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Semi-Numerical, Semi-Analytical Approximations of the Rayleigh Equation for Gas-Filled Hyper-Spherical Bubble

A new semi-numerical, semi-analytical approach based on the differential transform method is proposed to solve the problems of a gas-filled hyper-spherical bubble governed by the Rayleigh equation. Semi-numerical, semi-analytical approximations are constructed for the Rayleigh equation in the form of piecewise functions. The proposed approach is compared with the standard fourth-order Runge–Kutta method and the standard differential transform method, respectively. The results reveal two main benefits of the new approach, one is that it possesses result with higher precision than the standard fourth-order Runge–Kutta method, the other is that it remains valid and accurate for longer time compared to the standard differential transform method. In addition, we also consider the Rayleigh equation in [Formula: see text] dimensions when the surface tension is not zero.