A Numerical Method for Computing Double Integrals with Variable Upper Limits

Olha Chernukha, Yurii Bilushchak, Natalya Shakhovska and Rastislav Kulhánek
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Olha Chernukha: Centre of Mathematical Modelling, Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, 15 Dudayev Str., 79005 Lviv, Ukraine
Yurii Bilushchak: Centre of Mathematical Modelling, Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, 15 Dudayev Str., 79005 Lviv, Ukraine
Natalya Shakhovska: Institute of Computer Sciences and Information Technologies, Lviv Polytechnic National University, 12 Bandera Str., 79013 Lviv, Ukraine
Rastislav Kulhánek: Department of Information Systems, Faculty of Management, Comenius University, Odbojárov 10, 814 99 Bratislava, Slovakia

Mathematics, 2021, vol. 10, issue 1, 1-26

Abstract: We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

Keywords: double integral; variable upper limit; variable integration region; division element; variable grid; cubature; triangulation; Taylor series; absolute error; relative error (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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