Application of the Non-Polynomial Spline Approach to the Solution of the Burgers Equation'

Ramadan, Mohamed A.; El-Danaf, Talaat S.; Alaal, Faisal E.I. Abd

doi:10.2174/1874114200701010015

Cited by 27 publications

(12 citation statements)

References 18 publications

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“…Let

Z_{i}^{j}

be an approximation to u ( x i , t j ) obtained by the segment P i ( x , t j ) of the spline function passing through the points

((), x_{i} MathClass-punc, Z_{i}^{j})

and

((), x_{i MathClass-bin+ 1} MathClass-punc, Z_{i MathClass-bin+ 1}^{j})

. Each segment has the form

P_{i} (x MathClass-punc, t_{j}) MathClass-rel= a_{i} (t_{j}) {(x MathClass-bin- x_{i})}^{2} MathClass-bin+ b_{i} (t_{j}) (x MathClass-bin- x_{i}) MathClass-bin+ c_{i} (t_{j})

for each i = 0,1, … , N − 1. To obtain expressions for the coefficients of Equation in terms of

Z_{i MathClass-bin+ 1 MathClass-bin∕ 2}^{j}

D_{i}^{j}

, and

S_{i MathClass-bin+ 1 MathClass-bin∕ 2}^{j}

, we first define

P_{i} (x_{i MathClass-bin+ 1 MathClass-bin∕ 2} MathClass-punc, t_{j})

…”

Section: Derivation Of the Numerical Methodsmentioning

confidence: 99%

Computational method for solving space fractional Fisher's nonlinear equation

El-Danaf

Hadhoud

2013

Math. Meth. Appl. Sci.

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This paper aims to present a general framework of the quadratic spline functions to develop a numerical method for solving the nonlinear space fractional Fisher's equation. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm. The results reveal that the proposed approach is very effective, convenient, and quite accurate to such considered problems. Copyright © 2013 John Wiley & Sons, Ltd.

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“…Let

Z_{i}^{j}

be an approximation to u ( x i , t j ) obtained by the segment P i ( x , t j ) of the spline function passing through the points

((), x_{i} MathClass-punc, Z_{i}^{j})

and

((), x_{i MathClass-bin+ 1} MathClass-punc, Z_{i MathClass-bin+ 1}^{j})

. Each segment has the form

P_{i} (x MathClass-punc, t_{j}) MathClass-rel= a_{i} (t_{j}) {(x MathClass-bin- x_{i})}^{2} MathClass-bin+ b_{i} (t_{j}) (x MathClass-bin- x_{i}) MathClass-bin+ c_{i} (t_{j})

for each i = 0,1, … , N − 1. To obtain expressions for the coefficients of Equation in terms of

Z_{i MathClass-bin+ 1 MathClass-bin∕ 2}^{j}

D_{i}^{j}

, and

S_{i MathClass-bin+ 1 MathClass-bin∕ 2}^{j}

, we first define

P_{i} (x_{i MathClass-bin+ 1 MathClass-bin∕ 2} MathClass-punc, t_{j})

…”

Section: Derivation Of the Numerical Methodsmentioning

confidence: 99%

Computational method for solving space fractional Fisher's nonlinear equation

El-Danaf

Hadhoud

2013

Math. Meth. Appl. Sci.

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“…In [4], a method based the fractional shifted Legendre polynomials was applied to solve non-homogeneous space and time fractional partial differential equations (FPDEs), in which space and time fractional derivatives are described in the Caputo sense. In [5], some applications of the nonpolynomial spline approach to the solution of the Burgers' equation were studied. In [6], the non-…”

Section: -Introductionmentioning

confidence: 99%

An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

Hasan

Salim

2021

eijs

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The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

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“…The non-polynomial spline used for solving nonlinear partial differential equations was employed by many researchers. The most known and well-focused results are those presented by Ramadan et al (2005), who used a numerical method for approximation of Burger's equation [2].…”

Section: Introductionmentioning

confidence: 99%

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

Alaofi¹,

Ali²,

Alaal³

et al. 2021

AJCM

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In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.

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Application of the Non-Polynomial Spline Approach to the Solution of the Burgers Equation'

Cited by 27 publications

References 18 publications

Computational method for solving space fractional Fisher's nonlinear equation

Computational method for solving space fractional Fisher's nonlinear equation

An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

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