Classification of the Group Invariant Solutions for Contaminant Transport in Saturated Soils under Radial Uniform Water Flows

Table 2

Adjoint representation for the base vectors.

Ad	𝑋1	𝑋2	𝑋3	𝑋4	𝑋5	𝑋6
𝑋1	𝑋1	𝑋2	𝑋3	−2𝜖𝑋1+𝑋4	−2𝜖𝑋3+𝑋5	4𝜖^2𝑋1+2𝜖𝑋2−4𝜖𝑋4+𝑋6
𝑋2	𝑋1	𝑋2	𝑋3	𝑋4	𝑋5	𝑋6
𝑋3	𝑋1	𝑋2	𝑋3	−𝜖𝑋3+𝑋4	𝜖𝑋2+𝑋5	−𝜖^2𝑋2+𝑋6−2𝜖𝑋5
𝑋4	𝑒^2𝜖𝑋1	𝑋2	𝑒^𝜖𝑋3	𝑋4	𝑒^−𝜖𝑋5	𝑒^−2𝜖𝑋6
𝑋5	𝑋1+2𝜖𝑋3−𝜖^2𝑋2	𝑋2	−𝜖𝑋2+𝑋3	𝑋4+𝜖𝑋5	𝑋5	𝑋6
𝑋6	𝑋1−2𝜖𝑋2+4𝜖𝑋4+4𝜖^2𝑋6	𝑋2	𝑋3+2𝜖𝑋5	𝑋4+2𝜖𝑋6	𝑋5	𝑋6