... u(x) = sin(x).(1.1)Here (a), (b) and (c) are “first order” equations, (d) is second order, and(e) is fourth order. So the order of an equation refers to the highest orderderivative involved in ... thesimple problem we considered above, we note that convergence is obtainedas ∆t is reduced.Some further discussion concerning numerical methods for ordinary dif-ferential equations is given in Project ... studied in later chapters.1.4.1 First-Order Homogeneous Equations Consider the following first-order homogeneous partial differential equation,ut(x, t)+a(x, t)ux(x, t)=0,x∈ R, t>0, (1.20)with...