Sentence examples for formulas for solution from inspiring English sources

In the present paper, we not only elaborate the theory of Fredholm boundary value problems with data on logarithmic spirals, but also construct explicit formulas for solution.

This enables easier getting formulas for solutions to system (4).

Using such obtained formulas in (47) and (48), we get formulas for solutions to system (2) in this case.

Recently several papers presenting formulas for solutions to some difference equations and systems, but without mentioning any theory, have appeared.

Theorem 10 allows one to construct explicit formulas for solutions of the Dirichlet problem in domains bounded by logarithmic spirals.

Using the obtained formulas for these twelve sequences in (47) and (48), we obtain closed form formulas for solutions to system (2).

Moreover, we elaborated a constructive invertibility theory for Toeplitz operators and thus derive explicit solvability conditions as well as formulas for solutions.

We also derive the formulas for solutions of the boundary value problem for the Burgers equation (see (4.26) and the heat equation on a time-space scale.

We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.

Since (z_{n}) and (w_{n}) defined by (47) and (48) are solutions to (2) such that their values for (n=0,1) are the same, it follows that they are closed form formulas for solutions to system (2).

However, our method allows one not only to develop a Fredholm theory for the corresponding boundary value problems, but also to obtain formulas for solutions in a closed form.