| Citation: |
H. Dehestani, Y. Ordokhani, M. Razzaghi. EXECUTION OF A NOVEL DISCRETIZATION APPROACH FOR SOLVING VARIABLE-ORDER CAPUTO-RIESZ TIME-SPACE FRACTIONAL SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 235-262. doi: 10.11948/20230194
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EXECUTION OF A NOVEL DISCRETIZATION APPROACH FOR SOLVING VARIABLE-ORDER CAPUTO-RIESZ TIME-SPACE FRACTIONAL SCHRÖDINGER EQUATIONS
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Abstract
This work deals with the variable-order Caputo-Riesz (VO-CR) time-space fractional Schrödinger equations with the help of the Pell discretization method. For the first step, we separate the proposed problem into real and imaginary parts. Then, expanding the functions with respect to Pell polynomials and utilizing the required operational matrices. The operational matrices, together with the Pell discretization method, reduce the problem into a system of algebraic equations. It should be noted that the technique of obtaining the operational matrices strongly affects the precision of the numerical method process. Finally, we implement the proposed approach in several numerical experiments to confirm the theoretical scheme. And also, the comparison of obtained results with some existing methods is displayed in tables.
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References
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H. Dehestani, Y. Ordokhani, M. Razzaghi. EXECUTION OF A NOVEL DISCRETIZATION APPROACH FOR SOLVING VARIABLE-ORDER CAPUTO-RIESZ TIME-SPACE FRACTIONAL SCHRÖDINGER EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 235-262. doi: 10.11948/20230194
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Figure 1.
The absolute error of the real part (left) and imaginary part of the approximate solution (right) obtained with
$ \alpha(x,t)=0.2,\beta=1.8 $ $ M=N=3 $ -
Figure 2.
The absolute error of the real part (left) and imaginary part of the approximate solution (right) obtained with
$ \alpha(x,t)=0.2,\beta=1.8 $ $ M=3,N=7 $ -
Figure 3.
The approximation of the real part at different time (left) and absolute error of real part at
$ t=1 $ $ \alpha(x,t)=0.5 $ $ M=4,N=5 $ -
Figure 4.
The approximation of the imaginary part at different time (left) and absolute error of imaginary part at
$ t=1 $ $ \alpha(x,t)=0.5 $ $ M=4,N=5 $ -
Figure 5.
The approximation at different time (left) and absolute error at
$ t=1 $ $ \alpha(x,t)=0.5, $ $ \beta=1.5 $ $ M=4,N=5 $ -
Figure 6.
The approximation at different time (left) and absolute error at
$ t=1 $ $ \alpha(x,t)=\frac{2+\sin(xt)}{400}, $ $ \beta=1.5 $ $ M=4,N=5 $ -
Figure 7.
The absolute error of the real part (left) and imaginary part (right) of approximate solution obtained with
$ \alpha(x,t)=0.75+0.2\exp(-xt) $ $ M=6,N=3 $