Least-square-based transform methods for analytical solutions of fractional-order nonlinear (1+1)-dimensional dynamics equations

In this work, we consider the analysis of the nonlinear Burgers' equation as a fractional-order partial differential equation using the Caputo derivation. Two new, effective methods, the least-square residual power series transform method and the least-square variational iteration transform method, are introduced to deal with the fractional Burgers' equation. These techniques combine the merits of least-square approximation, residual power series, variational iteration, and integral transformations to obtain accurate and convergent solutions. The Caputo derivative describes the fractional-order dynamics, which provides the best robustness and accuracy to explain complex physical processes. The proposed schemes are systematically implemented for Burgers' equation with a nonlinear source and their performances are illustrated by a comprehensive comparison and numerical process. The results highlight the precision, efficiency, and reliability of the least-square residual power series transform method and least-square variational iteration transform method methods in the solution of fractional-order complex space partial differential equations and show that the least-square residual power series transform method and least-square variational iteration transform method are dependable methods for researchers and engineers in the field of applied mathematics and mathematical physics. This work has made an essential contribution to the relatively new field of fractional calculus and provides a powerful tool for solving other non-linear fractional differential equations in science and engineering.