Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels

Iranian Journal of Numerical Analysis and Optimization

مقاله 4، دوره 8، شماره 2 - شماره پیاپی 14، دی 2018، صفحه 95-118   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v8i2.60778

نویسنده

P. Mokhtary^*

Sahand University of Technology, Tabriz,

چکیده

An efficient discrete collocation method for solving Volterra type weakly singular integral equations with logarithmic kernels is investigated. One of features of these equations is that, in general the first erivative of solution behaves like as a logarithmic function, which is not continuous at the origin.
In this paper, to make a compatible approximate solution with the exact ones, we introduce a new collocation approach, which applies the M¨untz logarithmic polynomials(Muntz polynomials with logarithmic terms) as basis functions. Moreover, since implementation of this technique leads to integrals with logarithmic singularities that are often difficult to solve numerically, we apply a suitable quadrature method that allows the exact evaluation of integrals of polynomials with logarithmic weights. To this end, we first remind the well-known Jacobi–Gauss quadrature and then extend it to integrals with logarithmic weights. Convergence analysis of the proposed scheme are presented, and some numerical results are illustrated to demonstrate the efficiency and accuracy of the proposed method.

کلیدواژه‌ها

Discrete collocation method؛ Muntz-logarithmic polynomials؛ Quadrature method؛ Volterra type weakly singular integral equations with logarithmic kernels

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