Application of Haar wavelet to shear-wave equation and corresponding fractional differential equation

Abstract

Shear waves find applications in several branches of science, such as geophysics, earth science, medical science etc. The Haar wavelet (HW) scheme is employed to solve the governing equation of the horizontal component of the shear wave (SH). The solutions of SH waves obtained from HW are compared with the exact solutions and some of the available results from approximation methods, such as the homotopy perturbation method (HPM) and wavelet Galerkin method with Daubechies wavelet (WG). HW solutions are found to be more accurate than WG at points away from the resonance and at the proximity of the resonance. HW yields solutions with higher accuracy than HPM solutions. The SH wave equation is also studied using the concept of fractional calculus by introducing arbitrary parameter $\alpha$, especially in the vicinity of the resonance with the values of $\alpha$ around one. The solutions are found to be damped oscillatory for $ \alpha <1 $, and diverging oscillatory for $\alpha > 1 $, respectively. The solutions are insensitive to small variations $ \alpha $ at and around the resonance point corresponding to the ODE. At a point far from the resonance, the solution with $\alpha \approx 1 $ matches nicely with those for $ \alpha \ne 1 $. The amplitude of the solution for $ \alpha= 1 $ becomes very large at a point very close to the resonance. In contrast, amplitudes of the solutions for $\alpha \ne 1$ remain the same in the vicinity of the resonance, including it. Therefore, if necessary, the parameter $ \alpha $ may be the control to avoid resonance.