Purpose

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

Design/methodology/approach

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Findings

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

Research limitations/implications

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

Practical implications

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

Social implications

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

Originality/value

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

中文翻译：

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弱奇异积分方程插值方法的精度和效率

目的

本研究旨在讨论弱奇异 Volterra 和 Fredholm 积分方程的数值解，这些方程用于模拟工程中的热传导和静电势理论等问题，采用改进的拉格朗日多项式插值技术结合双共轭梯度稳定方法（ BiCGSTAB）。积分方程唯一解的存在性框架是在巴纳赫收缩原理和 Bielecki 范数的背景下提供的。

设计/方法论/途径

作者应用修正的拉格朗日多项式方法来逼近第二类弱奇异Volterra和Fredholm积分方程的数值解。

发现

使用上述方法对未知函数进行插值会生成一个代数方程组，该方程组可以通过适当的经典技术求解。进一步证明了该方法的收敛性和误差估计的一些定理。给出的算例证明了该方法的实用性、有效性和可靠性。与弱奇异型Fredholm积分方程相比，当前技术对于弱奇异型Volterra积分方程效果更好。此外，还提供了说明性示例和比较来显示该方法的有效性和实用性，这表明本方法与参考方法相比效果良好。计算由 MATLAB 软件执行。

研究局限性/影响

这些方法的收敛性取决于解的平滑度，在由非平滑核的积分方程建模的各种应用中找到解并通过计算逼近它是具有挑战性的。传统的分析技术（例如投影方法）在这些情况下效果不佳，因为生成的线性系统是无条件的且难以解决。此外，证明收敛性和估计误差可能很困难。它们的实施通常也很昂贵。

实际影响

对于这些类型的方程，非常需要快速、用户友好的数值技术。此外，多项式是最常用的数学工具，因为它易于表达、在现代计算机上计算快速且易于定义。因此，他们多年来分别对近似和数值等理论和分析做出了重大贡献。

社会影响

这项工作提出了一种处理弱奇异积分方程的有用方法，而不涉及任何变量变化过程以消除解的奇异性。

原创性/价值

据作者所知，作者声称他们的工作具有原创性和有效性，强调了其首次在解决弱奇异 Volterra 和 Fredholm 积分方程方面的成功应用。重要的是，该方法承认并保留了解决方案可能的奇异性，这是该领域研究人员尚未探索的一个新颖方面。