Numerical study of soliton behavior of generalised Kuramoto-Sivashinsky type equations with Hermite splines

Abdul-Majeed Ayebire; Saroj Sahani; Priyanka; Shelly Arora; Abdul-Majeed Ayebire; Saroj Sahani; Priyanka; Shelly Arora

doi:10.3934/math.2025099

AIMS Mathematics

2025, Volume 10, Issue 2: 2098-2130. doi: 10.3934/math.2025099

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Numerical study of soliton behavior of generalised Kuramoto-Sivashinsky type equations with Hermite splines

1.
Department of Mathematics, Punjabi University, Patiala, Punjab-147002, INDIA
2.
Department of Statistics, Bolgatanga Technical University, Bolgatanga, GHANA
3.
Department of Mathematics, South Asian University, New Delhi-110068, INDIA
4.
Department of Mathematics, Central University of Haryana, Jant-Pali, Mahendargarh, Haryana-123031, INDIA

Received: 30 September 2024 Revised: 11 December 2024 Accepted: 26 December 2024 Published: 08 February 2025
MSC : 35B30, 35B35, 35C10, 35B20, 65D07

The traveling wave behavior of the nonlinear third and fourth-order advection-diffusion equation has been elaborated. In this study, the effect of dispersion and dissipation processes was mainly analyzed thoroughly. In the thorough analysis, strictly permanent short waves to breaking waves, having comparative higher amplitudes, have been observed. The governed problem was employed with the space-splitting method for a coupled system of equations to conduct the computational process. For the time derivative, the Crank-Nicolson difference approximation was studied. An orthogonal collocation method using Hermite splines has been implemented to approximate the solution of the semi-discretized coupled problem. The proposed method reduces the equation to an iterative scheme of an algebraic system of collocation equations, which reduced the computational complexity. The proposed scheme is found to be unconditionally stable, and the numerical demonstrations and comparisons represented the computational efficiency.
- Crank-Nicolson technique,
- collocation scheme,
- KdV Burger equation,
- Gardner equation,
- soliton solution,
- shock wave
Citation: Abdul-Majeed Ayebire, Saroj Sahani, Priyanka, Shelly Arora. Numerical study of soliton behavior of generalised Kuramoto-Sivashinsky type equations with Hermite splines[J]. AIMS Mathematics, 2025, 10(2): 2098-2130. doi: 10.3934/math.2025099

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Abstract

The traveling wave behavior of the nonlinear third and fourth-order advection-diffusion equation has been elaborated. In this study, the effect of dispersion and dissipation processes was mainly analyzed thoroughly. In the thorough analysis, strictly permanent short waves to breaking waves, having comparative higher amplitudes, have been observed. The governed problem was employed with the space-splitting method for a coupled system of equations to conduct the computational process. For the time derivative, the Crank-Nicolson difference approximation was studied. An orthogonal collocation method using Hermite splines has been implemented to approximate the solution of the semi-discretized coupled problem. The proposed method reduces the equation to an iterative scheme of an algebraic system of collocation equations, which reduced the computational complexity. The proposed scheme is found to be unconditionally stable, and the numerical demonstrations and comparisons represented the computational efficiency.

References

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