A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel method
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A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel method Mehmet Giyas Sakar1 · Onur Saldır1 Received: 30 January 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, “good” Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics. Keywords Reproducing kernel method · Boussinesq equation · Fractional order · Convergence Mathematics Subject Classifications 65M12 · 35R11 · 46E22
1 Introduction In 1870’s, Boussinesq equation was introduced by Joseph Boussinesq to determine the motion of water waves[1–3]. Based on Scott Russell’s solitary wave phenomenon, Boussinesq obtained some water wave equations that can mathematically explain this phenomenon. The well-known generalized Boussinesq equation is defined as u ζ ζ (η, ζ ) − au ηη (η, ζ ) + wu ηηηη (η, ζ ) + [ f (u)]ηη = 0 (a > 0)
(1)
In Eq. (1), if a = 1, w = 1 and f (u) = u 2 are taken, then “good” Boussinesq equation is obtained as
B
Mehmet Giyas Sakar [email protected] Onur Saldır [email protected]
1
Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080 Van, Turkey
123
M. G. Sakar, O. Saldır
u ζ ζ (η, ζ ) − u ηη (η, ζ ) + u ηηηη (η, ζ ) + [u 2 ]ηη = 0
(2)
which describes the movement of long waves propagation on shallow water surface in both directions. And also, it balances between nonlinearity and dispersion that can cause the blow-up solutions or solitons. Moreover, this equation is linearly stable and it is possible to make computations for this equation. Good Boussinesq equation arises in wide range of areas such as ion-sound waves, anharmonic lattice waves, ocean and coastal engineering, biology, plasma waves and in the study of the water waves. On the other hand, specially if a = 1, w = −1 and f (u) = u 2 are taken, then bad Boussinesq equation is obtained u ζ ζ (η, ζ ) − u ηη (η, ζ ) − u ηηηη (η, ζ ) + [u 2 ]ηη = 0
(3)
which is linearly unstable equation and also numerical results are unreliable for this type. Kalantorov and Ladyzhenskaya showed that this equation unsolvable, but also they presented that it is solvable under some special conditions [4]. Like good Boussinesq equation, also bad Boussinesq eqution arises in physics engineering, plasma physics, optic etc. In this article, the numerical solution of time fractional order “good” Boussinesq equation and generalized Boussinesq equation with initial and boundary conditions wil
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