Practical analytical methods for partial differential equations (Q2817988)

A variety of known techniques for solving and approximating the solutions of partial differential equations (PDEs) is considered. In Section 2 the method of characteristics for first-order PDEs is given. In Section 3 the second-order linear PDEs NEWLINE\[NEWLINE A\frac{\partial^2f}{\partial t^2}+B\frac{\partial^2f}{\partial x\partial t}+C\frac{\partial^2f}{\partial x^2}=0 NEWLINE\]NEWLINE in two variables are classified. Then in Section 4 the second-order 1D wave equation \(\frac{\partial^2u}{\partial t^2} -c^2 \frac{\partial^2u}{\partial x^2}=0\) and the general 1D hyperbolic equation are discussed. Elliptic equations are briefly considered in Section 5. The method of separation of variables is discussed in Section 6. In Section 7 the theory of matched asymptotic expansions for the ODEs having a small parameter \(\varepsilon\) is developed. Ten exercises are presented in Section 8 and the solution to example 9 is given in Section 9.NEWLINENEWLINEFor the entire collection see [Zbl 1343.00028].