Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

In this research work, a new derived optimal quadrature formula is discussed, which includes the sum of the values of the function and its first and second order derivatives at the points located at the same distance on the interval [0,1] in the L2(m)(0,1) space. we first obtain an analytical representation of the error function norm, and a system of equations of the Wiener-Hopf type construct using the method of Lagrange unknown multipliers for finding the conditional extremum of multivariable functions. Optimal coefficients found by solving the system. Using the exact form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula for m=3 and m=4 calculate and the order of approximation was shown to be O(hm). The obtain theoretical conclusions confirmed by numerical experiments.

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