Higher order trigonometric B-spline algorithms to the solution of coupled Burgers’ equation

Onarcan, Aysun Tok; Hepson, Özlem Ersoy

doi:10.1063/1.5020493

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…According to Table 5, it can be observed that the suggested method generates lower error compared to (Mittal and Arora, 2011). Results of the suggested method are also compared with that of the differential quadrature method (Mittal and Jiwari, 2012), Galerkin method (Kutluay and Ucar, 2013), modified cubic B-spline collocation method (Mittal and Tripathi, 2014) and quintic trigonometric B-spline collocation method (Onarcan and Hepson, 2018). The program is run until t = 3 and parameters are selected as N = 50 and Δ t = 0.01 for comparison with the methods in the list.…”

Section: Computational Resultsmentioning

confidence: 99%

“…The difficulty in the numerical method of CBE arises due to the nonlinear terms and viscosity parameters. Many numerical techniques have constructed to get the solution of the CBE fourth order accurate compact ADI scheme (Radwan, 1999), spectral collocation method using chebyshev polynomials (Khater et al , 2008), the Fourier pseudospectral method (Rashid and Ismail, 2009), the differential transform method via Taylor series formula (Liu and Hou, 2011), generalized differential quadrature method (Mokhtari et al , 2011), a robust technique for solving optimal control of CBE (Sadek and Kucuk, 2011), a differential quadrature method (Mittal and Jiwari, 2012), B-spline finite element method (Kutluay and Ucar, 2013; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al , 2016; Shallal et al , 2019; Onarcan and Hepson, 2018), a mesh free interpolation method (Islam et al , 2009), a fully implicit finite-difference method (Srivastava et al , 2013), a composite numerical scheme based on finite difference (Kumar and Pandit, 2014), logarithmic finite-difference method (Srivastava et al , 2014). Also, finite difference and differential quadrature methods (Bashan et al , 2015; Bashan, 2020; Ersoy et al , 2018; Karakoc et al , 2014; Ucar et al , 2019) are applied to Burgers and modified Burgers equations.…”

Section: Introductionmentioning

confidence: 99%

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A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

Hepson

2020

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Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

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Section: Computational Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

Hepson

2020

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“…Salih et al 21 have presented a new approach and methodology to solve one dimensional coupled viscous Burgers' equation with Dirichlet boundary conditions using cubic trigonometric B‐spline collocation method. Onarcan and Hepson 22 stated that trigonometric B‐spline functions of higher degrees have advantages over lower ones since they can be used as approximate functions in the numerical methods if the differential equation include higher order derivatives. They have used quintic trigonometric B‐splines to get numerical solutions of the coupled Burgers equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of the coupled Burgers equation by trigonometric B‐spline collocation method

Uçar

Yağmurlu

YİĞİT

2022

Math Methods in App Sciences

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In the present study, the coupled Burgers equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which cubic trigonometric and quintic B-splines are used as approximate functions. In order to support the present study, three test problems given with appropriate initial and boundary conditions are going to be investigated. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the error norms L 2 and L ∞ . A linear stability analysis of the approximation obtained by the scheme shows that the method is unconditionally stable.

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“…Also, they applied collocation of cubic B‐splines finite element method to solve symmetric regularized long wave equations . Furthermore, there are lots of paper about collocation method to solve partial differential equations using B‐splines . In this paper, we use the collocation method and interpolation method via cubic B‐spline functions and bicubic B‐spline functions to solve one dimensional and two dimensional nonlinear stochastic quadratic integral equations, respectively.…”

Section: Introductionmentioning

confidence: 99%

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

Mirzaee

Alipour

2019

Math Methods in App Sciences

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This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient. KEYWORDS collocation method, cubic and bicubic B-spline functions, nonlinear quadratic integral equations, stochastic integral equations MSC CLASSIFICATION 60H20; 97N50; 65D30; 41A15; 65D05 wileyonlinelibrary.com/journal/mma

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Higher order trigonometric B-spline algorithms to the solution of coupled Burgers’ equation

Cited by 7 publications

References 12 publications

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

Numerical solution of the coupled Burgers equation by trigonometric B‐spline collocation method

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

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