Investigation of Some Time–Space M-Truncated Partial Differential Equations: Lie Symmetry Analysis, Exact Solutions and Conservation Laws

The Lie group analysis adapted to study the symmetry properties of the time–space M-truncated partial differential equations is discussed. Based on the infinitesimal approach, we have constructed a proper extension of the classical prolongation formulas of point transformations for the time–space truncated M-fractional derivatives. This analysis is successfully illustrated and employed to construct symmetry groups admitted by the Burgers and the combined Korteweg–de Vries-modified Korteweg–de Vries equations. Furthermore, similarity reductions are performed, and some exact solutions, including invariants, are derived based on different approaches. Finally, in accordance with the nonlinear self-adjointness property, conservation laws are formulated for the underlying equations.