Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation

Table 3

Conservation laws of (3) for some special choices of 𝑓 and 𝑔 (𝑛≠0).

𝑓	𝑔	Symmetries	Conserved vector with substitution 𝑣
			𝐶^𝑡=(𝑣𝑡−𝛼𝑣)(𝑢+𝑥𝑢𝑥)−𝑣(𝑢𝑡+𝑥𝑢𝑥𝑡)
1𝑢^2	ln𝑢	𝑥𝜕𝜕𝜕𝑥−𝑢𝜕𝑢	𝐶^𝑥=𝑥𝑣𝑢𝑡𝑡+𝛼𝑥𝑣𝑢𝑡+𝑣
			𝑣=𝐶1+𝐶2𝑒^𝛼𝑡
			𝐶^𝑡=2𝑛𝑢−𝑥𝑢𝑥𝛼𝑣−𝑣𝑡2+𝑣𝑛𝑢𝑡−𝑥𝑢𝑥𝑡
𝑢^𝑛	1	𝑥𝜕+𝜕𝑥2𝑢𝑛𝜕𝜕𝑢	𝐶^𝑥=𝑥𝑣𝑢𝑡𝑡+𝛼𝑣𝑢𝑡−𝑢^𝑛𝑢𝑥𝑣𝑥+2𝑛𝑢^𝑛+1𝑣𝑥−21+𝑛𝑣𝑢^𝑛𝑢𝑥
			𝑣=𝐶1+𝐶2𝑒^𝛼𝑡𝐶3+𝐶4𝑥
1	𝑢	𝑡𝜕𝜕𝜕𝑥+𝑥−𝜕𝑡(𝑡+𝛼𝑥)𝑢2𝜕𝜕𝑢	𝐶^𝑡=−𝑣(1+𝑥+𝛼𝑡)𝑢𝑥+12(𝑡+𝛼𝑥)𝑢𝑡+121+𝛼𝑡+𝛼^2𝑥𝑢+𝑡𝑢𝑥𝑡+𝑥𝑢𝑥𝑥+𝑣𝑡12(𝑡+𝛼𝑥)𝑢+𝑥𝑢𝑡+𝑡𝑢𝑥
			𝐶^𝑥=𝑣(1+𝑥+𝛼𝑡)𝑢𝑡+12(𝑡+𝛼𝑥)𝑢𝑥+12(𝑡+𝛼+𝛼𝑥)𝑢+𝑡𝑢𝑡𝑡+𝑥𝑢𝑥𝑡−𝑣𝑥12(𝑡+𝛼𝑥)𝑢+𝑥𝑢𝑡+𝑡𝑢𝑥
			𝑣=𝐶1+𝐶2𝑒^𝛼𝑡+(𝐶3+𝐶4𝑒^𝛼𝑡)𝑒^𝑥 or 𝑏(𝑡,𝑥)
1	1	𝑡𝜕𝜕𝜕𝑥+𝑥−𝜕𝑡𝛼𝑥𝑢2𝜕𝜕𝑢	𝐶^𝑡𝛼=−𝑣2𝑥𝑢𝑡+𝑥𝑢𝑥𝑥+𝛼^22𝑥𝑢+𝛼𝑡𝑢𝑥+𝑢𝑥+𝑡𝑢𝑥𝑡
			+𝑣𝑡𝑥𝑢𝑡+𝑡𝑢𝑥+𝛼2𝑥𝑢
			𝐶^𝑥𝛼=𝑣2𝑥𝑢𝑥+𝑡𝑢𝑡𝑡+𝛼2𝑢+𝛼𝑡𝑢𝑡+𝑢𝑡+𝑥𝑢𝑥𝑡
			−𝑣𝑥𝑥𝑢𝑡+𝑡𝑢𝑥+𝛼2𝑥𝑢
			𝑣=(𝐶1+𝐶2𝑒^𝛼𝑡)(𝐶3+𝐶4𝑥) or 𝐶1𝑒^𝛼𝑡𝑢+𝑏(𝑡,𝑥)