MIXED QUADRATURE RULE FOR DOUBLE INTEGRALS

A mixed quadrature rule of precision five for double integrals which is a linear combination of Simpson’s 3 8 th rule and Gauss-Legendre-2 point rule, where each constituent rule is of degree of precision three in two variables is formulated. The rule is numerically tested taking some suitable texts and the error bound is determined. AMS Subject Classification: 65D30, 65D31


Introduction
A mixed quadrature rule of higher degree of precision has been formed by different researchers [2], [3], [4].These rules were meant for single integral.In the same vein here we have developed a mixed quadrature rule of degree of precision-5 for double integrals taking the convex combination of Simpson's 3 8 th and Gauss-Legendre-2 point rule each of degree of precision 3.This paper has been designed as follows.Section 2 contains formulation of quadrature of con-stituent rules and the corresponding errors in two variables.Section 3 has developed for construction of mixed quadrature rules.The error analysis has been done in Section 4. In Section 5 the rule is nu merically verified by taking suitable examples.The conclusions are drawn in Section 6.

Construction of Quadrature Rule
For approximate evaluation of real definite integral The Simpson's 3 8 th rule is ( The Gauss-Legendre's two point rule is Here each rule, i.e. equation ( 2)and (3) has a precision 3. Hence where E S 3 8 (f ) and E GL2 (f ) are error in approximating the integrals I(f ) by equation (2) and equation (3) respectively.
As the error contains at least fourth derivative of the integral function, it vanishes for all polynomials of degree ≤ 3 in x and y.Thus the formula becomes exact for all polynomials of degree up to 3 i.e. degree of precision of the formula is 3.

Error in Gauss-Legendre's Two Point Rule
The error associated with the Gauss-Legendre 2-point rule is obtained substituting equation( 6)and equation (11) in equation ( 5), i.e.
In this case also, the error contains at least fourth derivative of the integral function.Thus the degree of the precision is 3.

Numerical Verification
The approximate value of integrals: are presented in Table 1.

Conclusion
From Table 1, we find that