Numerical Solution of Ninth Order Boundary Value Problems by Petrov-Galerkin Method with Quintic B-splines as Basis Functions and Septic B-splines as Weight Functions

Viswanadham, K. N. S. Kasi; Reddy, S. M.

doi:10.1016/j.proeng.2015.11.470

Cited by 18 publications

(11 citation statements)

References 5 publications

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“…The linear operator is chosen to be the homogeneous part of the nonlinear operator N and the value of h is chosen as h ¼ À0:2. Table 7 shows the approximate solution values and the comparison of the absolute errors using the proposed method with those obtained by Viswanadham and Ballem [31]. Figures 1, 2, 3 , 4, 5, 6, 7, 8, 9, and 10 show the comparison of the approximate solutions to the exact solutions and the variation of the absolute errors over the interval of domain for Examples 1-5.…”

Section: Examplementioning

confidence: 92%

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An improved adaptation of homotopy analysis method

Sadaf

Akram

2017

Math Sci

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An improved adaptation of the well-known homotopy analysis method (HAM) is proposed to approximate the solutions of strongly nonlinear differential problems in terms of a rapidly convergent series. The proposed method involves simpler integrals and less computations than the standard HAM. The method is illustrated using different numerical examples. The comparative analysis confirms the applicability and efficiency of the proposed technique.

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Section: Examplementioning

confidence: 92%

“…The proposed method is numerically illustrated for the solutions of different higher order boundary value problems. Viswanadham and Ballem [31] used Galerkin method with septic B-splines to approximate the solution of tenth order boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

An improved adaptation of homotopy analysis method

Sadaf

Akram

2017

Math Sci

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“…Consider the following ninth-order nonlinear differential equation [12,14]     The analytic solution and the approximate solution obtained are depicted in Fig.3. There is a very good agreement and relationship between the approximate solutions obtained by 8th iterations using Galerkin weighted residual method and the exact solutions which are shown in Table 3.…”

Section: Investigation By Third Examplementioning

confidence: 99%

Concepts of Bezier Polynomials and its Application in Odd Higher Order Non-linear Boundary Value Problems by Galerkin WRM

Islam¹

2021

IJMSC

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Many different methods are applied and used in an attempt to solve higher order nonlinear boundary value problems (BVPs). Galerkin weighted residual method (GWRM) are widely used to solve BVPs. The main aim of this paper is to find the approximate solutions of fifth, seventh and ninth order nonlinear boundary value problems using GWRM. A trial function namely, Bezier Polynomials is assumed which is made to satisfy the given essential boundary conditions. Investigate the effectiveness of the current method; some numerical examples were considered. The results are depicted both graphically and numerically. The numerical solutions are in good agreement with the exact result and get a higher accuracy in the solutions. The present method is quit efficient and yields better results when compared with the existing methods. All problems are performed using the software MATLAB R2017a.

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“…Shen [33] derived the 8th order differential equation by bending and axial vibrations of an elastic beam. In [34][35][36][37] many authors built-up Quintic B-splines Collocation method, Sextic B-Spline Collocation Method, Galerkin Method with Quintic B-splines and Galerkin method with Septic B-splines for 8th order BVP. Wazwaz [38] used modified decomposition method to solve special 8th-order BVP's.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells

et al. 2019

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This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.

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Numerical Solution of Ninth Order Boundary Value Problems by Petrov-Galerkin Method with Quintic B-splines as Basis Functions and Septic B-splines as Weight Functions

Cited by 18 publications

References 5 publications

An improved adaptation of homotopy analysis method

An improved adaptation of homotopy analysis method

Concepts of Bezier Polynomials and its Application in Odd Higher Order Non-linear Boundary Value Problems by Galerkin WRM

Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells

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