Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.