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

Table 3

Multipliers and conserved vectors for the partial differential equation (94).

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Case A. 0≤𝑛<∞,0≤𝑚<∞

Multiplier: Λ(𝜙)=1

𝑇¹=𝜙

𝑇²=exp𝑛𝜙−1+𝑚exp(𝑛−𝑚)𝜙−1𝜙_𝑡𝜙_𝑧−exp(𝑛−𝑚)𝜙−1𝜙_𝑡𝑧

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Case B.1. 𝑚+𝑛≠0,𝑚>0,𝑛>0

Multiplier: Λ(𝜙)=exp−(𝑚+𝑛)𝜙

𝑇¹1=−−()𝜙−1𝑚+𝑛exp𝑚+𝑛2(𝑛+𝑚)exp𝑚−𝑛−2𝑚𝜙𝜙²_𝑧

𝑇²𝑛=−𝑚exp−𝑚𝜙+𝑛+exp𝑚−𝑛−2𝑚𝜙(𝑚𝜙_𝑡𝜙_𝑧−𝜙_𝑡𝑧)

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Case B.2. 𝑚=0, 𝑛=0

Multiplier: Λ(𝜙)=𝜙

𝑇¹=12𝜙²

𝑇²=𝜙_𝑡−𝜙_𝑡𝑧

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Case B.3. 𝑚=0, 𝑛>0

Multiplier: Λ(𝜙)=exp(−𝑛𝜙)

𝑇¹1=−𝑛1exp(−𝑛𝜙)−2𝑛exp−𝑛𝜙𝜙²_𝑧

𝑇²=𝑛exp−𝑛𝜙−𝜙_𝑡𝑧

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Case B.4. 𝑛=0, 𝑚>0

Multiplier: Λ(𝜙)=exp−𝑚𝜙

𝑇¹1=−𝑚−1exp−𝑚𝜙2𝑚exp𝑚1−2𝜙𝜙²_𝑧

𝑇²=exp𝑚1−2𝜙(𝑚𝜙_𝑡𝜙_𝑧−𝜙_𝑡𝑧)

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Case C. 𝑚=𝑛, 𝑛≠0

Multiplier: Λ(𝜙,𝜙_𝑧,𝜙_𝑧𝑧)=exp−2𝑛𝜙𝜙_𝑧𝑧−12𝑛exp−2𝑛𝜙𝜙²_𝑧−14𝑛2𝑛𝜙+1exp1−𝑛−2𝑛𝜙

𝑇₁=12exp−2𝑛𝜙𝜙²_𝑧𝑧−12𝑛²exp−2𝑛𝜙𝜙⁴_𝑧+14𝑛(2𝑛𝜙+1)𝜙²_𝑧+18𝑛²2𝑛𝜙+1exp1−𝑛−2𝑛𝜙

𝑇₂=𝑛²exp−2𝑛𝜙𝜙³_𝑧−14𝑛2𝑛𝜙+1exp1−𝑛−2𝑛𝜙)𝜙_𝑧𝜙_𝑡−𝑛exp(−𝑛𝜙+1)𝜙_𝑡𝑧

   −142𝑛𝜙+1exp1−𝑛−2𝑛𝜙

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