Fourth-Order Finite-Difference Method for Third-Order Boundary-Value Problems

Salama, Amgad; Mansour, Awad R.

doi:10.1080/10407790590907903

Cited by 22 publications

(14 citation statements)

References 14 publications

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“…All numerical tests were performed using prescribed error Table 1, given to 6 significant figures, alongside values obtained by Salama and Mansour (2005b) and Asaithambi (2005Asaithambi ( , 2004a. The results obtained with the current explicit formulation are in almost exact agreement with those obtained in previous studies, in most cases coinciding up to 5 significant figures for the full range of b considered.…”

Section: Results For Explicit Third-order Finite Difference Schemesupporting

confidence: 77%

“…Figure 1 shows normalized velocity profiles obtained using the current explicit scheme, for select values of b, at a grid resolution of h = 0.00125. Visual inspection shows these profiles to be in close agreement with others presented in the literature, e.g., Salama and Mansour (2005b), Schlichting and Gersten (2000) and Cebeci and Bradshaw (1977). Close inspection of the data shows that for higher values of b, a smaller ∞ η produces an optimum solution, as noted in the literature (see, e.g., Asaithambi (2005); Salama and Mansour (2005b)).…”

Section: Results For Explicit Third-order Finite Difference Schemesupporting

confidence: 72%

“…In some cases, additional 'fictitious' end points are added, depending on the accuracy and range of applicability of the particular numerical model proposed. Also, in addition to the mathematical complexity often involved, the numerical methods proposed tend to require significant computational time, as noted by Asaithambi (1998) and Salama and Mansour (2005b).…”

Section: Paper Received 25 October 2010 Paper Accepted 21 June 2011mentioning

confidence: 99%

See 2 more Smart Citations

Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes

Duque-Daza¹,

Lockerby²,

Galeano³

2011

J. Braz. Soc. Mech. Sci. & Eng.

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Section: Results For Explicit Third-order Finite Difference Schemesupporting

confidence: 77%

Section: Results For Explicit Third-order Finite Difference Schemesupporting

confidence: 72%

Section: Paper Received 25 October 2010 Paper Accepted 21 June 2011mentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes

Duque-Daza¹,

Lockerby²,

Galeano³

2011

J. Braz. Soc. Mech. Sci. & Eng.

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“…Even though the problem is almost a century old, recent papers that employ the Blasius problem as an example include [2,1,5,6,11,15,16,21,18,17,23,25,26,27,28,29,30,32,33,34,36].…”

Section: Because All Fluid Flows Must Be Zero At a Solid Boundary Thmentioning

confidence: 99%

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Boyd¹

2008

SIAM Rev.

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Abstract. The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semiinfinite flat plate. The Blasius function is the solution to 2fxxxWe use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of "particularity." In spite of these triumphs, many properties of the Blasius function f (x) are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.

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“…Some numerical and analytical methods such as shooting method [5], Finite-element method [6], sincGalerkin method [7], finite-difference method [8], Adomian technique [9], variational iteration method (VIM) [10], homotopy perturbation method (HPM) [11], analysis method (HAM) [12], have been studied for obtaining approximate solutions to boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains

Biazar¹,

Ghanbari²

2011

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Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the following type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to adjust and control the convergence region and rate of convergence of the obtained series solutions, by defining the so-called control parameter , is provided. This paper aims to propose an efficient way of finding the proper values of . Such values of parameter can be determined at the any order of approximations of HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical results confirm that obtained series solutions agree very well with the exact solutions.

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Fourth-Order Finite-Difference Method for Third-Order Boundary-Value Problems

Cited by 22 publications

References 14 publications

Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes

Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

A New Analytical Approach for Solving Nonlinear Boundary Value Problems in Finite Domains

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