The Laplace Equation and Wave Equation

摘要: In this chapter we introduce the central linear partial differential equations of second order, Laplace equation $$\Delta u = f$$ (0.1) and wave $$\left (\frac{{\partial }^{2}} {\partial {t}^{2}} -\Delta\right )u f.$$ (0.2) For flat Euclidean space \(\mathbb{R}^{n}\), operator is defined by =\frac{{\partial }^{2}u} {x}_{1}^{2}} +\cdots +\frac{{\partial {x}_{n}^{2}}.$$ (0.3)