EXACT SOLUTIONS OF A GENERALIZED TIME-FRACTIONAL KDV EQUATION UNDER RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL OPERATORS

Xianbin Wu; Wan Fan; Xiaochun Hong; Wei Lu; Weiguo Rui

doi:10.11948/20240506

2025 Volume 15 Issue 6

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Xianbin Wu, Wan Fan, Xiaochun Hong, Wei Lu, Weiguo Rui. EXACT SOLUTIONS OF A GENERALIZED TIME-FRACTIONAL KDV EQUATION UNDER RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL OPERATORS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3576-3602. doi: 10.11948/20240506

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Xianbin Wu, Wan Fan, Xiaochun Hong, Wei Lu, Weiguo Rui. EXACT SOLUTIONS OF A GENERALIZED TIME-FRACTIONAL KDV EQUATION UNDER RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL OPERATORS[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3576-3602. doi: 10.11948/20240506

EXACT SOLUTIONS OF A GENERALIZED TIME-FRACTIONAL KDV EQUATION UNDER RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL DIFFERENTIAL OPERATORS

1.
School of Data Science, Zhuhai College of Science and Technology, Zhuhai 519040, China
2.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Author Bio: Email: wxb3210@zcst.edu.cn(X. Wu); Email: fw11061119@163.com(W. Fan); Email: luwei@zcst.edu.cn(W. Lu); Email: wgruihhu@163.com(W. Rui)
Corresponding author: Email: xchong@zcst.edu.cn(X. Hong)
Fund Project: This study was financially supported by the National Natural Science Foundation of China (Grant No. 11761075), the Key Research Platforms and Projects of Ordinary Universities under the Eduction Department of Guangdong Provincial (Grant No. 2023ZDZX1049), and the Research Project of Chongqing Education Commission (Grant No. CXQT21014)

Abstract

Abstract

It is well known that searching exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work. In this paper, based on a modified separation method of variables and the dynamic system method, a combinational method is proposed in order to develop new methods for solving nonlinear time-fractional PDEs. Compared with the traditional separation method of variables, the modified separation method of variables has some advantages in reducing nonlinear time-fractional PDEs. As an example for the application of this combinational method, a generalized nonlinear time-fractional KdV equation is studied under the Riemann-Liouville and Caputo fractional differential operators, respectively. In different parametric regions, different kinds of phase portraits of the dynamic systems derived from the generalized time-fractional KdV equation are presented. Existence and dynamic properties of solutions of the generalized time-fractional KdV equation are investigated. In some special parametric conditions, many exact solutions are obtained, some of them are parametric form. Such solutions of parametric form are usually unable to be obtained by other methods, which also shows an advantage of dynamic system method.
- Separation method of semi-fixed variables /
- dynamic system method /
- nonlinear time-fractional PDEs /
- generalized time-fractional KdV equation
MSC: 26A33, 34A05, 34K18, 35D05

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References

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