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

Table 1

Multipliers and conserved vectors for the partial differential equation (20).

Case A. 0≤𝑛<∞,0≤𝑚<∞	Case B.5. 𝑛+𝑚=1,𝑚≠1
Multiplier: Λ=1	Multiplier: Λ=𝜙^−2(𝑛−2)
𝑇^1=𝜙	𝑇^11=−2𝜙^−2𝑚𝜙^2𝑧+ln𝜙
𝑇^2=𝜙^𝑛(1+𝑚𝜙^−𝑚−1𝜙𝑡𝜙𝑧−𝜙^−𝑚𝜙𝑡𝑧)	𝑇^2=𝜙^1−𝑚1ln𝜙−𝜙1−𝑚^1−𝑚−(𝜙^1−2𝑚ln𝜙)𝜙𝑡𝑧
	+(𝑚𝜙^−2𝑚ln𝜙) 𝜙𝑡𝜙𝑧
Case B.1. 𝑚≠1,𝑚+𝑛≠1,𝑚+𝑛≠2	Case B.6. 𝑛+𝑚=2, 𝑚≠1
Multiplier: Λ=𝜙^1−𝑚−𝑛	Multiplier: 1Λ=𝜙
𝑇^1=12−𝑚−𝑛(𝜙^2−𝑚−𝑛1−1)+2(1−𝑚−𝑛)𝜙^−2𝑚𝜙^2𝑧	𝑇^11=−2𝜙^−2𝑚𝜙^2𝑧+ln𝜙
𝑇^2=𝑛𝜙1−𝑚^1−𝑚−𝜙^1−2𝑚𝜙𝑡𝑧+𝑚𝜙^−2𝑚𝜙𝑡𝜙𝑧	𝑇^2=2−𝑚𝜙1−𝑚^1−𝑚−𝜙^1−2𝑚𝜙𝑡𝑧+𝑚𝜙^−2𝑚𝜙𝑡𝜙𝑧
Case B.2. 𝑚=1,𝑛≠0, 𝑛≠1	Case C.1. 3𝑛=𝑚−1,𝑚≠2,𝑚≠1
Multiplier: Λ=𝜙^−𝑛	Multiplier: Λ=𝜙^1−2𝑚𝜙𝑧𝑧−12(𝑚+1)𝜙^−2𝑚𝜙^2𝑧
𝑇^1=1−𝑛1−𝑛2𝜙^−2𝜙^2𝑧	+𝑚−2𝜙3−2𝑚^3−2𝑚
𝑇^2=𝑛ln𝜙−𝜙^−1𝜙𝑡𝑧+𝜙^−2𝜙𝑡𝜙𝑧	𝑇^1=1𝜙1−𝑛^1−𝑛−𝑛2𝜙^−2𝜙^2𝑧
Case B.3. 𝑚=𝑛=1	+12(3−𝑚)𝜙^1−2𝑚𝜙^2𝑧+1𝜙2(3−2𝑚)^4−2𝑚
Multiplier: 1Λ=𝜙	𝑇^2=16𝑚(𝑚+5)𝜙^−2𝑚−2𝜙^3𝑧−(𝑚−3)(𝑚−1)()𝜙3−2𝑚^1−2𝑚𝜙𝑧𝜙𝑡
𝑇^11=−2𝜙^−2𝜙^2𝑧+ln𝜙	−12(𝑚+1)𝜙^−2𝑚−1𝜙^2𝑧+(2−𝑚)𝜙(3−2𝑚)^2−2𝑚𝜙𝑡𝑧
𝑇^2=ln𝜙−𝜙^−1𝜙𝑡𝑧+𝜙^−2𝜙𝑡𝜙𝑧	−12(𝑚−1)𝜙^−𝑚−1𝜙^2𝑧+(𝑚−1)𝜙(3−2𝑚)^2−𝑚
Case B.4. 𝑚=1,𝑛=0	Case C.2. 3𝑛=𝑚−1,𝑚=21,𝑛=2
Multiplier: Λ=𝜙^2	Multiplier: Λ=𝜙^−2𝜙^2𝑧𝑧−54𝜙^−3𝜙^2𝑧−12ln𝜙
𝑇^1=12𝜙^−2𝜙^2𝑧+𝜙ln𝜙−𝜙	𝑇^1=12𝜙^−3𝜙^2𝑧𝑧−78𝜙^−5𝜙^4𝑧+34𝜙^−2𝜙^2𝑧
𝑇^2=−(𝜙^−1ln𝜙) 𝜙𝑡𝑧+(𝜙^−2ln𝜙)𝜙𝑡𝜙𝑧	+121𝜙ln𝜙−2𝜙
	𝑇^2=138𝜙^−5𝜙^3𝑧−𝜙^−2𝜙𝑧+34(𝜙^−2ln𝜙)𝜙𝑧𝜙𝑡−54𝜙^−4𝜙^2𝑧+12𝜙^−1ln𝜙𝜙𝑡𝑧
	−14𝜙^−(5/2)𝜙^2𝑧+12𝜙^1/2ln𝜙