This paper aims to examine a newly proposed, more flexible, and thorough definition of a novel generalization of Chebyshev wavelet basis functions, possessing more computational flexibility and resolution by analyzing its properties. A proficient collocation technique is developed to examine the nonlinear Bratu boundary-value problem, which has recently emerged in combustion and chemical reaction theory. The differential equation is transformed into a system of nonlinear algebraic equations, after which the efficacy of the method is evaluated using the $\eta$-base wavelet in comparison to other numerical instances, exact solutions, and other analytical techniques. According to the results, this method offers a reliable and flexible tool for handling challenging boundary-value issues in scientific computing.

This study focuses on the efficient and accurate solution of time-fractional nonlinear partial differential equations (PDEs), which arise in various scientific and engineering applications but are often challenging to solve due to their complexity. We propose a novel method that integrates the Laplace transform with the accelerated Adomian decomposition method (AADM), forming the Laplace transform accelerated Adomian decomposition method (LAADM). The key innovation of this approach lies in its ability to handle the fractional time derivative expressed in the Caputo sense, while enhancing convergence speed and reducing computational effort. The methodology is systematically formulated, and several numerical experiments are conducted to validate its performance. The results demonstrate that LAADM provides highly accurate solutions and exhibits superior efficiency when compared to traditional solution techniques for fractional PDEs.