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On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations

Karel Van Bockstal (UGent) , Mahmoud A. Zaky and Ahmed Hendy
(2023) FRACTIONAL CALCULUS AND APPLIED ANALYSIS. 26. p.2175-2201
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Abstract
In this contribution, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is considered. Avoiding the circumstances of expressing the nonlinear variable-order fractional wave equations via closed-form expressions in terms of special functions, we investigate the existence and uniqueness of this problem with Rothe's method. First, the weak formulation for the considered wave problem is proposed. Then, the uniqueness of a solution is established by employing Gronwall's lemma. The Rothe scheme's basic idea is to use Rothe functions to extend the solutions on single-time steps over the entire time frame. Inspired by that, we next introduce a uniform mesh time-discrete scheme based on a discrete convolution approximation in the backward sense. By applying some reasonable assumptions to the given data, we can predict a priori estimates for the time-discrete solution. Employing these estimates side by side with Rothe functions leads to proof of the solution's existence over the whole time interval. Finally, the full discretisation of the problem is introduced by invoking Galerkin spectral techniques in the spatial direction, and numerical examples are given.
Keywords
Fractional calculus, Variable-order, Wave equation, Rothe's discretization, Galerkin spectral method, Existence and uniqueness, NUMERICAL-METHODS, REGULARITY

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Citation

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MLA
Van Bockstal, Karel, et al. “On the Rothe-Galerkin Spectral Discretization for a Class of Variable Fractional-Order Nonlinear Wave Equations.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS, vol. 26, 2023, pp. 2175–201, doi:10.1007/s13540-023-00184-x.
APA
Van Bockstal, K., Zaky, M. A., & Hendy, A. (2023). On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations. FRACTIONAL CALCULUS AND APPLIED ANALYSIS, 26, 2175–2201. https://doi.org/10.1007/s13540-023-00184-x
Chicago author-date
Van Bockstal, Karel, Mahmoud A. Zaky, and Ahmed Hendy. 2023. “On the Rothe-Galerkin Spectral Discretization for a Class of Variable Fractional-Order Nonlinear Wave Equations.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 26: 2175–2201. https://doi.org/10.1007/s13540-023-00184-x.
Chicago author-date (all authors)
Van Bockstal, Karel, Mahmoud A. Zaky, and Ahmed Hendy. 2023. “On the Rothe-Galerkin Spectral Discretization for a Class of Variable Fractional-Order Nonlinear Wave Equations.” FRACTIONAL CALCULUS AND APPLIED ANALYSIS 26: 2175–2201. doi:10.1007/s13540-023-00184-x.
Vancouver
1.
Van Bockstal K, Zaky MA, Hendy A. On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations. FRACTIONAL CALCULUS AND APPLIED ANALYSIS. 2023;26:2175–201.
IEEE
[1]
K. Van Bockstal, M. A. Zaky, and A. Hendy, “On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations,” FRACTIONAL CALCULUS AND APPLIED ANALYSIS, vol. 26, pp. 2175–2201, 2023.
@article{01HMC1A8V7CQ7Q5T3MJE4SA51Q,
  abstract     = {{In this contribution, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is considered. Avoiding the circumstances of expressing the nonlinear variable-order fractional wave equations via closed-form expressions in terms of special functions, we investigate the existence and uniqueness of this problem with Rothe's method. First, the weak formulation for the considered wave problem is proposed. Then, the uniqueness of a solution is established by employing Gronwall's lemma. The Rothe scheme's basic idea is to use Rothe functions to extend the solutions on single-time steps over the entire time frame. Inspired by that, we next introduce a uniform mesh time-discrete scheme based on a discrete convolution approximation in the backward sense. By applying some reasonable assumptions to the given data, we can predict a priori estimates for the time-discrete solution. Employing these estimates side by side with Rothe functions leads to proof of the solution's existence over the whole time interval. Finally, the full discretisation of the problem is introduced by invoking Galerkin spectral techniques in the spatial direction, and numerical examples are given.}},
  author       = {{Van Bockstal, Karel and  Zaky, Mahmoud A. and  Hendy, Ahmed}},
  issn         = {{1311-0454}},
  journal      = {{FRACTIONAL CALCULUS AND APPLIED ANALYSIS}},
  keywords     = {{Fractional calculus,Variable-order,Wave equation,Rothe's discretization,Galerkin spectral method,Existence and uniqueness,NUMERICAL-METHODS,REGULARITY}},
  language     = {{eng}},
  pages        = {{2175--2201}},
  title        = {{On the Rothe-Galerkin spectral discretization for a class of variable fractional-order nonlinear wave equations}},
  url          = {{http://doi.org/10.1007/s13540-023-00184-x}},
  volume       = {{26}},
  year         = {{2023}},
}
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