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1 ode 101.1 basic notions1.2 local existence 1.2.1 normed spaces and fixed point theorem 1.2.2 applications to ode system and linear algebraic system 1.3 critical point 1.4 plane analysis for the duffing equation 1.5 homoclinic orbit and limit cycle 1.6 stability and lyapunov function 1.7 bifurcation 1.8 chaos: lorenz equations and logistic map 2 parabolic equations 532.1 introduction: bvp and ibvp, equilibrium 2.2 dispersion relation, linear and nonlinear stability 2.3 invariant domain 2.4 perturbation method 2.5 traveling waves 2.6 burgers' equation and cole-hopf transform 2.7 evolutionary duffing equation 3 elliptic equations 853.1 sobolev spaces 3.2 variational formulation 3.3 neumann boundary value problem 4 hyperbolic equations 934.1 linear advection equation, characteristics method 4.2 nonlinear hyperbolic equations 4.3 discontinuities in inviscid burgers' equation 4.4 elementary waves in inviscid burgers' equation 4.5 wave interactions in inviscid burgers' equation 4.6 elementary waves in a polytropic gas 4.7 riemann problem in a polytropic gas 4.8 elementary waves in a polytropic ideal gas 4.9 soliton and inverse scattering transform