Coupled sub-equations method for seeking analytical solutions of fractional partial differential equations

Qinghua Feng; Qinghua Feng

doi:10.3934/math.2026889

AIMS Mathematics

2026, Volume 11, Issue 7: 21989-22003. doi: 10.3934/math.2026889

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Coupled sub-equations method for seeking analytical solutions of fractional partial differential equations

Qinghua Feng ^,

School of Mathematics and Statistics, Shandong University of Technology, Zhangzhou Road 12, Zibo, Shandong, 255049, China

Received: 27 April 2026 Revised: 14 June 2026 Accepted: 29 June 2026 Published: 23 July 2026
MSC : 35C07, 35Q51, 35Q53

In this paper, we considered seeking interaction exact solutions with variable coefficients for fractional partial differential equations involving the conformable fractional derivative. Combining with the properties of the conformable fractional calculus, and using a new ansatz structure, we proposed a new coupled sub-equations method for seeking interaction exact solutions of nonlinear fractional partial differential equations, that is, the coupled extended $ (\frac{ R'}{ R}) $ equation-Riccati equation method. In the process of the method, based on conformable fractional calculus, unknown transformed functions of a fractional system were expressed in certain polynomials with respect to some variables, which satisfied two known coupled sub-equations, and the homogeneous balance principle was used to determine the degree of the polynomials. The method was applied to solve the time fractional two-dimensional Boussinesq equation. With the aid of mathematical software Maple, an abundance of interaction exact solutions with variable coefficients for the equation were obtained by the ansatz structure.
- coupled sub-equations method,
- fractional partial differential equations,
- exact solutions,
- variable coefficients,
- soliton
Citation: Qinghua Feng. Coupled sub-equations method for seeking analytical solutions of fractional partial differential equations[J]. AIMS Mathematics, 2026, 11(7): 21989-22003. doi: 10.3934/math.2026889

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Abstract

In this paper, we considered seeking interaction exact solutions with variable coefficients for fractional partial differential equations involving the conformable fractional derivative. Combining with the properties of the conformable fractional calculus, and using a new ansatz structure, we proposed a new coupled sub-equations method for seeking interaction exact solutions of nonlinear fractional partial differential equations, that is, the coupled extended $ (\frac{ R'}{ R}) $ equation-Riccati equation method. In the process of the method, based on conformable fractional calculus, unknown transformed functions of a fractional system were expressed in certain polynomials with respect to some variables, which satisfied two known coupled sub-equations, and the homogeneous balance principle was used to determine the degree of the polynomials. The method was applied to solve the time fractional two-dimensional Boussinesq equation. With the aid of mathematical software Maple, an abundance of interaction exact solutions with variable coefficients for the equation were obtained by the ansatz structure.

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