数学7202。偏微分方程1。(4小时)
介绍了偏微分方程,其理论基础,和他们的应用程序,包括光学、波的传播(光、声音、和水),电场理论和扩散。主题包括一阶方程方法的特点;线性、拟线性和非线性方程组;应用交通流和几何光学;高阶方程的原则;幂级数和Cauchy-Kowalevski定理;二阶方程的分类;线性方程和广义的解决方案;波方程在不同空间维度;依赖域范围的影响; Huygens’ principle; conservation of energy, dispersion, and dissipation; Laplace’s equation; mean values and the maximum principle; the fundamental solution, Green’s functions, and Poisson kernels; applications to physics; properties of harmonic functions; the heat equation; eigenfunction expansions; the maximum principle; Fourier transform and the Gaussian kernel; regularity of solutions; scale invariance and the similarity method; Sobolev spaces; and elliptic regularity.