| Citation: |
Parisa Rahimkhani, Salameh Sedaghat. ALTERNATIVE WAVELETS FOR THE SOLUTION OF VARIABLE-ORDER FRACTAL-FRACTIONAL DIFFERENTIAL EQUATIONS SYSTEM WITH POWER AND MITTAG-LEFFLER KERNELS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 1830-1861. doi: 10.11948/20240117
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ALTERNATIVE WAVELETS FOR THE SOLUTION OF VARIABLE-ORDER FRACTAL-FRACTIONAL DIFFERENTIAL EQUATIONS SYSTEM WITH POWER AND MITTAG-LEFFLER KERNELS
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Abstract
In this paper, a procedure based on the fractional-order alternative Legendre wavelets (FALWs) is introduced for solving variable-order fractalfractional differential equations (VFFDEs) system with power and MittagLeffler kernels. An analytic formula is obtained for computing the variableorder fractal-fractional integral operator of the FALWs by employing the regularized beta functions. The presented method converts solving the primary problem to solving a system of nonlinear algebraic equations. To do this, the variable-order fractal-fractional (VFF) derivative of the unknown function is expanded in terms of the FALWs with unknown coefficients at first. Then, by employing the properties of the VFF derivative and integral, together with the collocation method, a system of algebraic equations is obtained, that can be easily solved by the Newton's iterative scheme. An error upper bound for the numerical solution in the Sobolev space is obtained. Finally, different chaotic oscillators of variable-order are solved in order to illustrate the accuracy and validity of the suggested strategy.
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References
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Parisa Rahimkhani, Salameh Sedaghat. ALTERNATIVE WAVELETS FOR THE SOLUTION OF VARIABLE-ORDER FRACTAL-FRACTIONAL DIFFERENTIAL EQUATIONS SYSTEM WITH POWER AND MITTAG-LEFFLER KERNELS[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 1830-1861. doi: 10.11948/20240117
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Figure 1.
The graphs of the fractional-order alternative wavelets with
$ \theta , t \in [0, 1] $ $ (k=2, M=10) $ $ (k=3, M=8) $ -
Figure 2.
The graphs of the CFF variable-order integral with
$ k=2, M=8, h=1, \alpha (t)= \tanh (t+5), \beta (t)=0.99+\frac{0.01}{2}(\sin (t)+1) $ $ (a): \theta =1 $ $ (b): \theta = \frac{1}{8} $ -
Figure 3.
The graphs of the CFF variable-order integral with
$ k=3, M=8, h=1, \alpha (t)= \tanh (t+5), \beta (t)=0.99+\frac{0.01}{2}(\sin (t)+1) $ $ (a): \theta =1 $ $ (b): \theta = \frac{2}{3} $ -
Figure 4.
The NR with
$ k=2, M=6, h=2, \theta =\frac{1}{4} $ $ \beta (t) \in (0, 1) $ -
Figure 5.
The EE with
$ k=2, M=8, h=2, \theta =\frac{1}{8} $ $ (a): $ $ (b): $ -
Figure 6.
The NR with
$ k=1, h=10, \theta =\frac{1}{8} $ $ M=4, 6, 8 $ -
Figure 7.
The NR with
$ k=2, M=9, h=2, \theta =1 $ $ \alpha (t) \in (0, 1) $ -
Figure 8.
The NR with
$ k=2, M=9, h=2, \theta =\frac{1}{8} $ $ \alpha (t) \in (0, 1) $ -
Figure 9.
The EE with
$ k=2, M=9, h=2, \theta =\frac{1}{8} $ $ (a): $ $ (b): $ -
Figure 10.
The NR with
$ k=1, h=200, \theta =1 $ $ M=5, 7, 9 $ -
Figure 11.
The NR with
$ \theta =1, $ $ k=1, h=1, $ -
Figure 12.
The NR for up:
$ \theta =\frac{2}{3} $ $ \theta =1 $ $ k=1, h=400, $