M.N. Imanova
On some connection between the definite and indefinite integrals


It is known that in the construction of the numerical methods for solving of initial-value problem of ODE in used the methods, which have applied to calculation of the definite integrals. However, here for the calculation of definite integrals have proposed to use the methods that have used in solving of initial-value problem for the ODEs. The definite integrals are expressed by the indefinite integrals that are the solution of the above-mentioned problem. For the construction of more exact methods of calculation of the definite integrals here proposed to use forward-jumping (advanced) and hybrid methods. Have established some connection between the Gauss and hybrid methods. And have determined some necessary conditions for the satisfying of the coefficients of the proposed methods. Constructed the stable methods with the degree p≤8. Have been shown that how received here results can be transferred to double definite integrals. For this aim, determines some connection between double integrals and single definite integrals. By using this relation have constructed methods that are applied to calculate the double integrals. Advantages of this method illustrated by calculation of model double integral by the constructed here methods.

Keywords: Numerical method, Single and double definite integrals, Indefinite integrals, Hybrid and stepwise methods, Initial-value problem, Nonlinear system of algebraic equations
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