Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels
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 th...
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Ferdowsi University of Mashhad
2018-10-01
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doaj-67d624a99e8442cc8540d89c4cc009572021-02-24T08:56:06ZengFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-69772423-69692018-10-01829511810.22067/ijnao.v8i2.6077824722Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernelsP. Mokhtary0Sahand University of Technology, Tabriz,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.https://ijnao.um.ac.ir/article_24722_f2a2b860955ac201cd28b207349de840.pdfdiscrete collocation methodmuntz-logarithmic polynomialsquadrature methodvolterra type weakly singular integral equations with logarithmic kernels |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
P. Mokhtary |
spellingShingle |
P. Mokhtary Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels Iranian Journal of Numerical Analysis and Optimization discrete collocation method muntz-logarithmic polynomials quadrature method volterra type weakly singular integral equations with logarithmic kernels |
author_facet |
P. Mokhtary |
author_sort |
P. Mokhtary |
title |
Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels |
title_short |
Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels |
title_full |
Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels |
title_fullStr |
Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels |
title_full_unstemmed |
Discrete collocation method for Volterra type weakly singular integral equations with logarithmic kernels |
title_sort |
discrete collocation method for volterra type weakly singular integral equations with logarithmic kernels |
publisher |
Ferdowsi University of Mashhad |
series |
Iranian Journal of Numerical Analysis and Optimization |
issn |
2423-6977 2423-6969 |
publishDate |
2018-10-01 |
description |
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. |
topic |
discrete collocation method muntz-logarithmic polynomials quadrature method volterra type weakly singular integral equations with logarithmic kernels |
url |
https://ijnao.um.ac.ir/article_24722_f2a2b860955ac201cd28b207349de840.pdf |
work_keys_str_mv |
AT pmokhtary discretecollocationmethodforvolterratypeweaklysingularintegralequationswithlogarithmickernels |
_version_ |
1724253090499002368 |