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

Table 4

Group invariant solutions for (4) given 𝐷(𝑣)=𝜆𝑒^2𝜙.

Symmetry	Reductions	𝑎∈ℝ	Group invariant solutions
𝑋5+𝑎𝑋2	𝑒𝐶=exp^−2𝜙−𝜏4𝜆+𝑎4𝜆(𝜏𝐺𝛾8𝜆ln),; where 𝛾=𝑒√^𝜙4𝜏 and 𝐺 satisfies; 𝜆^2𝛾^6𝐺^′′−14𝛾𝜆1−12𝛾𝜆^4𝐺^′−𝑎8𝐺=0.	𝑎<0	𝑒𝐶=exp^−2𝜙−𝜏4𝜆+(𝜏)4𝜆𝑎ln×𝑘8𝜆2𝑒^−2𝜙4√𝜆𝜏^1𝐹14+𝑎,38𝜆2;−𝑒^−4𝜙16𝜆𝜏+𝑘11𝐹1𝑎,18𝜆2;−𝑒^−4𝜙.16𝜆𝜏
		𝑎=0	𝑒𝐶=exp^−2𝜙−𝜏4𝜆𝑘4𝜆^2−√𝜆𝜋𝑘1𝑒erf^−2𝜙4√.𝜆𝜏
		𝑎>0	𝑒𝐶=exp^−2𝜙−𝜏4𝜆+𝑎4𝜆(𝜏×𝑘8𝜆ln)2𝑒^−2𝜙4√𝜆𝜏^1𝐹14−𝑎,38𝜆2𝑒;−^−4𝜙16𝜆𝜏+𝑘11𝐹1−𝑎,18𝜆2𝑒;−^−4𝜙.16𝜆𝜏
𝑋1+𝑋4	𝐶=exp8𝜆𝜏^33+𝜏𝑒^−2𝜙−𝜏^2𝐺(𝛾),; where 𝛾=4𝜏^2+𝑒^−2𝜙 and 𝐺 satisfies; 𝐺^′′−1𝐺2𝜆^′−𝛾4𝜆𝐺=0.	—	𝐶=exp8𝜆𝜏^33+𝜏𝑒^−2𝜙−𝜏^2×𝑒4𝜏^2+𝑒^−2𝜙𝑘4𝜆^1Ai1+44𝜏^2+𝑒^−2𝜙(4𝜆)^4/3+𝑘2Bi1+44𝜏^2+𝑒^−2𝜙(4𝜆)^4/3.
𝑋1+𝑎𝑋2	𝐶=𝑒^𝑎𝜏𝐺𝛾,; where 𝛾=𝜙 and 𝐺 satisfies; 𝐺^′′+𝑒2+^−2𝜙𝜆𝐺^′+𝑎𝜆𝑒^−4𝜙𝐺=0.	𝑎<0	𝐶𝜏,𝜙=𝑒^𝑎𝜏−1exp√4𝜆−1−1−4𝑎𝜆𝑒^−2𝜙𝑘1−1+exp√4𝜆−1+1−4𝑎𝜆𝑒^−2𝜙𝑘2.
		𝑎=0	𝐶=𝑘2𝑒+exp^−2𝜙2𝑘1.
		𝑎>0	𝐶=𝑒^𝑎𝜏1exp√4𝜆1−1+4𝑎𝜆𝑒^−2𝜙𝑘11+exp√4𝜆1+1+4𝑎𝜆𝑒^−2𝜙𝑘2.
𝑋1+𝑋6+𝑎𝑋2			Leading to reductions which are unsolvable