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A Simple Method For Interpolating and Integrating Discontinuous Functions ‎
  • Emil Shoukralla
Emil Shoukralla
Menoufia University

Corresponding Author:[email protected]

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Abstract

There are many scientific problems related to the topic of interpolation, especially for non-‎continuous functions, as well as the evaluating of the integrals of such non-continuous ‎‎functions. These needs appear clearly in the development of profit and loss plans in economic ‎‎projects, stock exchange transactions, and investment funds, solving ecological issues and ‎‎recording data on the evolution of epidemics and virology. We investigate a new straightforward ‎‎method for interpolating and integrating discontinuous valued functions. The method is based ‎‎on some modified ‎matrix-vector ‎barycentric Lagrange interpolation formulas. We developed ‎‎seven rules for the ‎optimum distribution of nodes ‎inside the domain of integration; five for ‎‎single-valued ‎discontinuous functions, and two rules for the two independent variables ‎‎discontinuous ‎functions. We designed these rules to be depending on the length of the ‎‎integration domain and ‎the degree of the interpolant polynomials. Thus, we obtained uniform ‎‎interpolation and the ‎minimum roundoff errors. Based on these rules with the application of the ‎‎modified matrix-vector barycentric formulas, we easily isolated the singularities of the ‎‎interpolant integrands and ‎evaluated the corresponding interpolant integral values with super ‎‎accuracy. The numerical ‎solutions of the given five examples show the ‎super accuracy, and ‎‎efficiency of the ‎presented method compared with a cited method.‎ ‎ Keywords- interpolation, singular integrals, computational methods‎