Shifted Jacobi collocation method for solving multi-dimensional fractional Stokes’ first problem for a heated generalized second grade fluid
- Mohamed A Abdelkawy^{1, 2}Email author and
- Rubayyi T Alqahtani^{1}
https://doi.org/10.1186/s13662-016-0845-z
© Abdelkawy and Alqahtani 2016
Received: 20 February 2016
Accepted: 18 April 2016
Published: 26 April 2016
Abstract
This paper reports a new spectral collocation technique for solving fractional Stokes’ first problem for a heated generalized second grade fluid (FSFP-HGSGF). We develop a collocation scheme to approximate FSFP-HGSGF by means of the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) and shifted Jacobi-Gauss-Radau collocation (SJ-GR-C) methods. The discussed numerical tests illustrate the capability and high accuracy of the proposed methodologies.
Keywords
1 Introduction
In recent years, spectral methods (see [1–8]) have often turned out to be efficient and highly accurate schemes when compared with the local methods. The speed of convergence is one of the great advantages of spectral methods. Besides, spectral methods have exponential rates of convergence; they also have a high level of accuracy. The main idea of all versions of spectral methods is to express the approximate solution of the problem as a finite sum of certain basis functions (orthogonal polynomials or a combination of them) and then choose the coefficients in order to minimize the difference between the exact and approximate solutions as well as possible. The spectral collocation method is a specific type of spectral methods, which is more applicable and widely used to solve almost all types of the differential equations [9–12].
The Stokes’ problem based on the Navier-Stokes theory was studied in several articles; see for example [13–21]. Recently, the first problem of Stokes has become more significant due to its applications. The flow of an Oldroyd-B fluid over a suddenly moved flat plate has been described by Stokes’ first problem in [22]. The first problem of Stokes for Oldroyd-B fluid in a porous half-space and a heated boundary second grade fluid in a porous half-space has been discussed by Tan et al. [23, 24]. Shen et al. [25] introduced the fractional derivative model of the Rayleigh-Stokes problem for a heated generalized second grade. The numerical study based on Laguerre-Galerkin method [26] has been introduced for the first problem of Stokes, which describes a Newtonian fluid in a non-Darcian porous half-space.
Fractional calculus [27–36] is a branch of calculus theory, which makes partial differential equations more convenient to describe many phenomena in several fields such as fluid mechanics, chemistry [33, 34], biology [35], viscoelasticity [36], engineering, finance, and physics [37] fields. Bhrawy et al. [38] proposed an accurate Jacobi collocation algorithm for the systems of high-order linear differential-difference equations with mixed initial conditions. The Jacobi pseudospectral method has been discussed by Bhrawy et al. [39] to solve a class of functional-differential equations. Moreover, Bhrawy et al. [40] introduced a combination of Jacobi Gauss-Lobatto and Gauss-Radau collocation algorithms for solving the fractional Fokker-Planck equations. In this paper, the SJ-GL-C and SJ-GR-C methods are proposed to solve multi-dimensional FSFP-HGSGF. The solution of such equation is approximated by means of a finite expansion of shifted Jacobi polynomials for independent variables. The proposed collocation scheme is investigated for both temporal and spatial discretizations. Then we evaluate the residuals of the FSFP-HGSGF at the shifted Jacobi-Gauss-Lobatto (SJ-GL) and shifted Jacobi-Gauss-Radau (SJ-GR) quadrature points. Thereby, the problem is reduced to a system of algebraic equations which is far easier to solve. Indeed, with the freedom to select the shifted Jacobi indices \(\alpha >-1\) and \(\beta >-1\), the method can be calibrated for a wide variety of problems. To the best of our knowledge, there are no results on SJ-GL-C and SJ-GR-C methods for FSFP-HGSGF.
This paper is organized as follows. A few facts of shifted Jacobi polynomials and fractional calculus are listed in Section 2. In Section 3, we introduce a new collocation method for the one-dimensional space FSFP-HGSGF. In Section 4, the proposed scheme is successfully extended to solve the two-dimensional space FSFP-HGSGF. Section 5 is used to solve several problems. A conclusion is given in the last section.
2 Mathematical preliminaries
2.1 Fractional calculus
The fractional integration definition of order \(\nu>0\), can be expressed by several formulas and in general they are not equal to each other. The most often used definitions are the Caputo and Riemann-Liouville definitions.
Definition 2.1
Definition 2.2
Definition 2.3
2.2 Properties of shifted Jacobi polynomials
3 One-dimensional space of fractional Stokes problem
In this section, we introduce a numerical algorithm based on the SJ-GR-C and SJ-GL-C methods for solving numerically one-dimensional FSFP-HGSGF. The collocation points are selected at the SJ-GR and SJ-GL interpolation nodes for temporal and spatial variables, respectively. The core of the proposed method consists of discretizing the one-dimensional FSFP-HGSGF to create a system of algebraic equations of the unknown coefficients. This system can then easily be solved with a standard numerical scheme.
We are interested in using the SJ-GL-C and SJ-GR-C methods to transform the previous FSFP-HGSGF into a system of algebraic equations. In order to do this, we approximate the independent space variable x using the SJ-GL-C method at the \(x^{(\alpha_{1},\beta_{1})}_{L,N,i}\) nodes, while the independent temporal variable t was approximated by the SJ-GR-C methods. The nodes are the set of points in a specified domain where the dependent variable values are to be approximated. In general, the choice of the location of the nodes is optional. However, taking the roots of the shifted Jacobi orthogonal polynomials, referred to as shifted Jacobi collocation points, gives particularly accurate solutions for the spectral methods.
4 Two-dimensional space of fractional Stokes problem
5 Numerical results and comparisons
This section listed several numerical examples to demonstrate the accuracy of the proposed method. Also, we compare our numerical results with the existing numerical results [41–44]. The obtained results of these examples show that the proposed method, by selecting a few number nodes, has a high level of accuracy.
Example 1
The MAEs of Example 1
(N,M) | Our method with \(\alpha _{1}=\beta _{1}=\frac{1}{2}\), \(\alpha _{2}=\beta _{2}=0\) and several choices of γ | ||
---|---|---|---|
0.5 | 0.6 | 0.7 | |
(8,8) | 4.40262 × 10^{−5} | 3.40111 × 10^{−5} | 2.36798 × 10^{−5} |
(8,16) | 9.00183 × 10^{−7} | 6.07094 × 10^{−7} | 3.68766 × 10^{−7} |
(8,32) | 2.3328 × 10^{−8} | 1.92534 × 10^{−8} | 1.35477 × 10^{−8} |
\(\tau=h^{2}\) | Implicit numerical approximation scheme [41] | ||
---|---|---|---|
0.5 | 0.6 | 0.7 | |
\(\frac{1}{64}\) | 2.9530 × 10^{−3} | 3.2886 × 10^{−3} | 3.6359 × 10^{−3} |
\(\frac{1}{256}\) | 7.6212 × 10^{−4} | 8.4259 × 10^{−4} | 9.2593 × 10^{−4} |
\(\frac{1}{1024}\) | 2.0744 × 10^{−4} | 2.2061 × 10^{−4} | 2.2671 × 10^{−4} |
Example 2
The MAEs of Example 2
(N,M) | Our method with \(\alpha _{1}=\beta _{1}=\frac{1}{2}\), \(\alpha _{2}=\beta _{2}=0\) and several choices of γ | ||
---|---|---|---|
0.4 | 0.5 | 0.6 | |
(4,4) | 7.27394 × 10^{−3} | 6.35126 × 10^{−3} | 5.08519 × 10^{−3} |
(4,12) | 1.81744 × 10^{−4} | 1.29398 × 10^{−4} | 8.43394 × 10^{−4} |
(4,36) | 6.87762 × 10^{−6} | 5.42605 × 10^{−6} | 3.79045 × 10^{−6} |
\(\tau=h^{2}\) | Fourier method and an extrapolation technique [42] | ||
---|---|---|---|
0.4 | 0.5 | 0.6 | |
\(\frac{1}{4}\) | 7.0342 × 10^{−3} | 1.0336 × 10^{−2} | 1.3420 × 10^{−2} |
\(\frac{1}{64}\) | 8.2629 × 10^{−4} | 1.0360 × 10^{−3} | 1.1898 × 10^{−3} |
\(\frac{1}{1024}\) | 8.2731 × 10^{−5} | 7.5748 × 10^{−5} | 1.3471 × 10^{−4} |
Example 3
The MAEs of Example 3
γ | Our method at (N,M,K) \(\alpha _{1}=\beta _{1}=\alpha _{2}=\beta _{2}=\frac {1}{2}\), \(\alpha _{3}=\beta _{3}=0\) | Explicit numerical approximation scheme [43] | ||
---|---|---|---|---|
(4,4,6) | (4,4,12) | \((\frac{1}{4},\frac {1}{4},\frac{1}{900})\) | \((\frac{1}{8},\frac{1}{8},\frac{1}{4900})\) | |
0.7 | 2.147706 × 10^{−3} | 1.351622 × 10^{−4} | 1.823187 × 10^{−3} | 4.963875 × 10^{−4} |
0.8 | 1.246139 × 10^{−3} | 8.543486 × 10^{−5} | 1.825094 × 10^{−3} | 4.959106 × 10^{−4} |
0.9 | 5.288470 × 10^{−4} | 6.502807 × 10^{−5} | 1.826525 × 10^{−3} | 4.968643 × 10^{−4} |
γ | Our method at (N,M,K) \(\alpha _{1}=\beta _{1}=\alpha _{2}=\beta _{2}=\frac {1}{2}\), \(\alpha _{3}=\beta _{3}=0\) | Implicit numerical approximation scheme [43] | ||
---|---|---|---|---|
(6,6,6) | (6,6,24) | \((\frac{1}{8},\frac {1}{8},\frac{1}{64})\) | \((\frac{1}{32},\frac{1}{32},\frac {1}{1024})\) | |
0.7 | 1.351622 × 10^{−4} | 7.517080 × 10^{−6} | 1.350407 × 10^{−3} | 1.456738 × 10^{−4} |
0.8 | 6.878345 × 10^{−5} | 3.349460 × 10^{−6} | 1.451373 × 10^{−3} | 2.348423 × 10^{−4} |
0.9 | 2.558291 × 10^{−5} | 1.090555 × 10^{−6} | 1.657724 × 10^{−3} | 3.523827 × 10^{−4} |
The MAEs of Example 3
γ | Our method at (N,M,K) with \(\alpha _{1}=\beta _{1}=\alpha _{2}=\beta _{2}=\frac{1}{2}\), \(\alpha _{3}=\beta _{3}=0\) | |||
---|---|---|---|---|
(4,4,12) | (4,4,18) | \((\frac{1}{6},\frac {1}{6},\frac{1}{24})\) | \((\frac{1}{6},\frac{1}{6},\frac{1}{28})\) | |
0.55 | 2.871835 × 10^{−4} | 8.137372 × 10^{−5} | 1.9487123 × 10^{−5} | 1.059474 × 10^{−5} |
γ | Compact finite difference scheme [44] | |||
---|---|---|---|---|
\((\frac{1}{8},\frac{1}{8},\frac {1}{10})\) | \((\frac{1}{8},\frac{1}{8},\frac{1}{80})\) | \((\frac {1}{8},\frac{1}{8},\frac{1}{320})\) | \((\frac{1}{8},\frac{1}{8},\frac {1}{640})\) | |
0.55 | 1.2126 × 10^{−3} | 1.4427 × 10^{−4} | 3.5483 × 10^{−5} | 1.7808 × 10^{−5} |
6 Conclusion
We have presented a new space-time spectral algorithm based on the shifted Jacobi-Gauss-Lobatto and the shifted Jacobi-Gauss-Radau collocation schemes. Based on the numerical results given in Section 5, it has been concluded that the obtained results are excellent in terms of accuracy for all tested problems. We have outlined the implementation of spectral collocation method for solving similar problems with a one- or two-dimensional space. In addition, this method may be extended to related problems. Furthermore, the proposed spectral method might be further developed by considering the generalized Laguerre or modified generalized Laguerre polynomials to solve similar problems in semi-infinite spatial intervals.
Declarations
Acknowledgements
We acknowledge the Editorial Board and the referees for their efforts and constructive criticism, which have improved the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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