The Differential Transformation Method and Pade Approximant for a Form of Blasius Equation

Peker, Haldun Alpaslan; Karaoğlu, Onur; Oturanç, Galip

doi:10.3390/mca16020507

Cited by 20 publications

(16 citation statements)

References 22 publications

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“…Praks and Brkić [24] recently showed a Newton-Raphson iterative solution of the Colebrook equation based on Padé approximants [25][26][27][28][29]. Based on their solution, one simplified approach and a novel starting point that significantly reduces numerical error will be offered herein.…”

Section: 51mentioning

confidence: 99%

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Brkić

Praks

2019

Computation

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The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and Wright Ω functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence.

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Section: 51mentioning

confidence: 99%

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Brkić

Praks

2019

Computation

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“…These power series are often approximated by polynomials, nevertheless, polynomials tend to exhibit oscillations that may produce error bounds, also, the singularities of polynomials cannot be observed clearly in a finite plane [16,17]; hence, the transformation of the power series for numerical approximation using Pade approximants.…”

Section: Pade Approximant Of a Power Series Solutionmentioning

confidence: 99%

Numerical solutions of nonlinear biochemical model using a hybrid numerical-analytical technique

Edeki¹,

Owoloko²,

Osheku³

et al. 2015

ijma

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In this paper, a hybrid numerical-analytical technique resulting from the combination of the differential transformation method and the Pade approximation technique; hereby referred to as differential transformation-Pade approximation technique (DTPAT) is introduced and applied for numerical solutions to the nonlinear biochemical reaction model. The obtained numerical results via the DTPAT are in excellent agreement with those obtained using ADM, PIM, RK4, HPM, and MPHM. The DTPAT increases the convergent rate of the series solutions obtained via the DTM, is showed to be very effective; it requires less computational work, and hence a promising technique for both linear and nonlinear systems in other areas of medical and biomedical sciences.

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“…It is now well known that Pade approximants Baker (1975), Baker and Graves-Morris (1981) have the advantage of manipulating the polynomial approximation into rational functions of polynomials. It is therefore essential to combination of the series solution, obtained by the DTM with the Pade approximant Peker et al (2011), Rashidi and Erfani (2011) to provide an effective tool to handle boundary value problems at infinite domains.…”

Section: Introductionmentioning

confidence: 99%

DTM-Pade Approximants for MHD Flow with Suction/Blowing

2013

JAFM

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In this paper, we study theoretically the magnetic effect of Blasius equation with suction/blowing. The similarity transformations are applied to reduce the governing partial differential equations to a set of nonlinear ordinary differential equations in dimensionless form. A mathematical technique, namely the Differential Transform Method (DTM), is used to solve the nonlinear differential equations under appropriate boundary conditions, in the form of series with easily computable terms. Then, Pade approximants are applied to the solutions to increase the convergence of the given series. The combined DTM-Pade procedure is implemented directly without requiring linearization, discretization or perturbation. Graphical results are presented to investigate influence of the Magnetic field on the velocity profiles.

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The Differential Transformation Method and Pade Approximant for a Form of Blasius Equation

Cited by 20 publications

References 22 publications

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Colebrook’s Flow Friction Explicit Approximations Based on Fixed-Point Iterative Cycles and Symbolic Regression

Numerical solutions of nonlinear biochemical model using a hybrid numerical-analytical technique

DTM-Pade Approximants for MHD Flow with Suction/Blowing

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