Approximate solution of MRLW equation in B-spline environment
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ORIGINAL RESEARCH
Approximate solution of MRLW equation in B‑spline environment Saumya Ranjan Jena1 · Archana Senapati1 · Guesh Simretab Gebremedhin1 Received: 5 March 2020 / Accepted: 23 July 2020 © Islamic Azad University 2020
Abstract In this paper, the numerical solution of the modified regularized long wave equation is obtained using a quartic B-spline approach with the help of Butcher’s fifth-order Runge–Kutta (BFRK) scheme. Here, any kind of transformation or linearization technique is not implemented to tackle the nonlinearity of the equation. The BFRK scheme is applied to solve the systems of first-order ordinary differential equations of time-dependent variables. Three invariants of the motion are evaluated to justify the conservative properties of the recommended scheme. Three examples are illustrated for comparing the present work with the exact solution and the results of others. The stability of the quartic B-spline collocation scheme is found to be unconditionally stable. The main advantage of the proposed scheme is to obtain better approximate solutions by applying the BFRK scheme to solve the systems of first-order ordinary differential equations without transformation or linearization technique. Keywords BFRK method · B-spline · Invariants · MRLW equation · Single solitary wave Mathematics Subject Classification 35Q20 · 65A30 · 65N35
Introduction Nonlinear partial differential equations are important in describing a variety of phenomena across a range of disciplines. In fact, it is not always possible to obtain the exact solutions of all partial differential equations. At that time, different numerical methods have been applied to approximate the solutions to such problems. Let us consider the generalized regularized long wave (GRLW) equation of the form
ut + ux + 𝛾uq ux − 𝛼uxxt = 0,
(1)
where 𝛾 and 𝛼 are positive constants and q is a positive integer. This equation is important in the propagation of * Saumya Ranjan Jena [email protected] Archana Senapati [email protected] Guesh Simretab Gebremedhin [email protected] 1
Department of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar, Odisha 24, India
nonlinear dispersive waves. Their solutions are kinds of solitary waves which are called solitons, and their shapes are not affected by collisions. The solitary waves are packets or pulses, which propagate in nonlinear dispersive media. Due to the dynamical balance between the nonlinear and dispersive effects, these waves retain a stable waveform. In the past years, several authors have proposed different approaches to obtain the approximate solution of Eq. (1) such as lumped Galerkin method with cubic B-spline [1], basis of reproducing kernel space [2], Chebyshev–Chebyshev spectral collocation scheme [3], elementfree kp-Ritz method [4], an approximate quasilinearization approach [5], Petrov–Galerkin scheme [6], parabolic Monge–Ampere moving mesh and uniform [7], septic B-spline collocation scheme with two different linearization techniques [8], P
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