Approximate Solution Fredholm Integral Equation of the Second Kind by the Optimal Quadrature Method
Keywords:
Sobolev space, Fredholm integral equations, derivative optimal quadrature formula, error functional, Sobolev method, optimal coefficientsAbstract
In the world, special attention is paid to the creation of various optimal calculation methods. This article is devoted to the construction of derivative optimal quadrature formulas in the space of differentiable functions using the Sobolev method. This quadrature formula consists of a linear combination of the values of the interval [0, 1] up to the second derivative of the function at all nodes. The error of the quadrature formulas is estimated by the norm of the error function. We obtain the optimal quadrature formula by minimizing the norm of the error functional by the coefficients of the derivative quadrature formula. The resulting optimal quadrature formulas are exact for all degree functions. In addition, some methods for the numerical solution of Fredholm integral equations of the second type are given. These methods are derivative optimal quadrature formulas and Simpson’s 1/3 method. Numerical examples are provided to demonstrate the effectiveness and accuracy of the work presented.
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