B-Spline Solution and the Chaotic Dynamics of Troesch's Problem

A B-spline method is presented for solving the Troesch problem. The numerical approximations to the solution are calculated and then their behavior is studied and commenced. The chaotic dynamics exhibited by the solutions of Troesch's problem as they were derived by the decomposition method approximation are examined and an approximate critical value for the parameter lambda is introduced also in this study. For the parameter value slightly less than lambda approximate to 2.2, the solutions begin to show successive bifurcations, finally entering chaotic regimes at higher lambda values. The effectiveness and accuracy of the B-spline method is verified for different values of the parameter, below its critical value, where the first bifurcation occurs.