The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach

This work scrutinizes the well-known nonlinear non-classical Sobolev-type wave model which addresses the fluid flow via fractured rock, thermodynamics and many other fields of modern sciences. The nonlinear non-classical Sobolev-type wave model provides a more comprehensive and accurate description of wave phenomena in a wide range of fields. By incorporating both nonlinearity and the complexities of dispersive waves, these models enhance our understanding of natural phenomena and enable more precise predictions and applications in various scientific and engineering disciplines. Therefore, this study is investigating it. Prior to this study, no previous research has performed Lie symmetry analysis and achieved invariant solutions of this kind. The symmetry generators are taking into account the Lie invariance criteria. The suggested approach produces the three dimensional Lie algebra, where translation symmetries in space and time are associated with mass conservation and conservation of energy, respectively and the other symmetries are scaling or dilation. The nonlinear non-classical Sobolev-type wave partial differential equation is transformed into a system of highly nonlinear ordinary differential equations by employing appropriate similarity transformations through using Lie group methodology. The power series technique is used to generate exact wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The graphical behaviour of certain solutions is demonstrated in 3-D and 2-D for particular quantities of the physical factors in the investigated equation.