Numerical Solution of Nonlinear Fifth-Order KdV-Type Partial Differential Equations via Haar Wavelet
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Numerical Solution of Nonlinear Fifth-Order KdV-Type Partial Differential Equations via Haar Wavelet Sidra Saleem1
· Malik Zawwar Hussain1
Accepted: 28 September 2020 © Springer Nature India Private Limited 2020
Abstract In this research article, the numerical solution of different forms of widely used onedimensional Korteweg–de Vries equation is discussed. For this purpose, Haar wavelet collocation method is implemented. A simple algorithm is constructed which is based on the proposed method. The presented method is tested on fifth-order Lax equation, Sawada–Kotera equation, Caudrey–Dodd–Gibbon equation, Kaup–Kuperschmidt equation and Ito equation. The obtained approximate results are displayed using tables and figures. The numerical results show good accuracy of the proposed method. Keywords Haar wavelet collocation method · Lax equation · Sawada–Kotera equation · Caudrey–Dodd–Gibbon equation · Kaup–Kuperschmidt equation · Ito equation Mathematics Subject Classification 65M06 · 65M12 · 65M70
Introduction The nonlinear partial differential equations (PDEs) with distinct orders are involved in the scientific fields such as chemical kinetics, wave propagation, quantum mechanics, electro magnetism, heat conduction, plasma physics, mathematical physics and biology, nonlinear optics, fluid dynamics and modeling of different phenomena. These PDEs are extensively used to describe physical systems. Mostly, important features of physical systems are hidden in their nonlinear nature. The analytic solutions of these nonlinear PDEs may not be available or very hard to find. With the help of appropriate methods for solution of nonlinear systems, these phenomena can be studied [1–12]. In 1895, for modeling of Russell’s phenomena of solitons Kortweg and de Vries investigated a PDE (called KdV equation afterwards). According to, them these solitons are substantial solitary waves and behave like particles. To deal with the dissipative wave phenomena in many fields of science like fluid dynamics, optics, plasma physics and quantum mechanics etc, the KdV equations are entertained as the models of Mathematics. The fifth-
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Sidra Saleem [email protected]; [email protected] Department of Mathematics, University of the Punjab, Lahore, Pakistan 0123456789().: V,-vol
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order KdV type equation is normally used to study various nonlinear phenomena in plasma physics and it has a vital role in the wave propagations [4]. The KdV type equations involve dissipative terms of order three and five, which are related to the problem of magneto-acoustic wave in such plasma which is cold collision free and dissipative terms represent generation near critical angle. Plasma is defined as an electrically conductive fluid that is quasi neutral and complex as well. It consists of electric and magnetic fields because of its electrical conductive behavior. When the particles and fields interact, different types of waves are involved. So, for heating plasmas, fluctuations an
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