| Citation: |
Weijun He, Weiguo Rui, Xiaochun Hong. AN EXTENSIONAL CONFORMABLE FRACTIONAL DERIVATIVE AND ITS EFFECTS ON SOLUTIONS AND DYNAMICAL PROPERTIES OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1790-1819. doi: 10.11948/20230418
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AN EXTENSIONAL CONFORMABLE FRACTIONAL DERIVATIVE AND ITS EFFECTS ON SOLUTIONS AND DYNAMICAL PROPERTIES OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
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Abstract
Investigations have shown that the conformable fractional derivative is very different from the classical fractional derivatives, it does not have function of memory like the classical fractional derivatives, so it is more appropriate to be called as cognate derivative of the classical integer-order derivative. In this paper, following the idea of constructing the conformable fractional derivative, an extensional conformable fractional derivative named sech-fractional derivative is proposed. The effects of the new conformable fractional differential operator on dynamical properties of nonlinear partial differential equations (PDEs) are discussed. As example, by using the dynamical system method, traveling wave solutions and their dynamical properties of a nonlinear fractional Schrödinger equation are investigated under the sech-fractional differential operator. The solutions and their dynamical properties of the nonlinear Schrödinger equation are compared under three kinds of differential operators, their distinction and connection are revealed. Some interesting phenomena are found and deserve attentions further.
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References
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Weijun He, Weiguo Rui, Xiaochun Hong. AN EXTENSIONAL CONFORMABLE FRACTIONAL DERIVATIVE AND ITS EFFECTS ON SOLUTIONS AND DYNAMICAL PROPERTIES OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1790-1819. doi: 10.11948/20230418
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Figure 1.
Graphs of the (2.35) and (2.37) under the
$ A=2, \ k=0.8, \ c=0.5, \ \alpha=0.25, \ \beta=0.75, \ t=1. $ -
Figure 2.
Graphs of phase portraits of the system (3.10).
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Figure 3.
Comparison graph of modules of the solutions (3.18) and (3.19).
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Figure 4.
Comparison graph of modules of solutions (3.20) and (3.19).
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Figure 5.
Comparison graphs of modules of solutions (3.35), (3.36) and (3.37).
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Figure 6.
Comparison graphs of modules of solutions (3.48), (3.49) and (3.50).
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Figure 7.
Graphs of derivatives defined by (5.2) under two operators.
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Figure 8.
Graphs of functions defined by (5.3) and (5.4) under two operators.
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Figure 9.
Graphs of functions defined by (5.5) and (5.6) under two operators.
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Figure 10.
Graphs of functions defined by (5.7) and (5.8) under two operators.