- Research
- Open Access
An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays
- Ali H Bhrawy^{1, 2}Email author,
- Mohamed A Abdelkawy^{2} and
- Fouad Mallawi^{1}
- Received: 23 March 2015
- Accepted: 29 May 2015
- Published: 19 June 2015
Abstract
In this paper, the Chebyshev Gauss-Lobatto pseudospectral scheme is investigated in spatial directions for solving one-dimensional, coupled, and two-dimensional parabolic partial differential equations with time delays. For the one-dimensional problem, the spatial integration is discretized by the Chebyshev pseudospectral scheme with Gauss-Lobatto quadrature nodes to provide a delay system of ordinary differential equations. The time integration of the reduced system in temporal direction is implemented by the continuous Runge-Kutta scheme. In addition, the present algorithm is extended to solve the coupled time delay parabolic equations. We also develop an efficient numerical algorithm based on the Chebyshev pseudospectral algorithm to obtain the two spatial variables in solving the two-dimensional time delay parabolic equations. This algorithm possesses spectral accuracy in the spatial directions. The obtained numerical results show the effectiveness and highly accuracy of the present algorithms for solving one-dimensional and two-dimensional partial differential equations.
Keywords
- two-dimensional parabolic differential equations
- delay system of differential equation
- pseudospectral scheme
- Chebyshev Gauss-Lobatto quadrature
- continuous Runge-Kutta method
1 Introduction
In recent years there has been a high level of interest in employing spectral methods for numerically solving many types of integral and differential equations, due to their high accuracy and ease of applying them for finite and infinite domains [1–9]. The spectral collocation method is a specific type of spectral methods that is more applicable and widely used to solve most types of differential equations [10–13].
In biology, physics, engineering, and computer design, many models can be attributed to time-delay partial differential equations (PDEs) (see, e.g., [14–17]). In the last few years, various analytical and numerical methods have been proposed for solving delay integral and PDEs (see, e.g., [18–20]). The Legendre pseudospectral algorithm [21] was implemented successfully to provide very accurate solutions of parabolic integro-differential equations in bounded and unbounded intervals. Kaushik et al. [22] proposed the uniform difference scheme for time-delayed PDEs. In the same direction, the singularly perturbed delay PDEs have been solved by an operator finite difference scheme [23]. The Chebyshev wavelet method was improved in [24] to obtain numerical solutions of time-varying delay systems.
Recently, Rashidinia et al. [25] proposed an efficient numerical technique for solving parabolic convection-diffusion problems, in which a finite difference scheme was used in the temporal direction to reduce the problem to a system of ordinary differential equations (ODEs) in a spatial direction, and the Sinc-Galerkin scheme was then used to solve this system. The Bessel collocation approximation was investigated in [26] to reduce the parabolic convection-diffusion equation to a system of algebraic equations. Moreover, Bhrawy et al. [27] proposed the Jacobi pseudospectral scheme for numerically solving the time-delayed Burgers’ equations and the same method was applied successfully for the generalized Fitzhugh-Nagumo equation in [28]; additionally, the Jacobi Gauss-Lobatto pseudospectral scheme has been proposed and extended to solve the complex generalized Zakharov system [29]. Ashyralyev and Agirseven [30] presented a difference scheme for the delay parabolic differential equation and studied the convergence of this scheme. Furthermore, Tian [31] discussed the asymptotic stability and convergence of three difference schemes which were applied to solve delay parabolic PDEs. More recently, the authors of [32] and [33] proposed a linearized compact multi-splitting method and compact difference method combined with an extrapolation scheme to solve convection-reaction-diffusion and neutral parabolic PDEs with delay, respectively.
As pointed out above, most researchers used finite difference schemes for the spatial discretizations of time-delay partial differential equations to get a system of ODEs with time delays. However, the accuracy of such methods is poor in the spatial directions. This motivated our interest to investigate the Chebyshev pseudospectral method for spatial discretizations for the initial-boundary value problems with time delays in one and two dimensions. This method is known for its ease in implementation along with the high accuracy and exponential convergence that can be achieved.
The main aim of this article is to investigate the Chebyshev pseudospectral scheme to solve the one-dimensional parabolic PDEs. We focus primarily on implementing this scheme in spatial independent variable. The spatial direction is collocated at \((N-1)\) collocation points of the Chebyshev Gauss-Lobatto quadrature nodes. This scheme has the advantage of reducing the one-dimensional parabolic PDEs into a system of \((N-1)\) ODEs with time delays in the time direction, that can be solved by continuous Runge-Kutta (RK) method.
We also extend the application of this algorithm to solve the coupled time delay parabolic equations at \((N-1)\) collocation nodes, which provides a system of \((2N-2)\) ODEs with time delays. Moreover, this algorithm is developed to solve the two-dimensional time delay parabolic equations, in which the two spatial variables are collocated at \((N-1)\times (M-1)\) Chebyshev Gauss-Lobatto quadrature nodes. This provides a system of \((N-1)\times(M-1)\) ODEs with time delays. Finally, several numerical simulations are given to confirm the high accuracy of the proposed algorithm.
This paper is outlined as follows: In the next section, we present some properties of Chebyshev polynomials. The Chebyshev pseudospectral approximation in spatial directions is obtained for solving the one-dimensional time-dependent PDEs, coupled parabolic PDEs, and two-dimensional time-dependent parabolic PDEs with time delays. In Section 4, some illustrative numerical experiments are given and some comparisons are made between our method and other methods. The paper ends with some conclusions and observations in Section 5.
2 Chebyshev polynomials interpolation
Theorem 2.1
3 Chebyshev pseudospectral scheme
We propose a pseudospectral algorithm based on Chebyshev polynomials to integrate the spatial variable for the \((1+1)\) parabolic PDEs with discrete time delay. The problem is then transformed into a system of ODEs with time delay. The algorithm is extended to treat the coupled parabolic PDEs and \((2+1)\) parabolic PDEs with time delay. The spatial variables are integrated based on the Chebyshev pseudospectral scheme, in which we use the Gauss-Lobatto interpolation nodes, to convert the problems to those of ODEs with time delay. The continuous RK scheme is employed to solve the resulting systems of ODEs.
3.1 \((1+1)\) Parabolic PDEs with time delay
The previous system of first-order ODEs with time delay (22) is integrated by the continuous RK scheme [35, 36].
3.2 Coupled \((1+1)\) parabolic PDEs with time delay
3.3 \((2+1)\) Parabolic PDEs with time delay
This system of first-order ODEs can be solved by the continuous RK scheme.
4 Numerical results
To demonstrate the accuracy and applicability of the proposed method in the present paper, four numerical examples are carried out by the algorithms presented in the previous section with a comparison with a linearized compact difference scheme [39] in the one-dimensional case. In each of these examples, we shall highlight the accuracy and efficiency of the method.
Example 4.1
Comparison between the absolute errors of problem ( 61 )
x | t | \(\boldsymbol {M_{E}}\) | LCDM [39] |
---|---|---|---|
1.5 | 0.1 | 1.77 × 10^{−9} | 6.83 × 10^{−4} |
0.2 | 5.85 × 10^{−10} | 5.54 × 10^{−4} | |
0.3 | 1.73 × 10^{−10} | 1.59 × 10^{−4} | |
0.4 | 6.27 × 10^{−11} | 4.12 × 10^{−4} | |
0.5 | 2.32 × 10^{−11} | 1.12 × 10^{−3} | |
1.5 | 0.6 | 8.86 × 10^{−12} | 1.93 × 10^{−3} |
0.7 | 3.55 × 10^{−12} | 2.80 × 10^{−3} | |
0.8 | 1.35 × 10^{−12} | 3.65 × 10^{−3} | |
0.9 | 6.42 × 10^{−13} | 4.41 × 10^{−3} | |
1.0 | 1.79 × 10^{−13} | 4.98 × 10^{−3} |
MAEs using the Chebyshev collocation method for problem ( 61 )
N | 4 | 6 | 8 | 10 |
---|---|---|---|---|
\(M_{E}\) | 2.26 × 10^{−3} | 5.03 × 10^{−5} | 9.84 × 10^{−8} | 1.77 × 10^{−9} |
Example 4.2
MAEs of problem ( 65 )
N | 4 | 6 | 8 | 10 | 12 |
---|---|---|---|---|---|
\(M^{1}_{E}\) | 1.03 × 10^{−2} | 2.20 × 10^{−4} | 2.47 × 10^{−6} | 1.71 × 10^{−8} | 1.22 × 10^{−9} |
\(M^{2}_{E}\) | 3.51 × 10^{−3} | 4.21 × 10^{−5} | 3.82 × 10^{−7} | 2.47 × 10^{−9} | 2.96 × 10^{−10} |
Absolute errors for problem ( 65 ) at \(\pmb{N=12}\)
x | t | \(\boldsymbol {E_{1}(x,t)}\) | \(\boldsymbol {E_{2}(x,t)}\) |
---|---|---|---|
0.1 | 1 | 7.26 × 10^{−11} | 3.63 × 10^{−11} |
0.2 | 1.43 × 10^{−10} | 5.98 × 10^{−11} | |
0.3 | 1.99 × 10^{−10} | 6.42 × 10^{−11} | |
0.4 | 2.35 × 10^{−10} | 5.95 × 10^{−11} | |
0.5 | 2.47 × 10^{−10} | 4.23 × 10^{−11} | |
0.6 | 1 | 2.35 × 10^{−10} | 2.09 × 10^{−11} |
0.7 | 1.99 × 10^{−10} | 3.83 × 10^{−12} | |
0.8 | 1.43 × 10^{−10} | 1.08 × 10^{−11} | |
0.9 | 7.29 × 10^{−11} | 1.07 × 10^{−11} | |
1.0 | 5.45 × 10^{−12} | 1.29 × 10^{−15} |
Example 4.3
x | y | t | E ( x , y , t ) |
---|---|---|---|
0.1 | 0.1 | 0 | 1.20835 × 10^{−6} |
0.2 | 0.2 | 1.37925 × 10^{−6} | |
0.3 | 0.3 | 1.90777 × 10^{−7} | |
0.4 | 0.4 | 7.16319 × 10^{−7} | |
0.5 | 0.5 | 3.46945 × 10^{−18} | |
0.6 | 0.6 | 7.16319 × 10^{−7} | |
0.7 | 0.7 | 1.90777 × 10^{−7} | |
0.8 | 0.8 | 1.37925 × 10^{−6} | |
0.9 | 0.9 | 1.20835 × 10^{−6} | |
0.1 | 0.1 | 0.5 | 3.00614 × 10^{−7} |
0.2 | 0.2 | 3.22082 × 10^{−7} | |
0.3 | 0.3 | 2.01012 × 10^{−8} | |
0.4 | 0.4 | 1.74909 × 10^{−7} | |
0.5 | 0.5 | 6.71257 × 10^{−12} | |
0.6 | 0.6 | 1.74909 × 10^{−7} | |
0.7 | 0.7 | 2.01036 × 10^{−8} | |
0.8 | 0.8 | 3.22081 × 10^{−7} | |
0.9 | 0.9 | 3.00614 × 10^{−7} | |
0.1 | 0.1 | 1 | 6.24895 × 10^{−8} |
0.2 | 0.2 | 6.69564 × 10^{−8} | |
0.3 | 0.3 | 4.1809 × 10^{−9} | |
0.4 | 0.4 | 3.63598 × 10^{−8} | |
0.5 | 0.5 | 2.93906 × 10^{−13} | |
0.6 | 0.6 | 3.63598 × 10^{−8} | |
0.7 | 0.7 | 4.18141 × 10^{−9} | |
0.8 | 0.8 | 6.69562 × 10^{−8} | |
0.9 | 0.9 | 6.24895 × 10^{−8} |
Example 4.4
5 Concluding remarks and future work
The Chebyshev Gauss-Lobatto pseudospectral method was investigated successfully in spatial discretizations to get accurate approximate solutions of one-dimensional, coupled, and two-dimensional parabolic PDEs with time delay. All of these problems were transformed to systems of ODEs with time delays, greatly simplifying the problems. The continuous RK scheme is then applied to the resulting semidiscrete delay systems. From the numerical experiments, by the obtained results the effectiveness and highly accuracy were demonstrated of the Chebyshev Gauss-Lobatto pseudospectral method for solving the mentioned problems. The present algorithm is also available for approximating the solution for singularly perturbed delay parabolic equations.
We have outlined the implementation of a Chebyshev pseudospectral approximation based on the Chebyshev Gauss-Lobatto quadrature nodes for solving the two-dimensional time delay parabolic PDEs. The technique can be extended to more sophisticated problems with variable and distributed time delays. In principle, this method may be extended to related problems in mathematical physics. It is possible to use other orthogonal polynomials, say Legendre polynomials, or Jacobi polynomials to solve the mentioned problems in this article. Furthermore, the proposed spectral method might be developed by considering the Chebyshev pseudospectral approximation in both temporal and spatial discretizations. We should note that, as a numerical method, we are restricted to solving problems over a finite domain. Also, the pseudospectral approximation might be employed based on generalized Laguerre or modified generalized Laguerre polynomials to solve similar problems in semi-infinite spatial intervals.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions, which have improved the paper.
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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