Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons
Abstract
:1. Introduction
2. Applications of Fractional Calculus
- Definition of Conformable Derivative:
- Some Properties of Conformable Derivative:
3. Formation of Proposed Methods
3.1. Extended Simple Equation Method
3.2. -Expansion Method
- CASE I: When
- CASE II: When
- CASE III: When
3.3. The -Expansion Method
3.4. Modified F-Expansion Method
4. Applications
4.1. Application of Extended Simple Equation Method
- CASE 1: ,
- FAMILY-I
- FAMILY-II
- CASE 2: ,
- CASE 3: ,
- FAMILY-I
- FAMILY-II
- FAMILY-III
4.2. The -Expansion Method
- CASE I:
- CASE II:
- CASE III:
4.3. The -Expansion Method
- CASE II: ,
- CASE III: ,
- CASE IV: ,
- CASE V:
4.4. Modified F-Expansion Method
- FAMILY-I
- FAMILY-II
- FAMILY-III
- FAMILY-I
- FAMILY-II
- FAMILY-III
- FAMILY-I
- FAMILY-II
- FAMILY-III
- FAMILY-I
- FAMILY-II
- FAMILY-III
- FAMILY-I
- FAMILY-II
- FAMILY-III
- A = B = 0
- B = C = 0
- C = 0
5. Discussion of the Results
- Our solutions and are likely similar to solutions and mentioned in Equations (3.12–3.13) respectively in [37].
- Our solutions and are likely similar to solutions and mentioned in Equations (4.4–4.5) respectively in [38].
- Our solution is likely similar to solution mentioned in [39].
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
References
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Seadway, A.R.; Ali, A.; Bekir, A.; Cevikel, A.C. Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons. Fractal Fract. 2024, 8, 515. https://doi.org/10.3390/fractalfract8090515
Seadway AR, Ali A, Bekir A, Cevikel AC. Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons. Fractal and Fractional. 2024; 8(9):515. https://doi.org/10.3390/fractalfract8090515
Chicago/Turabian StyleSeadway, Aly R., Asghar Ali, Ahmet Bekir, and Adem C. Cevikel. 2024. "Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons" Fractal and Fractional 8, no. 9: 515. https://doi.org/10.3390/fractalfract8090515