We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the ...We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.展开更多
The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error function...The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function.Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_(2)^((m,0)) is obtained,moreover the existence and uniqueness of the solution of this system are discussed.Next,the discrete analogue D_(m)(hβ)of the differential operatord^(2m)/dx^(2m)-1 is constructed.Further,Sobolev's method of construction of optimal quadrature formulas in the space W_(2)^((m,0)),which based on the discrete analogue D_(m)(hβ),is described.Next,for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Finally,at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_(2)^((3,0))for the cases m=1 and m=3 are presented.展开更多
文摘We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.
基金supported by the “Korea Foundation for Advanced Studies”/“Chey Institute for Advanced Studies” International Scholar Exchange Fellowship for academic year of 2018–2019
文摘The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function.Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_(2)^((m,0)) is obtained,moreover the existence and uniqueness of the solution of this system are discussed.Next,the discrete analogue D_(m)(hβ)of the differential operatord^(2m)/dx^(2m)-1 is constructed.Further,Sobolev's method of construction of optimal quadrature formulas in the space W_(2)^((m,0)),which based on the discrete analogue D_(m)(hβ),is described.Next,for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Finally,at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_(2)^((3,0))for the cases m=1 and m=3 are presented.
