| Citation: |
Shazia Sadiq, Mujeeb ur Rehman. NUMERICAL TECHNIQUE BASED ON GENERALIZED LAGUERRE AND SHIFTED CHEBYSHEV POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1977-2001. doi: 10.11948/20220504
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NUMERICAL TECHNIQUE BASED ON GENERALIZED LAGUERRE AND SHIFTED CHEBYSHEV POLYNOMIALS
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Abstract
In this study, we present a numerical scheme for solving a class of fractional-order partial differential equations. First, we introduce $ \psi $-Laguerre polynomials, then, we employ $ \psi $-shifted Chebyshev and $ \psi $-Laguerre polynomials for the solution of space-time fractional-order differential equations. In our approach, we project $ \psi $-shifted Chebyshev and $ \psi $-Laguerre polynomials to develop operational matrices of fractional-order integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provides the unknown matrix which is then used to obtain the approximated numerical solution. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.
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References
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Shazia Sadiq, Mujeeb ur Rehman. NUMERICAL TECHNIQUE BASED ON GENERALIZED LAGUERRE AND SHIFTED CHEBYSHEV POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1977-2001. doi: 10.11948/20220504
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Figure 1.
Exact and proposed solution for
$ \psi_1 $ $ \gamma=1.8, \zeta=1.3 $ $ N=5 $ -
Figure 2.
Absolute error for
$ \psi_2 $ $ \gamma=1.8, \zeta=1.3 $ $ N=5 $ -
Figure 3.
Absolute error for
$\psi_2$ $\gamma=$ $\zeta=1.3$ $N=5$ -
Figure 4.
Absolute error for
$\psi_3$ $\gamma=$ $1.8, \zeta=1.3$ $N=5$ -
Figure 5.
Exact and proposed solution for
$ u(x, 1) $ $ u(x, 5) $ $ \gamma=2, \zeta=1.5 $ -
Figure 6.
Absolute error for
$u(x, 1)$ $u(x, 5)$ $\gamma=2, \zeta=1.5$ -
Figure 7.
Exact and proposed solution for
$ \gamma=\zeta=1 $ $ N=8 $ -
Figure 8.
Absolute error at
$\gamma=\zeta=1$ $N=8$ -
Figure 9.
Exact and proposed solution for
$ \gamma=0.5, \zeta=2/3 $ $ N=4 $ -
Figure 10.
Absolute error at
$\gamma=$ $0.5, \zeta=2 / 3$ $N=4$ -
Figure 11.
Exact and proposed solution at
$ \gamma=2, \zeta=0.5 $ $ N=6 $ -
Figure 12.
Absolute error at
$\gamma=2, \zeta=0.5$ $N=6$ -
Figure 13.
Exact and proposed solution at
$ \gamma=2, \zeta=0.5 $ $ N=10 $ -
Figure 14.
Absolute error at
$\gamma=2, \zeta=0.5$ $N=10$ -
Figure 15.
Exact and proposed solution at
$ \gamma=1.8, \zeta=1 $ $ N=8 $ -
Figure 16.
Absolute error at
$\gamma=1.8, \zeta=1$ $N=8$