Shanks extrapolation method and exact solutions of equations of nonlinear mathematical physics

We propose a procedure for constructing exact solutions of equations of nonlinear mathematical physics based on the application of the Shanks extrapolation method to a segment of a perturbation series in powers of exponents that are solutions of a sequence of linear problems. We assume that a sequence of partial sums of the power series belongs to the Shanks transformation kernel. In the Shanks method, the initial value of the order of the linear combination is chosen to be one greater than the order of the pole of the solution to the original equation. The efficiency of the method is demonstrated in the construction of exact localized solutions of a nonlinear heterogeneous ordinary differential equation, the generalized Tzitzéica equation, as well as its difference and differential–difference analogues.