Discussion of numerical and analytical techniques for the emerging fractional order murnaghan model in materials science

The Murnaghan model of the doubly dispersive equation, which is well-known in the field of materials research, is taken into consideration in this work. This equation is resolved using three different mathematical methods. The analytical method can produce traveling wave solutions by utilizing the wave transform. While bright-dark soliton and bright optical soliton solutions are produced by using auxiliary equation method, hyperbolic type traveling wave solutions are produced with the 1/G ' -expansion method. The solutions produced by both analytical methods are different from the literature. A new discussion area is created using traveling wave solutions to this problem, which also has a Conformable derivative operator. It is a cognitive fact that the solutions of partial differential equations shed light on the physical phenomenon. In light of this scientific fact, it comprises scientific debates that take into account the material's density difference, Poisson's ratio, material-specific wave velocity, and laboratory predefined values. These discussions are supported by the solutions obtained by numerical technique. In addition, the accuracy of the results obtained with the numerical technique is analyzed with the L-2 and L-infinity norm errors and the datas have been compared with the table. After discussing the advantages and disadvantages of both mathematical techniques, scientific interpretations that will shed light on physical and chemical phenomena, the effects of parameters on the solution function through traveling wave solutions are discussed and supported by graphics.