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.
Highlights
The numerical method for calculating doubles integrals with variable upper limits was developed. It can be divided into several stages as determining the variable region of integration; overlaying the square or rectangular grid on the integration region; separating the integration subregions consisting of square and triangular elements; applying the cubatures in the subregion with square elements; triangulation partition along variable boundary; calculating the volumes of elementary elements with triangular basis, calculating the reference integral and establishing the calculation error
The variable region of integration leads to the necessity to change the grid of its division into elementary volumes
We considered an assignment of increasing and decreasing functions describing the change in the number of integration nodes on specific examples
Summary
When solving various engineering and scientific problems, we fail to deal with the necessity to calculate double integrals with variable integration limits. Approximations of the double integral f ( x, y)dx dy are obtained under the asa c sumption that the partial derivatives of the integrand belong to the Lebesgue space L p for certain 1 ≤ p ≤ ∞ [33] In this case, it is not enough to satisfy the condition that integrand f ( x, y) is a simple integrated function. It is necessary to impose additional restrictions: if the integrand is real in the domain Ω = [ a; b] × [c; d] the formulas for numerical integration of the double integral exist under the assumption that the mixed partial derivative f xy belongs to one of the Lebesgue spaces L p (Ω) for some 1 ≤ p ≤ ∞ (if p = ∞, f ( x, y) is a continuously differentiable function of both variables). Note that zero values of the lower limits of integration can be obtained using the additive property of an integral
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