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Theory and Modern Applications

Table 1 Some similarity reduced equations

From: Some new exact solutions of \((3+1)\)-dimensional Burgers system via Lie symmetry analysis

\(OP_{i} \)

P

\(U_{i}\)

Similarity reduced equations

\(X_{3}, X_{2}, X_{5}\)

\(\frac{z}{x}\)

xu

\(\varphi ''+\varphi ''p^{2}+2p\varphi '\eta +4p\varphi '+2\varphi =2p\varphi '\psi +2\psi \varphi \)

xv

\(\varphi 'p+\varphi =0\)

xw

\(\varphi '=0\)

\(X_{1}, X_{3}, X_{4}\)

\(\frac{z}{x}\)

\(\frac{x(2tu+y)}{2t}\)

\(\varphi ''+p^{2}\varphi ''+2p\varphi '\eta +4p\varphi '+2\varphi =2p\varphi '\psi +2\psi \varphi \)

\(\frac{x(2tv+x)}{2t}\)

\(\varphi 'p+\varphi =0\)

\(\frac{x(2tw+z)}{2t}\)

\(\varphi '=0\)