Scholarlink

<p>This paper develops an L2-type semi-discrete approach for solving time-fractional nonlocal Sobolev-type equations. To handle the initial singularity, both the analysis and computation are performed on nonuniform temporal meshes. By utilizing the relationship between the Caputo fractional derivative and Riemann-Liouville fractional integral involving mixed derivatives, we apply the novel L2 formula and thereby construct a time semi-discrete scheme. Several auxiliary lemmas are employed to establish the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^{1}\)</EquationSource> </InlineEquation>-stability of this method under mild conditions on the time-step ratios. Moreover, under reasonable regularity assumptions, a rigorous error analysis on graded meshes is carried out and then several bounds for the total truncation errors are obtained. Owing to the lack of monotonicity in the coefficients of L2 formula, we are unable to use the well-established discrete Grönwall inequality and thus fail to demonstrate the convergence of propoesd scheme. In this case, we instead present a conjecture consistent with the general expectations and provide verification through numerical examples.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

An L2-type time-discrete method for mixed fractional Sobolev-type equations on general nonuniform meshes

  • 期刊论文

  • 发表日期：

    2026年2月10日

查看全文
  • Mingchao Zhao,
  • Omid Nikan

摘要

This paper develops an L2-type semi-discrete approach for solving time-fractional nonlocal Sobolev-type equations. To handle the initial singularity, both the analysis and computation are performed on nonuniform temporal meshes. By utilizing the relationship between the Caputo fractional derivative and Riemann-Liouville fractional integral involving mixed derivatives, we apply the novel L2 formula and thereby construct a time semi-discrete scheme. Several auxiliary lemmas are employed to establish the \(H^{1}\) -stability of this method under mild conditions on the time-step ratios. Moreover, under reasonable regularity assumptions, a rigorous error analysis on graded meshes is carried out and then several bounds for the total truncation errors are obtained. Owing to the lack of monotonicity in the coefficients of L2 formula, we are unable to use the well-established discrete Grönwall inequality and thus fail to demonstrate the convergence of propoesd scheme. In this case, we instead present a conjecture consistent with the general expectations and provide verification through numerical examples.