A discrete method for the logarithmic-kernel integral equation on an open arc (original) (raw)
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Ferdowsi University of Mashhad, 2018
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 derivative 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 Müntz-logarithmic polynomials(Müntz 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 inte-grals 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.
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The theory of integral equations has been an active field of research for many years and is inextricably related with other areas of Mathematics such as complex and mathematical analysis, function theory, integral transforms and functional analysis. Integral Equations arise naturally in applications, in many areas of Mathematics, Engineering, Science and Technology and have been studied extensively both at the theoretical and practical level. It is significant to note that a MathSciNet keyword search on Integral Equations returns more than eleven thousand items. In this paper, we do a brief survey of the existing literature on methods of solving integral equations of Volterra and Fredholm type of the first, second and third kind, Cauchy type singular integral equations and integral equations over an infinite interval. The objective is to classify the selected methods and evaluate their applicability while discussing challenges faced by individual researchers in this field. We also provide a rather extensive bibliography for the reader who would be interested in learning more about various theoretical and computational aspects of Integral Equations.
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