authenticated by comparison with the results gained by the existing method—optimal homotopy perturbation method. The method consists of few steps and gives better results. The easy applicability and fast convergence are goals of the applied technique. The applied technique has fewer limitations and can be used for the phenomena containing ordinary differential equation, partial differential equation, integro-differential equation, and their systems.

A Novel Method for Nonlinear Boundary Value Problems Based on Multiscale Orthogonal Base

2021 ◽  
pp. 2150036
Author(s):  
Yingchao Zhang ◽  
Liangcai Mei ◽  
Yingzhen Lin
Keyword(s):  
Differential Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Boundary Value Problems ◽  
Numerical Examples ◽  
Second Order Differential Equations ◽  
Orthogonal Base ◽  
Novel Method

In this paper, a new algorithm is presented to solve the nonlinear second-order differential equations. The approach combines the Quasi-Newton’s method and the multiscale orthogonal bases. It is worth mentioning that the convergence of Newton’s method is verified for solving the nonlinear differential equations. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and feasibility of the proposed method.

scholarly journals Approximate Solutions of Fractional Riccati Equations Using the Adomian Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Fei Wu ◽  
Lan-Lan Huang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Riccati Equations ◽  
Crucial Step ◽  
Adomian Decomposition

The fractional derivative equation has extensively appeared in various applied nonlinear problems and methods for finding the model become a popular topic. Very recently, a novel way was proposed by Duan (2010) to calculate the Adomian series which is a crucial step of the Adomian decomposition method. In this paper, it was used to solve some fractional nonlinear differential equations.

A NEW ALGORITHM FOR SOLVING NONLINEAR BOUNDARY VALUE PROBLEMS ARISING IN HEAT TRANSFER

2011 ◽  
Vol 02 (03) ◽  
pp. 355-373 ◽  
Author(s):  
S. S. MOTSA
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Closed Form ◽  
Boundary Value Problems ◽  
Nonlinear Equations ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Analytic Solutions ◽  
Nonlinear Boundary Value Problems

In this paper, a very efficient and easy-to-use successive linearization approach for solving nonlinear differential equations is proposed. The implementation of the method is demonstrated by solving three nonlinear differential equations of different complexities arising in heat transfer. New closed form explicit analytic solutions of some of the governing nonlinear equations are obtained and compared with the results of the proposed method and with numerical solutions from the MATLAB in-built routine bvp4c.

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

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