Conditional Lie–Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations

Conditional Lie–Bäcklund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations

“…CLBS of INDE (1.8) is also studied in [45]- [47], where the determining equation used is within the framework of Lie's operator. Solving the nonlinear form of determining equation is as difficult as solving the original equation, thus we used special forms of CLBS (3.8)-(3.11) there.…”

“…obtained in [45] and [46]. It is noted that all the results there can be derived by using the method of linear determining equation here.…”

“…(1.8) are also considered in [45]- [47]. The special cases of the second-order CLBSs (1.6) are derived in [45,46], where the sign-invariants and two-dimensional invariant subspaces of the considered equations can be got due to the CLBSs admitted by the corresponding equations. Third-order CLBSs (1.7) related to three-dimensional invariant subspaces of Eq.…”

“…It is noted that all the results there can be derived by using the method of linear determining equation here. The cases which are not appeared in [45,46] are listed in Table 1.…”

“…As a consequence, the substitution of u into INDE (1.8) will finally identify the time evolutions of those constants. Here several examples are presented to illustrate the reduction procedure and these examples have not been given in [45] and [46]. Example 1.…”