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- W2415692080 abstract "This paper first presents a Gauss Legendre quadrature method for numerical integration of I ¼ R RT f ðx; yÞdxdy, wheref(x,y) is an analytic function in x, y and T is the standard triangular surface: {(x,y)j0 6 x, y 6 1, x + y 6 1} in the Cartesiantwo dimensional (x,y) space. We then use a transformation x = x(n,g), y = y(n,g) to change the integral I to an equivalentintegral R RS f ðxðn; gÞ; yðn; gÞÞ oðx; yÞoðn;gÞ dndg, where S is now the 2-square in (n,g) space: {(n,g)j 1 6 n,g 6 1}. We thenapply the one dimensional Gauss Legendre quadrature rules in n and g variables to arrive at an efficient quadrature rulewith new weight coefficients and new sampling points. We then propose the discretisation of the standard triangular surfaceT into n2 right isosceles triangular surfaces Ti (i = 1(1)n2) each of which has an area equal to 1/(2n2) units. We haveagain shown that the use of affine transformation over each Ti and the use of linearity property of integrals lead to theresult:I ¼ Xnni¼1Z ZT if ðx; yÞdxdy ¼ 1n2Z ZTHðX; Y ÞdX dY ;where HðX; Y Þ ¼ Pnni¼1 f ðxiðX; Y Þ; yiðX; Y ÞÞ and x = xi(X,Y) and y = yi(X,Y) refer to affine transformations which mapeach Ti in (x,y) space into a standard triangular surface T in (X,Y) space. We can now apply Gauss Legendre quadratureformulas which are derived earlier for I to evaluate the integral I ¼ 1n2RRT HðX; Y ÞdX dY . We observe that the above procedurewhich clearly amounts to Composite Numerical Integration over T and it converges to the exact value of the integralRRT f ðx; yÞdxdy, for sufficiently large value of n, even for the lower order Gauss Legendre quadrature rules. We havedemonstrated this aspect by applying the above explained Composite Numerical Integration method to some typicalintegrals." @default.
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- W2415692080 date "2007-01-01" @default.
- W2415692080 modified "2023-09-22" @default.
- W2415692080 title "Symmetric Gauss Legendre quadrature formulae for composite numerical integration over a tetrahedral region." @default.
- W2415692080 hasPublicationYear "2007" @default.
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