A new cubic B-spline method for approximating the solution of a class of nonlinear second-order boundary value problem with two dependent variables

Lang, Feng-Gong; Xu, Xiaoping

doi:10.2306/scienceasia1513-1874.2014.40.444

Cited by 7 publications

(6 citation statements)

References 15 publications

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“…This section describes the classical cubic B-spline approximation and the new secondorder approximation invented by Lang and Xu [42]. Let the finite interval [a, b], where a = x 0 < .…”

Section: Preliminary Conceptsmentioning

confidence: 99%

“…The second derivatives, W (x j ) and Z (x j ) can be simplified as S j and R j , respectively. Subsequently, the new approximation for second-order derivatives can be represented as follows [42,49]:…”

Section: Preliminary Conceptsmentioning

confidence: 99%

“…The new symmetric cubic B-spline method (NCBM) was first studied by Lang and Xu in [42] to solve non-linear second-order BVPs with two dependent variables. The NCBM is a typical CBM, equipped with a new second-order derivative approximation.…”

Section: Introductionmentioning

confidence: 99%

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An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

et al. 2023

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This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method’s accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient.

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“…This section describes the classical cubic B-spline approximation and the new secondorder approximation invented by Lang and Xu [42]. Let the finite interval [a, b], where a = x 0 < .…”

Section: Preliminary Conceptsmentioning

confidence: 99%

Section: Preliminary Conceptsmentioning

confidence: 99%

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An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

et al. 2023

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“…The truncation error in Ẅm is O(h 2 ). Therefore, instead of using (8), we shall apply the following O(h 3 ) approximation for a second-order derivative [22,23]:…”

Section: Cubic B-spline Functionsmentioning

confidence: 99%

Numerical Solutions of Third-Order Time-Fractional Differential Equations Using Cubic B-Spline Functions

et al. 2022

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Numerous fields, including the physical sciences, social sciences, and earth sciences, benefit greatly from the application of fractional calculus (FC). The fractional-order derivative is developed from the integer-order derivative, and in recent years, real-world modeling has performed better using the fractional-order derivative. Due to the flexibility of B-spline functions and their capability for very accurate estimation of fractional equations, they have been employed as a solution interpolating polynomials for the solution of fractional partial differential equations (FPDEs). In this study, cubic B-spline (CBS) basis functions with new approximations are utilized for numerical solution of third-order fractional differential equation. Initially, the CBS functions and finite difference scheme are applied to discretize the spatial and Caputo time fractional derivatives, respectively. The scheme is convergent numerically and theoretically as well as being unconditionally stable. On a variety of problems, the validity of the proposed technique is assessed, and the numerical results are contrasted with those reported in the literature.

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“…The truncation error in € U m is O(h 2 ). Therefore, instead of using equation (2.5), we shall use the following O(h 3 ) approximation for the second-order derivative (Lang and Xu, 2014;Iqbal et al, 2018):…”

Section: Cubic B-spline Functionsmentioning

confidence: 99%

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

Nazir

Abbas

Iqbal

2020

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Purpose The purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory. Design/methodology/approach The system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously. Findings A different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations. Originality/value The current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

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A new cubic B-spline method for approximating the solution of a class of nonlinear second-order boundary value problem with two dependent variables

Cited by 7 publications

References 15 publications

An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs

Numerical Solutions of Third-Order Time-Fractional Differential Equations Using Cubic B-Spline Functions

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

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