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On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method
Author(s) -
Moradi V.,
Sayevand K.
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6788
Subject(s) - mathematics , lipschitz continuity , fractional calculus , mathematical analysis , uniqueness , hadamard transform , finite difference , backward euler method , order (exchange) , euler equations , finance , economics
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.