In heat transfer, the melting of a PCM is one of the most applicable processes in energy storage systems. Obtaining a fast conversion rate from solids into liquids with minimum cost is the major demand of the modern technology, which can be made possible by reducing operational time. While there are already available works on the melting of a PCM, there is still insufficient modelling of the quadratic form of temperature-dependent thermal conductivity and the generalized form of temperature-dependent convection. This paper attempts to address these gaps. We have successfully developed a wavelet-based approximation function for the temperature profile in HBIM. A comparison of the proposed temperature profile with a previous available temperature profile in HBIM is presented in particularly, and the obtained results show excellent acceptance. The effect of various problem parameters like Stefan number, Péclet number, and coefficient of variation on temperature and melting front is extensively analyzed. In each case of temperature-dependent convection, it is obtained that there is an increment in the temperature field with increasing the values of the Stefan number, Péclet number, and coefficient of variation. As a result, melting process gets accelerated. Moreover, the exponential profile of temperature-dependent convection is the best fit to achieve fast melting of a PCM.