A New Numerical Simulation for Modified Camassa-Holm and Degasperis-Procesi Equations via Trigonometric Quintic B-spline

Abstract

In this study, the soliton solutions of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations, called modified b-equations with important physical properties, were obtained. The soliton waves' movement and positions formed by solving the mCH and mDP equations were calculated. Ordinary differential equation systems were obtained using trigonometric quintic B-spline bases for position and time direction derivatives in the equations to obtain numerical solutions. An algebraic equation system was then created by writing Crank-Nicolson type approximations for time and position-dependent terms. The stability analysis of this system was examined using the von-Neumann Fourier series method. L₂, L_{∞} and absolute error norms were used to measure the numerical results' convergence to the real solution. The numerical results calculated were compared with the exact solution and some studies in the literature.

Keywords

• Modified Camassa-Holm
• Modified Degasperis-Procesi Equation
• Soliton Waves
• Trigonometric Quintic B-spline bases
• Collocation Method

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