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<p>Our research analyzes a refined iterative method within a new framework involving enriched non-expansive mappings in uniformly convex Banach spaces. We establish weak convergence using the well-known Opial’s condition and prove strong convergence under various domain or mapping assumptions. Through a detailed example of an enriched non-expansive mapping that is not non-expansive and graphical analysis, we demonstrate that the convergence rate of our iteration scheme is more effective than those proposed by Thakur, Ali, and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{D}^{\varvec{**}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="bold-italic">D</mi> </mrow> <mrow /> <mrow> <mrow /> <mo mathvariant="bold">∗</mo> <mrow /> <mo mathvariant="bold">∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> in a new setting. Furthermore, we investigate the impact of various parameters on the convergence behavior of our proposed method. Finally, we apply our fundamental findings to practical problems by estimating solutions for a partial differential equation, which is a model of various physical and biological processes such as chemical reactions, population dynamics, heat transfer, and the spread of diseases. These applications demonstrate the practical utility of our method and highlight its potential for addressing complex challenges in applied mathematics.</p>

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Analysis of a refined iterative method with a new setting and applications to various models of partial differential equations

  • 期刊论文

  • 发表日期：

    2025年7月7日

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  • Khairul Habib Alam,
  • Yumnam Rohen,
  • S. Surendra Singh

摘要

Our research analyzes a refined iterative method within a new framework involving enriched non-expansive mappings in uniformly convex Banach spaces. We establish weak convergence using the well-known Opial’s condition and prove strong convergence under various domain or mapping assumptions. Through a detailed example of an enriched non-expansive mapping that is not non-expansive and graphical analysis, we demonstrate that the convergence rate of our iteration scheme is more effective than those proposed by Thakur, Ali, and \(\varvec{D}^{\varvec{**}}\) D ∗ ∗ in a new setting. Furthermore, we investigate the impact of various parameters on the convergence behavior of our proposed method. Finally, we apply our fundamental findings to practical problems by estimating solutions for a partial differential equation, which is a model of various physical and biological processes such as chemical reactions, population dynamics, heat transfer, and the spread of diseases. These applications demonstrate the practical utility of our method and highlight its potential for addressing complex challenges in applied mathematics.