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<p>The Clannish Random Walker’s Parabolic (CRWP) equation is conveniently applicable in the fields of scientific research, as are the empirical studies of ecology, sociology, urban planning, and materials science, where comprehending the congregate behavior of individuals within a particular spatial context is necessary. By solving this equation, researchers gain insights into how individuals interact, form groups, and navigate within their environment, leading to a better understanding of emergent patterns and more efficient system design. This study incorporates the time <i>M</i>-fractional CRWP equation using two powerful and efficient techniques with the modified <i>M</i>-fractional derivative. To get the more solitary wave solutions of the <i>M</i>-fractional CRWP equation, the New form of modified Kudryashov (NMK) and simplest equation (SE) techniques are utilized. The NMK and SE techniques explore the solitary wave solutions of the types comprising exponential, hyperbolic, and trigonometric functions. Using Python programming, we have demonstrated some novel and significant phenomena from the numerical condition on the obtained soliton solutions of the <i>M</i>-fractional CRWP model. To define the characteristics of the <i>M</i>-fractional CRWP equation, we show some solitary wave patterns with three-dimensional plots. Also, to check the behavior of the <i>M</i>-fractional parameters, the two-dimensional plots are captured. By these plot systems, we compare the effect of the <i>M</i>-fractional derivative with the original classical derivative form. The cross periodic lump wave, kink wave, anti-kink wave, kinky periodic wave, interaction wave between kink and periodic lump wave, interaction wave between anti-kink and periodic lump wave, multi-lump wave, periodic lump wave, interaction wave between kink and cross periodic wave, cross periodic lump wave, curved-type kink wave, anti-kink wave, periodic wave are visualized by the proposed techniques. Thus, it is concluded that the previously mentioned methods could potentially be a helpful instrument to generate unique, accurate soliton solutions for various types of applications, critical to the fields of materials science, ecology, sociology, and urban planning, where understanding the collective behavior of individuals within a spatial context is essential and can effectively illustrate the core ingredients of the climate change.</p>

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Simulation of lump, interaction of lump and kink, anti-kink wave, and solitary wave propagation of M-fractional Clannish Random Walker’s Parabolic model with two analytic techniques

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Md. Mamunur Roshid,
  • Mahtab Uddin,
  • Ashek Ahmed,
  • Harun -Or- Roshid,
  • Mohammad Safi Ullah,
  • Hadi Rezazadeh

摘要

The Clannish Random Walker’s Parabolic (CRWP) equation is conveniently applicable in the fields of scientific research, as are the empirical studies of ecology, sociology, urban planning, and materials science, where comprehending the congregate behavior of individuals within a particular spatial context is necessary. By solving this equation, researchers gain insights into how individuals interact, form groups, and navigate within their environment, leading to a better understanding of emergent patterns and more efficient system design. This study incorporates the time M-fractional CRWP equation using two powerful and efficient techniques with the modified M-fractional derivative. To get the more solitary wave solutions of the M-fractional CRWP equation, the New form of modified Kudryashov (NMK) and simplest equation (SE) techniques are utilized. The NMK and SE techniques explore the solitary wave solutions of the types comprising exponential, hyperbolic, and trigonometric functions. Using Python programming, we have demonstrated some novel and significant phenomena from the numerical condition on the obtained soliton solutions of the M-fractional CRWP model. To define the characteristics of the M-fractional CRWP equation, we show some solitary wave patterns with three-dimensional plots. Also, to check the behavior of the M-fractional parameters, the two-dimensional plots are captured. By these plot systems, we compare the effect of the M-fractional derivative with the original classical derivative form. The cross periodic lump wave, kink wave, anti-kink wave, kinky periodic wave, interaction wave between kink and periodic lump wave, interaction wave between anti-kink and periodic lump wave, multi-lump wave, periodic lump wave, interaction wave between kink and cross periodic wave, cross periodic lump wave, curved-type kink wave, anti-kink wave, periodic wave are visualized by the proposed techniques. Thus, it is concluded that the previously mentioned methods could potentially be a helpful instrument to generate unique, accurate soliton solutions for various types of applications, critical to the fields of materials science, ecology, sociology, and urban planning, where understanding the collective behavior of individuals within a spatial context is essential and can effectively illustrate the core ingredients of the climate change.