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A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation

  • * Corresponding author: Tel.:+904223773745

    * Corresponding author: Tel.:+904223773745
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  • In this paper, we developed a unified method to solve time fractional Burgers' equation using the Chebyshev wavelet and L1 discretization formula. First we give the preliminary information about Chebyshev wavelet method, then we describe time discretization of the problems under consideration and then we apply Chebyshev wavelets for space discretization. The performance of the method is shown by three test problems and obtained results compared with other results available in literature.

    Mathematics Subject Classification: Primary: 65T60, 65N35; Secondary: 35R11.

    Citation:

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  • Figure 1.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0025$, $m' = 10$ and $\nu = 1$ at $t = 1$

    Figure 2.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.0005$ and $\nu = 1$ at $t = 0.1$

    Figure 3.  Numerical solution and exact solution for $\alpha = 0.5$, $\Delta t = 0.005$ and $\nu = 1$ at $t = 0.5$

    Table 1.  Error norms for various values of $\alpha$ and for $\Delta t = 0.00025$ at $t = 1$

    $\alpha=0.1$ $\alpha=0.25$
    [6]Present[6]Present
    $N=40$ $m'=10$ $N=40$ $m'=10$
    $L_{2}\times10^{3}$0.0967330.0751460.0900530.073586
    $L_{\infty}\times10^{3}$0.2729430.1063400.2586230.104141
    $\alpha=0.75$
    [6]Present
    $N=40$ $m'=10$
    $L_{2}\times10^{3}$0.0354480.069536
    $L_{\infty}\times10^{3}$0.1245690.098312
     | Show Table
    DownLoad: CSV

    Table 2.  Error norms for various values of $\Delta t$ and for $\nu = 1$, $\alpha = 0.5$ at $t = 1$

    $\Delta t=0.002$ $\Delta t=0.001$
    [6]Present[6]Present
    $N=40$ $m'=10$ $N=40$ $m'=10$
    $L_{2}\times10^{3}$0.4345860.5705090.1761950.284035
    $L_{\infty}\times10^{3}$ 0.6420030.8072750.2654190.401953
    $\Delta t=0.0005$
    [6]Present
    $N=40$ $m'=10$
    $L_{2}\times10^{3}$0.0688690.141630
    $L_{\infty}\times10^{3}$0.2118830.200442
     | Show Table
    DownLoad: CSV

    Table 3.  Error norms for various values of $\nu$ and for $\Delta t = 0.0005$, $\alpha = 0.5$ at $t = 0.1$

    $\nu=1$ $\nu=0.5$
    [6]Present[6]Present
    $N=80$ $m'=10$ $N=80$ $m'=10$
    $L_{2}\times10^{3}$0.0065280.0069800.0058350.006492
    $L_{\infty}\times10^{3}$ 0.0091640.0095470.0082500.008854
    $\nu=0.1$
    [6]Present
    $N=80$ $m'=10$
    $L_{2}\times10^{3}$ 0.0031050.004288
    $L_{\infty}\times10^{3}$ 0.0048470.005714
     | Show Table
    DownLoad: CSV

    Table 4.  Error norms for various collocation points and for $\Delta t = 0.00025$, $\alpha = 0.5$ at $t = 1$

    [6] Present [6] Present
    $N=10$ $m'=10$ $N=20$ $m'=20$
    $L_{2}\times10^{3}$1.7872780.0242520.4403050.024212
    $L_{\infty}\times10^{3}$2.4155890.0328240.5835830.033666
    [6]Present
    $N=40$ $m'=40$
    $L_{2}\times10^{3}$0.0927350.024210
    $L_{\infty}\times10^{3}$0.1204950.033727
     | Show Table
    DownLoad: CSV

    Table 5.  Error norms for various values of $\Delta t$ and for $\nu = 1$, $\alpha = 0.5$ at $t = 1$

    $\Delta t=0.002$ $\Delta t=0.001$
    [6]Present[6]Present
    $N=120$ $m'=16$ $N=120$ $m'=16$
    $L_{2}\times10^{3}$1.2201231.1537600.5324360.466776
    $L_{\infty}\times10^{3}$1.7257651.5637580.7531710.609456
    $\Delta t=0.0005$
    [6]Present
    $N=120$ $m'=16$
    $L_{2}\times10^{3}$0.1887100.126335
    $L_{\infty}\times10^{3}$0.2675460.180767
     | Show Table
    DownLoad: CSV
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