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In this paper, we consider the problem of constructing new optimal explicitand implicit Adams-type difference formulas for finding an approximate solutionto the Cauchy problem for an ordinary differential equation in a Hilbert space.In this work, I minimize the norm of the error functional of the differenceformula with respect to the coefficients, we obtain a system of linearalgebraic equations for the coefficients of the difference formulas. Thissystem of equations is reduced to a system of equations in convolution and thesystem of equations is completely solved using a discrete analog of adifferential operator $d^2/dx^2-1$. Here we present an algorithm forconstructing optimal explicit and implicit difference formulas in a specificHilbert space. In addition, comparing the Euler method with optimal explicitand implicit difference formulas, numerical experiments are given. Experimentsshow that the optimal formulas give a good approximation compared to the Eulermethod.

Optimization of Adams-type difference formulas in Hilbert space $W_2^{(2,1)}(0,1)$