Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

Saima Noor; Azzh Saad Alshehry; Asfandyar Khan; Imran Khan; Saima Noor; Azzh Saad Alshehry; Asfandyar Khan; Imran Khan

doi:10.3934/math.20231422

AIMS Mathematics

2023, Volume 8, Issue 11: 27775-27819. doi: 10.3934/math.20231422

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Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations

1.
Department of Basic Sciences, Preparatory Year Deanship, King Faisal University, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan
4.
Department of Mathematics and Statistics, Bacha Khan University Charsadda 24420, Pakistan

Received: 20 August 2023 Revised: 10 September 2023 Accepted: 19 September 2023 Published: 09 October 2023
MSC : 33B15, 34A34, 35A20, 44A10, 35A22

The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.
Citation: Saima Noor, Azzh Saad Alshehry, Asfandyar Khan, Imran Khan. Innovative approach for developing solitary wave solutions for the fractional modified partial differential equations[J]. AIMS Mathematics, 2023, 8(11): 27775-27819. doi: 10.3934/math.20231422

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Abstract

The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.

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