Analysis of IVPs and BVPs on Semi‐Infinite Domains via Collocation Methods

Maleki, Mohammad; Hashim, Ishak; Abbasbandy, Saeid

doi:10.1155/2012/696574

Cited by 15 publications

(12 citation statements)

References 41 publications

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“…The reported digits are within the convergence tolerance of the shooting iteration. For comparison, the previous benchmark values reported by [19] and [54] Table 5. The intercept at 1/(n + 1)=0 is 2.611541077.…”

Section: Numerical Solution Of the Flierl-petviashvili Problemmentioning

confidence: 99%

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Barlow¹,

Stanton²,

Hill³

et al. 2017

Q J Mechanics Appl Math

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A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the 1 method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.

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Section: Numerical Solution Of the Flierl-petviashvili Problemmentioning

confidence: 99%

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

Barlow¹,

Stanton²,

Hill³

et al. 2017

Q J Mechanics Appl Math

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“…Only Abbasbandy and Maleki et al. among previous authors obtained results accurate to three decimal places or better.…”

Section: Introductionmentioning

confidence: 89%

The Kidder Equation:

Iacono

Boyd

2014

Stud Appl Math

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The Kidder problem is u x x + 2x(1 − αu) −1/2 u x = 0 with u(0) = 1 and u(∞) = 0 where α ∈ [0, 1]. This looks challenging because of the square root singularity. We prove, however, that |u(x; α) − erfc(x)| ≤ 0.046 for all x, α. Other very simple but very accurate curve fits and bounds are given in the text; |u(x; α) − erfc(x + 0.15076x/(1 + 1.55607x 2 ))| ≤ 0.0019. Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are O(10 −12 ) when the coefficients a n ∼ constant/n −6 . An initial-value problem is obtained if u x (0, α) is known; the slope Chebyshev series has only a fourth-order rate of convergence until a simple change-of-coordinate restores a geometric rate of convergence, empirically proportional to exp(−n/8). Kidder's perturbation theory (in powers of α) is much inferior to a delta-expansion given here for the first time. A quadratic-over-quadratic Padé approximant in the exponentially mapped coordinate z = erf(z) predicts the slope at the origin very accurately up to about α ≈ 0.8. Finally, it is shown that the singular case u(x; α = 1) can be expressed in terms of the solution to the Blasius equation.

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“…The partial integration and the p th derivative of the function at any points over 1 [ , ] m m t t t + ∈ can be approximated using (6) - (8), which come from their exact derivation using (4) and (5).…”

Section: Unified Quadrature Formulas Using Spline Interpolationmentioning

confidence: 99%

“…This possibility will be demonstrated using two different kinds of node. First, more evenly distributed nodes generated using the coordinate transformation method shown in [8] are used. Second, uniform nodes are selected to show that arbitrary nodes can be utilized in real applications without limits on the number of nodes and without numerical failures using the spline interpolation.…”

Section: Extension To Spline Interpolationmentioning

confidence: 99%

“…CQFs for spectral methods are generally derived using the orthogonal properties among interpolating polynomials. Many of them, expressed in an analytic form, are easily accessible in the literature [8][9][10]. Derivation of CQFs generally requires rigorous application of complex function theories and is strictly dependent on the types of interpolating functions and quadrature nodes, which may cause inconvenience when they are required for other interpolating functions and quadrature nodes different from the standard ones.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

Kim¹,

Park²,

Sung³

2015

Journal of Electrical Engineering and Technology

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-A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

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Analysis of IVPs and BVPs on Semi‐Infinite Domains via Collocation Methods

Cited by 15 publications

References 41 publications

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

The Kidder Equation:

A New Unified Scheme Computing the Quadrature Weights, Integration and Differentiation Matrix for the Spectral Method

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