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<p>Uncertain delay differential equations are important mathematical tools for studying the evolution laws of dynamic systems with delay characteristics over time in uncertain environments. However, the current research on the numerical solution methods for this type of equations is still insufficient, and there are also several paradoxes in the related theoretical system that need to be clarified. In order to improve the theoretical and algorithmic research on numerical solutions of uncertain delay differential equations, this paper focuses on a specific form of uncertain delay differential equations and systematically explores the mathematical properties of their <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> paths and the relevant applications in numerical solutions. Specifically, this paper first provides a complete proof of the continuity and monotonicity of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> paths of the specific form of uncertain delay differential equations with respect to the parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation>. Utilizing these properties, this paper further demonstrates the relationship between the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> paths and the solution of the original equation, that is, the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> paths are the inverse uncertainty distributions of the corresponding solutions. Based on the above conclusions, this paper finally proposes a numerical algorithm based on the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> paths that can be used to solve this type of uncertain delay differential equations.</p>

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Numerical approach for solutions of uncertain delay differential equations

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Shize Ning,
  • Yang Liu

摘要

Uncertain delay differential equations are important mathematical tools for studying the evolution laws of dynamic systems with delay characteristics over time in uncertain environments. However, the current research on the numerical solution methods for this type of equations is still insufficient, and there are also several paradoxes in the related theoretical system that need to be clarified. In order to improve the theoretical and algorithmic research on numerical solutions of uncertain delay differential equations, this paper focuses on a specific form of uncertain delay differential equations and systematically explores the mathematical properties of their \(\alpha\) paths and the relevant applications in numerical solutions. Specifically, this paper first provides a complete proof of the continuity and monotonicity of the \(\alpha\) paths of the specific form of uncertain delay differential equations with respect to the parameter \(\alpha\) . Utilizing these properties, this paper further demonstrates the relationship between the \(\alpha\) paths and the solution of the original equation, that is, the \(\alpha\) paths are the inverse uncertainty distributions of the corresponding solutions. Based on the above conclusions, this paper finally proposes a numerical algorithm based on the \(\alpha\) paths that can be used to solve this type of uncertain delay differential equations.