Abstract
Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.
Highlights
In this paper we consider the one-dimensional sine-Gordon equation ∂2u ∂t2 = ∂2u ∂x2 −sin (u), x ∈ (L1, L2), t ≥ 0, (1)with initial conditions u (x, 0) = φ1 (x), ut (x, 0) = φ2 (x) . (2)The Dirichlet boundary conditions are given by u (L1, t) = ψ1 (t), u (L2, t) = ψ2 (t), t ≥ 0. (3)The nonlinear sine-Gordon equation arises in many different applications such as propagation of fluxion in Josephson junctions [1], differential geometry, stability of fluid motion, nonlinear physics, and applied sciences [2]
The sine-Gordon equation (1) is a particular case of Klein-Gordon equation, which plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [3], given by
Numerical solution of nonlinear sine-Gordon equation has been obtained without using any transformation or without linearizing the nonlinear term
Summary
Dehghan and Shokri [13] solved the equation using collocation points and approximate the solution using radial basis functions; Dehghan and Mirzaei [14] used a boundary integral equation method; Rashidinia and Mohammadi [15] developed two implicit finite difference schemes, by using spline function approximations. Approximate solutions of sine-Gordon equation are obtained using a modified cubic B-spline collocation method (MCBCM) in space and strong stability preserving Runge-Kutta (SSP-RK54) scheme [20] in time. We use SSP-RK54 scheme to solve the obtained system of ODEs. Numerical solution of nonlinear sine-Gordon equation has been obtained without using any transformation or without linearizing the nonlinear term.
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