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<p>In this article, we focus on the analytical and numerical analysis of Cauchy’s nonlinear integral equation. We establish a sufficient condition ensuring the existence and uniqueness of the solution in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2[0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> space. In order to overcome the singularity problem, a technique for regularising the equation is introduced. Next, Newton’s method is used to linearise the equation, which allows Kantorovich’s method to be applied in order to construct a numerical solution. Theoretical results are presented to demonstrate the convergence of the numerical solution towards the exact solution. Finally, numerical examples illustrate the effectiveness of the proposed method.</p>

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Examining nonlinear Cauchy-Fredholm integral equations: numerical and analytical approaches

  • 期刊论文

  • 发表日期：

    2026年3月11日

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  • Asma Ayachi,
  • Boutheina Tair,
  • Hamza Guebbai

摘要

In this article, we focus on the analytical and numerical analysis of Cauchy’s nonlinear integral equation. We establish a sufficient condition ensuring the existence and uniqueness of the solution in the \(L^2[0,1]\) L 2 [ 0 , 1 ] space. In order to overcome the singularity problem, a technique for regularising the equation is introduced. Next, Newton’s method is used to linearise the equation, which allows Kantorovich’s method to be applied in order to construct a numerical solution. Theoretical results are presented to demonstrate the convergence of the numerical solution towards the exact solution. Finally, numerical examples illustrate the effectiveness of the proposed method.