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Numerical solutions and conservation laws for nonlinear evolution equations

  • Anisha und Rajni Rohila EMAIL logo
Veröffentlicht/Copyright: 10. Dezember 2024
Zeitschrift für Naturforschung A
Aus der Zeitschrift Zeitschrift für Naturforschung A Band 80 Heft 1

Abstract

This paper presents numerical solutions of nonlinear evolution equations using a hybrid collocation method. Nonlinear evolution equations, including the regularized long wave (RLW) equation and the modified regularized long wave (MRLW) equation, play a crucial role in modeling various physical phenomena. A hybrid collocation technique is used for estimating and examining the characteristics of the solitary waves, including their shape, structure, and propagation. The Crank–Nicolson method is used for time discretization and the hybrid collocation method for space discretization. The Fourier series analysis has been used to analyze the stability of the proposed method, and it is established that the hybrid collocation method is unconditionally stable. The accuracy of the proposed scheme is checked by computing the error norm L∞ and the three invariants. The novelty of the method lies in deriving new approximations for the second derivative and applying it on time-dependent nonlinear partial differential equations. A comparison with existing techniques in the literature is conducted to check the improvements in results. The numerical outcomes show that the proposed scheme effectively depicts the conservation laws of solitary waves. The values of three invariants at different time levels have been shown to coincide with their analytical values. The propagation of one, two, and three solitary waves, development of the Maxwellian initial condition into one, two, and more solitary waves, and wave undulations have been illustrated graphically. The method captures the collisions between solitary waves very accurately. Our findings demonstrate that the new cubic B-spline approach offers an accurate and effective solution for the nonlinear evolution equations.

Keywords: MRLW and RLW equations; cubic B-spline functions; finite-difference method; stability analysis
Corresponding author: Rajni Rohila, Department of Applied Sciences, The Northcap University, Gurugram, India, E-mail: 

  1. Research ethics: Not applicable.

  2. Informed consent: Informed consent was obtained from all individuals included in this study, or their legal guardians or wards.

  3. Author contributions: Dr. Rajni Rohila developed the methodology and C++ codes. Codes were simulated by Ms Anisha. Paper was written by Anisha, and final editing was done by Dr. Rajni Rohila.

  4. Use of Large Language Models, AI and Machine Learning Tools: Not applicable.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: No funding.

  7. Data availability: Data will be made available on reasonable request.

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Received: 2024-07-02
Accepted: 2024-11-25
Published Online: 2024-12-10
Published in Print: 2025-01-29

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Bright and dark optical chirp waves for Kundu–Eckhaus equation using Lie group analysis
  4. Numerical solutions and conservation laws for nonlinear evolution equations
  5. Solid State Physics & Materials Science
  6. On the structural, electronic, and thermoelectric properties of EuMg2X2 (X = P, As, Sb, Bi) zintl phase; A first principles investigations
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https://doi.org/10.1515/zna-2024-0148
Schlagwörter für diesen Artikel
MRLW and RLW equations; cubic B-spline functions; finite-difference method; stability analysis

Artikel in diesem Heft

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Bright and dark optical chirp waves for Kundu–Eckhaus equation using Lie group analysis
  4. Numerical solutions and conservation laws for nonlinear evolution equations
  5. Solid State Physics & Materials Science
  6. On the structural, electronic, and thermoelectric properties of EuMg2X2 (X = P, As, Sb, Bi) zintl phase; A first principles investigations
  7. Implications of incorporating CrFeNiTiZn high-entropy alloy on the tribomechanical and microstructure morphology behaviour of A356 composites processed by friction stir processing
  8. Improving the functionality of biosensors through the use of periodic and quasi-periodic one-dimensional phononic crystals
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