Hi, I need to send an abstract of my paper to the journal, but I have doubts about the grammatical correctness of the text. I am asking for your help, please check it.

In this paper, the axisymmetric dynamic problem of determining the stress state in the region of a circular crack in the finite cylinder is solved. The bottom end of the cylinder is rigidly fixed, and a load in the form of tangential stresses, which depend on time, is attached to the top end of the cylinder. Unlike traditional analytical methods based on the use of the Laplace transform, the proposed method consists in the difference approximation only of the time derivative. For this purpose, specially selected non-equidistant nodes and the special representation of the solution in these nodes are used. Such an approach makes it possible to reduce the original problem to a sequence of boundary problems for the homogeneous Helmholtz equation. Each of these problems is solved by using finite integral Fourier and Hankel transforms, with their subsequent reversal. As a result, an integral representation was obtained for the angular displacement through an unknown jump of this displacement in the plane of the crack. For the derivative of this jump from the boundary condition on a crack, an integral equation is obtained, which, as a result of using the Weber-Sonin integral operator and a series of transformations, is reduced to the Fredholm integral equation of the second kind with respect to the unknown function associated with the jump. An approximate solution of this equation is implemented by the collocation method, and the integrals are approximated by Gauss–Legendre quadrature formulas. The computed numerical solution made it possible to obtain an approximate formula for calculating the stress intensity factor (SIF). Using this formula, we studied the effect of the nature of the load and the geometric parameters of the cylinder on the time dependence of the SIF. Analysis of the results showed that for all the types of loading considered, the maximum of the SIF values was observed during the transient process. When a sudden constant load is applied, this maximum is 2-2.5 times the static value of the SIF. With a sudden harmonic load, the maximum SIF also significantly exceeds the values that it acquires with steady-state oscillations, in the absence of resonance. Increasing the height of the cylinder and a decreasing the crack area lead to an increase in the transient period and a decrease in the magnitude of the maximum SIF. The same effect is observed when the crack plane approaches the fixed end of the cylinder.

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