A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically

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A Strang Splitting Approach Combined with Chebyshev Wavelets to Solve the Regularized Long-Wave Equation Numerically ¨ Omer Oru¸c , Alaattin Esen and Fatih Bulut Abstract. In this manuscript, a Strang splitting approach combined with Chebyshev wavelets has been used to obtain the numerical solutions of regularized long-wave (RLW) equation with various initial and boundary conditions. The performance of the proposed method measured with three different test problems. To measure the accuracy of the method, L2 and L∞ error norms and the I1 , I2, I3 invariants are computed. The results of the computations are compared with the existing numerical and exact solutions in the literature. Mathematics Subject Classification. 65T60, 65N35. Keywords. Chebyshev wavelet method, Strang splitting, regularized long-wave equation, nonlinear phenomena, numerical solution.

1. Introduction The regularized long-wave (RLW) equation which was first proposed by Peregrine [1] in 1966 to calculate the development of an undular bore is one of the model partial differential equations (PDEs) of the nonlinear dispersive waves that governs a large number of important physical phenomena such as the nonlinear transverse waves in shallow water, ion-acoustic, and magnetohydrodynamic waves in plasma and phonon packets in nonlinear crystals, the anharmonic lattice longitudinal dispersive waves in elastic rods, pressure waves in liquid–gas bubble mixtures, rotating flow down a tube, the lossless propagation of shallow water waves, thermally exited phonon packets in low-temperature nonlinear crystals [2–4], and it is given by the equation:   ∂u ∂u ∂ ∂2u ∂u + + u −μ = 0, (1.1) ∂t ∂x∗ ∂x∗ ∂t ∂x2∗ 0123456789().: V,-vol

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with the boundary conditions u → 0 as x∗ → ±∞, where , μ are positive parameters and the subscripts t and x∗ denote time and space differentiation, respectively. Since the exact solution of this equation can be found only for a specific initial condition and conditions given at the boundaries, researchers have been trying different approaches to find numerical solution of this equation. Such as; Refs. [5,6] utilized finite-difference methods, Gardner et al. [7] have used least-squares technique with linear space–time finite elements. Zaki [8] used cubic B-spline finite-element method with time splitting technique. Least square quadratic and cubic B-spline finite-element method are used by Da˘ g et al. [9,10]. A lumped Galerkin method employed by Esen and Kutluay [11]. Do˘ gan [12] applied linear Galerkin finite-element method. Cubic splines and splitting technique are used by Jain et al. [13]. Raslan [14] used cubic Bspline finite-element method. Da˘ g et al. [15] adopted differential quadrature method which is based on cosine expansion. Saka et al. [3,16–19] used collocation and Galerkin finite-element methods. A meshfree method is employed by Islam et al. [20]. Irk et al. [21] made use of quartic trigonometric B-spline method. Exponential B-spline Galerkin method is employed by Zorsahin et al. [22]. Yagmurlu e