On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method. (12th August 2020)

• Record Type: Journal Article
• Authors: Moradi, V.; Sayevand, K.
• Other Names: Debbouche Amar guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 15(2021)
• Page Start: 11944
• Page End: 11960
• Publication Date: 2020-08-12
• Subjects: backward Euler differences -- finite difference method -- fractional water wave equation -- Riemann–Liouville fractional derivatives -- trapezoidal quadrature rule; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6788
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this study, a reliable implicit finite difference method based on the modified trapezoidal quadrature rule, backward Euler differences, nonstandard central approximations, and the Hadamard finite‐part integral is being considered to solve a viscous asymptotical model named as fractional Kakutani–Matsuuchi water wave model. The fractional derivative is used in the Riemann–Liouville sense. Based on the properties of Brouwer's fixed‐point theorem, the existence, uniqueness, convergence, and stability of the proposed method are proved. Furthermore, we show that the global convergence order of the method in maximum norm is O ( τ, h m i n { β, 3 − α } ), where 0 < α ≤ 1 and β > 0 are the order of fractional derivative and the Lipschitz constant. Also, τ and h are the time step and space step, respectively. Finally, several examples are used to illustrate the accuracy and performance of the method.