Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity

Table 4

Lie point symmetries of the partial differential equation (94).

Case 1. 𝑚,𝑛≠0,𝑚≠𝑛	Case 2. 𝑛=0,𝑚≠0
𝑋1=𝑚+𝑛𝑡𝜕𝑚−𝑛𝜕𝜕𝑡+𝑧−2𝜕𝑧𝜕𝑚−𝑛𝜕𝜙	𝑋1𝜕=𝜉(𝑡)𝜕𝑡
𝑋2=𝜕𝜕𝑡	𝑋2=𝜕𝜕𝑧
𝑋3=𝜕𝜕𝑧	𝑋3𝜕=𝑧−2𝜕𝑧𝑚𝜕𝜕𝜙
Case 3. 𝑛≠0,𝑚≠0,𝑚=𝑛	Case 4. 𝑚=0,𝑛≠0
𝑋1𝜕=𝑡−1𝜕𝑡𝑛𝜕𝜕𝜙	𝑋1𝜕=−𝑡𝜕𝜕𝑡+𝑧+2𝜕𝑧𝑛𝜕𝜕𝜙
𝑋2=𝜕𝜕𝑡	𝑋2=𝜕𝜕𝑧
𝑋3=𝜕𝜕𝑧	𝑋3=𝜕𝜕𝑡