Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

Algorithm 1

Consider the nonlinear generalized time fractional sixth order partial differential equation (13) along with the initial condition (14). Let be a solution of model (13) and (14) that has th order partial derivatives in conformable sense at any point . Then, to obtain the th approximation, execute the underlying steps:

Step A. Expand the solution of model (13) about as follows

Step B. Give a definition of the th-truncated solution of in view of the initial condition (14) as follows

Step C. Do truncate the th residual error of so

where .

Step D. Invoke the series solution obtained in Step B to the th-truncated residual error obtained in Step C as follows

Step E. Employ for every to the obtained equation in Step D to get

Step F. To obtain the first few terms for with the aid of execute the following subroutine:

F1. In Step E, put n=1, compute and find solution for to get .

F2. Once again, in Step E, set n=2, compute and find solution for to get .

F3. Once again, n Step E, set , compute and find solution for to get .

F4. Proceed for arbitrary order by setting , computing and establishing the new equation to get the th coefficients .

Step G. Keep the new components in an infinite series form. In fact, the closed form of the solution can be established in such way, that is, , if the relation of the pattern is very regular. If it was not the case, the solutions can be approximately obtained. Then, Stop.