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

Table 3

Group invariant solutions for (4) with constant 𝐷.

Symmetry	Reductions	𝑎∈ℝ	Group invariant solutions
𝑋4+𝑎𝑋2	𝐶=𝜏^𝑎/2𝐺(𝛾), where 𝛾=𝜏𝑒^2𝜙 and 𝐺 satisfies 𝛾^3𝐺^′′+14𝛾6𝛾−1𝐺^′′−18𝑎𝐺=0.	𝑎<0	𝐶=𝜏^𝑎/2121√𝜏𝑒^𝜙𝑘21𝐹1𝑎+12,32;−14𝜏𝑒^2𝜙+𝑘11𝐹1𝑎2,12;−14𝜏𝑒^2𝜙.
		𝑎=0	𝐶=𝑘2−2√𝜋𝑘11erf2𝜏𝑒^2𝜙.
		𝑎>0	𝐶=𝜏^𝑎/2121√𝜏𝑒^𝜙𝑘21𝐹11−𝑎2,32;−14𝜏𝑒^2𝜙+𝑘11𝐹1−𝑎2,12;−14𝜏𝑒^2𝜙.
𝑋1+𝑎𝑋2	𝐶=𝑒^𝑎𝜏𝐺(𝛾), where 𝛾=𝜙 and 𝐺 satisfies; 𝐺^′′+𝐺^′−𝑎𝑒^−2𝜙𝐺=0.	𝑎<0	𝐶=𝑒^𝑎𝜏𝑘1cos√𝑎𝑒^−𝜙−𝑘2sin√𝑎𝑒^−𝜙.
		𝑎=0	𝐶=−𝑒^−𝜙𝑘1+𝑘2.
		𝑎>0	𝐶𝜏,𝜙=𝑒^𝑎𝜏𝑘1cosh√𝑎𝑒^−𝜙−𝑖𝑘2sinh√𝑎𝑒^−𝜙.
𝑋1+𝑋5	𝐶=𝑒^(𝜏𝑒−𝜙+(2/3)𝜏3)𝐺(𝛾), where 𝛾=𝜏^2+𝑒^−𝜙 and 𝐺 satisfies; 𝐺^′′−𝛾𝐺=0.	—	𝐶=exp𝜏𝑒^−𝜙+23𝜏^3Ai𝜏^2+𝑒^−𝜙𝑘1+Bi𝜏^2+𝑒^−𝜙𝑘2.
𝑋1+𝑋6+𝑎𝑋2	−𝜏𝐶=exp𝑒^2𝜙(1+4𝜏2)−14ln1+4𝜏^2+𝑎2tan^−1(2𝜏)𝐺𝛾,  where 𝛾=𝑒^𝜙√1+4𝜏^2 and 𝐺 satisfies 𝛾^6𝐺^′′+2𝛾^5𝐺^′+(1−𝑎𝛾^2)𝐺=0.	𝑎<0	−𝜏𝐶=exp𝑒^2𝜙1+4𝜏^2−14ln1+4𝜏^2+𝑎2tan^−1(×𝑒2𝜏)−𝑖/2𝑒2𝜙(1+4𝜏2)12𝑒^2𝜙1+4𝜏^2𝑘21𝐹13+𝑖𝑎4,32;𝑖𝑒^2𝜙1+4𝜏^2+𝑒^−𝑖/2𝑒2𝜙(1+4𝜏2)12𝑒^2𝜙1+4𝜏^2𝑘1𝑈3+𝑖𝑎4,32;𝑖𝑒^2𝜙1+4𝜏^2.
		𝑎=0	−𝜏𝐶=exp𝑒^2𝜙1+4𝜏^2−14ln1+4𝜏^2×12𝑒^𝜙√1+4𝜏^2𝐽−1/412𝑒^2𝜙1+4𝜏^2𝑘1Γ34+12𝑒^𝜙√1+𝜏^2𝐽1/412𝑒^2𝜙(1+4𝜏^2)𝑘2Γ54.
		𝑎>0	−𝜏𝐶=exp𝑒^2𝜙1+4𝜏^2−14ln1+4𝜏^2+𝑎2(×𝑒arctan2𝜏)−𝑖/2𝑒2𝜙(1+4𝜏2)12𝑒^2𝜙1+4𝜏^2𝑘21𝐹13−𝑖𝑎4,32;𝑖𝑒^2𝜙1+4𝜏^2+𝑒^−𝑖/2𝑒2𝜙(1+4𝜏2)12𝑒^2𝜙1+4𝜏^2𝑘1𝑈3−𝑖𝑎4,32;𝑖𝑒^2𝜙1+4𝜏^2.