Wavelet-based techniques for Hygro-Magneto-Thermo vibration of nonlocal strain gradient nanobeam resting on Winkler-Pasternak elastic foundation

The primary objective of this article is to investigate the vibration characteristics of a nanobeam embedded in a Winkler-Pasternak elastic foundation adopting wavelet-based two relatively recent techniques, namely the Haar Wavelet Method (HWM) and the Higher Order Haar Wavelet Method (HOHWM). While subjected to a longitudinal magnetic field, the nanobeam is also exposed to hygroscopic and thermal environments. Hamilton's principle is being used to model the nanobeam within the frameworks of Euler-Bernoulli beam theory and nonlocal strain gradient theory. The effects of several parameters such as, the small scale, length scale, magnetic field intensity, hygroscopic, thermal, Winkler modulus and shear modulus on frequency parameters, are thoroughly investigated for PP (Pined-Pined) and CC (Clamped-Clamped) boundary conditions. Richardson's formula is used to determine the rate of convergence of both the Haar wavelet and higher order Haar wavelet methods, and the results obtained by the Haar wavelet technique are also extrapolated, revealing that the rate of convergence of extrapolated HWM is comparable to the higher order Haar wavelet method. The effectiveness of the higher order Haar wavelet method over the Haar wavelet method is demonstrated using PP (Pined-Pined) boundary condition as a test case. Furthermore, the results of this study have been cross-validated with previously published literature in specific circumstances, exhibiting exceptional agreement.