New homotopic algorithm following solution curve

The homotopic method for solving nonlinear problems is very effective. The new algorithm was proposed, in which the homotopic differential equations were integrated first by Runge-Kutta method to obtain the first approximate solutions, as initial values of next integration, then the equations continuously were integrated to get better approximate solutions until the solutions enter into Newton's convergence range, finally the Newton correction was employed to get the desired solutions. The algorithm overcomes shortcomings in Li-Yorke algorithm in correction time, calculation time and integral step. The algorithm is more convenient, efficient for solving engineering problems.