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

Table 2

Lie point symmetries of the partial differential equation (20).

Case 1. 𝑚≠1,𝑚≠𝑛,𝑛≠0	Case 4. 4𝑛=0,𝑚≠0,𝑚≠3
𝑋1𝜕=(2−𝑛−𝑚)𝑡𝜕𝜕𝑡+(𝑛−𝑚)𝑧𝜕𝜕𝑧+2𝜙𝜕𝜙	𝑋1𝜕=𝜉(𝑡)𝜕𝑡
𝑋2=𝜕𝜕𝑡	𝑋2𝑚=−2𝑧𝜕𝜕𝜕𝑧+𝜙𝜕𝜙
𝑋3=𝜕𝜕𝑧	𝑋3=𝜕𝜕𝑧
Case 2. 𝑛≠0,𝑚=1,𝑛≠1	Case 5. 𝑚=0,𝑛=0
𝑋1𝜕=−𝑡𝜕𝜕𝑡+𝑧+𝜕𝑧2𝜙𝜕𝑛−1𝜕𝜙	𝑋1𝜕=𝜉(𝑡)𝜕𝑡
𝑋2=𝜕𝜕𝑡	𝑋2=𝜕𝜕𝑧
𝑋3=𝜕𝜕𝑧	𝑋3𝜕=sinh(2𝑧)𝜕𝜕𝑧+𝜙cosh(2𝑧)𝜕𝜙
Case 3. 4𝑚=3,𝑛=0	𝑋4𝜕=cosh(2𝑧)𝜕𝜕𝑧+𝜙sinh(2𝑧)𝜕𝜙
𝑋11=−3𝑧^2𝜕𝜕𝜕𝑧+𝑧𝜙𝜕𝜙	𝑋5𝜕=𝜙𝜕𝜙
𝑋2𝜕=𝜉(𝑡)𝜕𝑡	𝑋𝐴𝜕=𝐴(𝑡,𝑧)𝜕𝜙
𝑋3𝜕=𝑧−3𝜕𝑧2𝜙𝜕𝜕𝜙	𝐴(𝑡,𝑧) satisfies (20) with 𝑛=𝑚=0
𝑋4=𝜕𝜕𝑧