Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel

IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED
M.H. Heydari , D. Baleanu , M. Bayram
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引用次数: 0

Abstract

This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.
分段对数切比雪夫基数函数:具有对数奇异核的非线性积分方程的应用
本文介绍了一类具有对数奇异核的非线性积分方程。严格地分析了这些方程解的存在唯一性。为了便于求解,我们构造了分段对数切比雪夫基数函数(CCFs),这是一种通用的基函数族。在此框架下,推导了PLCCFs的Hadamard分数阶积分的运算矩阵。通过利用这类对数奇点与Hadamard分数积分算子之间的联系,我们开发了一种简单而强大的数值方法来求解这些方程。在所提出的方法中,首先使用带未知系数的分段对数ccf的有限展开式来逼近解。然后,通过插值和分数阶积分运算矩阵的应用,将原积分方程重新表述为一个非线性代数方程组,其解决定展开系数。通过理论和数值研究验证了该方案的收敛性分析。通过求解一些具有解析解和非解析解的说明性实例,评价了所开发方法的准确性。
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来源期刊
Applied Numerical Mathematics
Applied Numerical Mathematics 数学-应用数学
CiteScore
5.60
自引率
7.10%
发文量
225
审稿时长
7.2 months
期刊介绍: The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.
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