One-dimensional Optimal Systems And Similarity Reductions Of A Few Nonlinear Partial Differential Equations

It is one of the important ways to solve the accurate solution of partial diferential equa-tion by reducing diferential equation through looking for symmetry. Therefore, we need tostudy more symmetry of diferential equation to get more accurate solutions.Using Lie symmetric method, this paper analyzes two single parameter symmetry groupLi of (2+1)-dimensional nonlinear partial diferential equations and the corresponding ex-pression of the group. And on this basis, we construct one dimensional optimal system ofthe Lie point symmetry group. And we also similarly reduce the original equation by usingeach element of the one dimensional optimal system.Firstly, we get the allowed Lie point symmetry cluster of (2+1)-dimensional nonlinearZK equation(ut+uux+uxxx+uxyy=0)and(2+1)-dimensional nonlinear difusion equation (ut uxuy2uuxy=0). Meanwhile, we give the commutation relation of Lie symmetry cluster andcorresponding expression. Using this corresponding expression, we obtain the correspondingadjoint matrix of Lie point symmetry cluster. So we set up its one-dimensional optimalsystem. According to all the elements of the one-dimensional optimal system, we similarlyreduce the two (2+1)-dimensional nonlinear partial diferential equations. In the reducingprocess, we get the reduced equations and all solutions of the same group.Secondly, we studythe expanded group symmetry of Black-Scholes equation of (u2t+12a2xuxx+bxuxuxx cu=0)that puts the coefcient as independent variables. We use the new symmetry expanded tosimplify equation, and thus we simplify the equation.Finally, it is the summary and outlookof the whole contents.