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<p>A nonlinear time fractional degenerate hyperbolic equation (TFDHE) for both weakly and strongly degenerate cases with Caputo fractional derivative is considered. The solution of such problems is not possible with traditional numerical methods on uniform grids because of the spatial singularity in the equation. A higher-order augmented finite volume method, which is an advancement of traditional finite volume method is used to find the solution of nonlinear TFDHE on uniform grids. In this method, the whole spatial domain split into two subdomains (singular and regular) by choosing a suitable intermediate point near the singular point. Then, the solution on singular subdomain is obtained by the Puisuex series and the solution on the regular subdomain is found by standard higher-order finite volume method. The main step is to recover the Puisuex series solution by Picard iteration method on the singular subdomain and combine the both subdomains that are also main difficulties because of the involvement of the time fractional derivative. This method combines the both subdomains by the unknown variables involved in Puiseuex series solution by finding them with the finite volume method over the regular subdomain. This method find the solution on the whole domain without creating mesh points inside the singular subdomain, which is also the main advantage. In this work, finite volume schemes having second-order in time while second and fourth-orders in space are used. Moreover, by using the discrete energy method the schemes are proved temporal second-order while spatial second and fourth-orders in discrete <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-norm. Finally, the efficiency of the method is tested by numerical examples for weakly degenerate case, strongly degenerate case, blow-up coefficient at degenerate point and for the solution having singularity at the left spatial boundary. The comparison of the new method with the traditional finite volume method is also given for each numerical example.</p>

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Hybrid finite volume methods for nonlinear time fractional degenerate hyperbolic equation

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Muhammad Aamir Ali,
  • Zhiyue Zhang

摘要

A nonlinear time fractional degenerate hyperbolic equation (TFDHE) for both weakly and strongly degenerate cases with Caputo fractional derivative is considered. The solution of such problems is not possible with traditional numerical methods on uniform grids because of the spatial singularity in the equation. A higher-order augmented finite volume method, which is an advancement of traditional finite volume method is used to find the solution of nonlinear TFDHE on uniform grids. In this method, the whole spatial domain split into two subdomains (singular and regular) by choosing a suitable intermediate point near the singular point. Then, the solution on singular subdomain is obtained by the Puisuex series and the solution on the regular subdomain is found by standard higher-order finite volume method. The main step is to recover the Puisuex series solution by Picard iteration method on the singular subdomain and combine the both subdomains that are also main difficulties because of the involvement of the time fractional derivative. This method combines the both subdomains by the unknown variables involved in Puiseuex series solution by finding them with the finite volume method over the regular subdomain. This method find the solution on the whole domain without creating mesh points inside the singular subdomain, which is also the main advantage. In this work, finite volume schemes having second-order in time while second and fourth-orders in space are used. Moreover, by using the discrete energy method the schemes are proved temporal second-order while spatial second and fourth-orders in discrete \(L_{2}\) L 2 -norm. Finally, the efficiency of the method is tested by numerical examples for weakly degenerate case, strongly degenerate case, blow-up coefficient at degenerate point and for the solution having singularity at the left spatial boundary. The comparison of the new method with the traditional finite volume method is also given for each numerical example.