Rational Solutions of Riccati-like Partial Differential Equations

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

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Exact Combined Solutions for the (2+1)-Dimensional Dispersive Long Water-Wave Equations

Journal of Function Spaces ◽

10.1155/2020/3707924 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Yi Wei ◽

Xing-Qiu Zhang ◽

Zhu-Yan Shao ◽

Lu-Feng Gu ◽

Xiao-Feng Yang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Water Wave ◽

Wave Equations ◽

Nonlinear Partial Differential Equations ◽

Soliton Solutions ◽

Traveling Wave Solutions ◽

Rational Solutions ◽

Partial Differential ◽

Undetermined Coefficient

The homogeneous balance of undetermined coefficient (HBUC) method is presented to obtain not only the linear, bilinear, or homogeneous forms but also the exact traveling wave solutions of nonlinear partial differential equations. Linear equation is obtained by applying the proposed method to the (2+1)-dimensional dispersive long water-wave equations. Accordingly, the multiple soliton solutions, periodic solutions, singular solutions, rational solutions, and combined solutions of the (2+1)-dimensional dispersive long water-wave equations are obtained directly. The HBUC method, which can be used to handle some nonlinear partial differential equations, is a standard, computable, and powerful method.

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Meromorphic solutions of the (2 + 1)- and the (3 + 1)-dimensional BLMP equations and the (2 + 1)-dimensional KMN equation

Demonstratio Mathematica ◽

10.1515/dema-2021-0009 ◽

2021 ◽

Vol 54 (1) ◽

pp. 129-139

Author(s):

Guoqiang Dang

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Complex Plane ◽

Nonlinear Partial Differential Equations ◽

Complex Method ◽

Meromorphic Solutions ◽

Rational Solutions ◽

Partial Differential ◽

Elliptic Solutions

Abstract The complex method is systematic and powerful to build various kinds of exact meromorphic solutions for nonlinear partial differential equations on the complex plane C {\mathbb{C}} . By using the complex method, abundant new exact meromorphic solutions to the ( 2 + 1 ) \left(2+1) -dimensional and the ( 3 + 1 ) \left(3+1) -dimensional Boiti-Leon-Manna-Pempinelli equations and the ( 2 + 1 ) \left(2+1) -dimension Kundu-Mukherjee-Naskar equation are investigated. Abundant new elliptic solutions, rational solutions and exponential solutions have been constructed.

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Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

Advances in Mathematical Physics ◽

10.1155/2020/3758353 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Kai Fan ◽

Cunlong Zhou

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Order ◽

Water Waves ◽

Expansion Method ◽

Auxiliary Equation ◽

Rational Solutions ◽

Partial Differential ◽

Wave Solutions ◽

Model Equations

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

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