Partial Differential EquationsFirst-Order PDE: The method of characteristics.Anton DzhamayDepartment of MathematicsThe University of MichiganAnn Arbor, MI 48109wPage: http://www.math.lsa.umich.edu/~adzhamemail: adzham@umich.eduCopyright \251 2003 by Anton DzhamayAll rights reserved

In this worksheet we give some examples on how to use the method of characteristics for first-order linear PDEs of the form NiMvLCgqJi0lImFHNiQlInhHJSJ0RyIiIi0lJWRpZmZHNiQtJSJ1R0YoRipGK0YrKiYtJSJiR0YoRistRi02JEYvRilGK0YrKiYtJSJjR0YoRitGL0YrRistJSJoR0Yo . The main idea of the method of characteristics is to reduce a PDE on the (NiQlInhHJSJ0Rw==)-plane to an ODE along a parametric curve (called the characteristic curve) parametrized by some other parameter NiMlJHRhdUc=. The characteristic curve is then determined by the condition that NiMvLSUlZGlmZkc2JC0lInVHNiQtJSJ4RzYjJSR0YXVHLSUieUdGLEYtLCYqJi1GJTYkLSUidEdGLEYtIiIiLUYlNiQtRig2JEYrRjVGNUY2RjYqJi1GJTYkRipGLUY2LUYlNiRGOUYrRjZGNg== =NiMsJiomLSUiYUc2JCUieEclInRHIiIiLSUlZGlmZkc2JC0lInVHRidGKUYqRioqJi0lImJHRidGKi1GLDYkRi5GKEYqRio= and so we need to solve another ODE to find the characteristic. In the examples below we always take NiMvLSUiYUc2JCUieEclInRHIiIi, and so we can use NiMlInRH instead of NiMlJHRhdUc=. In this case the characteristics are given by the equation NiMvLSUlZGlmZkc2JCUieEclInRHLSUiYkdGJg==. On the characteristic we then get an equation NiMvLCYtJSVkaWZmRzYkLSUidUc2IyUidEdGKyIiIiomLSUiY0dGKkYsRihGLEYsLSUiaEdGKg==, which is again an ODE. Solving both ODEs, choosing the constants of integration to match the initial data, and going from the characteristic to the whole plane then gives us the solution NiMtJSJ1RzYkJSJ4RyUidEc= of the PDE.

Some packages that we use in this worksheet:restart: with(plottools): with(plots):In all of the examples below we use the same initial profile given by a localized "hump":f_ini:=x->piecewise(x<-1,0,x<1,(cos(Pi/2*x))^2,0):plot(f_ini(x),x=-2..2,scaling=constrained,thickness=2,title="Initial profile");Note that this initial profile has a continuous first derivative:plot([f_ini(x),diff(f_ini(x),x)],x=-2..2,scaling=constrained,thickness=2,color=[red,blue], title="Initial profile and its derivative");

Our first example is a trivial one. We consider the equation NiMvLSUlZGlmZkc2JC0lInVHNiQlInhHJSJ0R0YrIiIh. pde:=diff(u(x,t),t)=0;The equation of the characteristics is then simplychar_eq:=diff(x(t),t)=0;The characteristic curves are then given by the equationchar_curve:=dsolve({char_eq,x(0)=x[0]});And so the characteristics are straight vertical lines (the nch variable below denotes the number of characteristics per unit interval on the NiMlInhH-axis)nch:=3: char_plot:=display([seq(plot([subs(x[0]=x/nch,eval(x(t),char_curve)),t,t=0..4], color=black,thickness=2),x=-4*nch..2*nch)],view=[-2..2,0..4]): display(char_plot);On the characteristics we need to solve the following ordinary differential equation:ode:=diff(w(t),t)=0;which coincides with our PDE upon the substitution NiMvLSUid0c2IyUidEctJSJ1RzYkLSUieEdGJkYn (restricted to a characteristic curve):eval(subs(w(t)=u(eval(x(t),char_curve),t),ode));This ODE is easy to solve:char_sol:=dsolve({ode,w(0)=f(x[0])});our solution of the PDE then is NiMvLSUidUc2JCUieEclInRHLSUid0c2JEYoLSZGJzYjIiIhRiY=:u:=unapply(subs(x[0]=solve(subs(x(t)=x,char_curve),x[0]),eval(w(t),char_sol)),x,t):'u(x,t)'=u(x,t);This result is not surprising since the equation means that the initail profile does not change in time and therefore is simply shifted in the NiMlInRH-direction. We color the plot according to the values, so same color represents same values of the function NiMtJSJ1RzYkJSJ4RyUidEc=.sol_plot:=plot3d(subs(f=f_ini,u(x,t)),x=-2..2,t=0..4,axes=boxed,scaling=constrained,numpoints=2000, shading=zhue): display(sol_plot,style=patchnogrid,view=-0.1..1,orientation=[-65,55]);We now combine the plot of the characteristic curves with the graph of the solution. We make the resulting graph a "movie" with two frames, the first frame plots the solution using the patchnogrid style, the second frame uses the wireframe style and so allows us to see the characteristics. Below is the graph of the solution as viewed from above, to get a better understanding of the solution you should rotate it. shift:=transform((x,y)->[x,y,-0.1]):movie:=display([display(sol_plot,style=patchnogrid),display(sol_plot,style=wireframe)],insequence=true):display([movie,shift(char_plot)],orientation=[-90,0],view=[-2..2,0..4,-0.1..1],scaling=constrained);Clean-up:u:='u': f:='f': w:='w':

Our second example is the transport equation. We consider the equation NiMvLCYtJSVkaWZmRzYkLSUidUc2JCUieEclInRHRiwiIiIqJiUiY0dGLS1GJjYkRihGK0YtRi0iIiE=. pde:=diff(u(x,t),t)+c*diff(u(x,t),x)=0;The equation of the characteristics is then simplychar_eq:=diff(x(t),t)=c;The characteristic curves are then given by the equationchar_curve:=dsolve({char_eq,x(0)=x[0]});And so the characteristics are straight lines with the slope of NiMqJiIiIkYkJSJjRyEiIg== (we choose NiMvJSJjRy0lJkZsb2F0RzYkIiImISIi for the plot)nch:=3: char_plot:=display([seq(plot([subs({c=0.5,x[0]=x/nch},eval(x(t),char_curve)),t,t=0..4], color=black,thickness=2),x=-4*nch..2*nch)],view=[-2..2,0..4]): display(char_plot);On the characteristics we need to solve the following ordinary differential equation:ode:=diff(w(t),t)=0;which coincides with our PDE upon the substitution NiMvLSUid0c2IyUidEctJSJ1RzYkLSUieEdGJkYn (restricted to a characteristic curve):eval(subs(w(t)=u(eval(x(t),char_curve),t),ode));This ODE is easy to solve:char_sol:=dsolve({ode,w(0)=f(x[0])});our solution of the PDE then is NiMvLSUidUc2JCUieEclInRHLSUid0c2JEYoLSZGJzYjIiIhRiY=:u:=unapply(subs(x[0]=solve(subs(x(t)=x,char_curve),x[0]),eval(w(t),char_sol)),x,t):'u(x,t)'=u(x,t);sol_plot:=plot3d(subs({c=0.5,f=f_ini},u(x,t)),x=-2..2,t=0..4,axes=boxed,scaling=constrained,numpoints=2000, shading=zhue): display(sol_plot,style=patchnogrid,view=-0.1..1,orientation=[-65,55]);shift:=transform((x,y)->[x,y,-0.1]):movie:=display([display(sol_plot,style=patchnogrid),display(sol_plot,style=wireframe)],insequence=true):display([movie,shift(char_plot)],orientation=[-90,0],view=[-2..2,0..4,-0.1..1],scaling=constrained);And this is how the initial profile evolves in time. This animation helps to see why our PDE is called the transport equation.animate(plot,[subs({c=0.5,f=f_ini},u(x,t)),x=-2..2,color=red,thickness=2],t=0..6,scaling=constrained);Clean-up:u:='u': f:='f': w:='w':

For the next example we consider the equation with non-constant coefficients, NiMvLCYtJSVkaWZmRzYkLSUidUc2JCUieEclInRHRiwiIiIqJkYrRi0tRiY2JEYoRitGLUYtIiIhpde:=diff(u(x,t),t)+x*diff(u(x,t),x)=0;The equation of the characteristics is char_eq:=diff(x(t),t)=x(t);The characteristic curves are then given by the equationchar_curve:=dsolve({char_eq,x(0)=x[0]});And so the characteristics are graphs of the exponential functionnch:=10: char_plot:=display([seq(plot([subs({c=0.5,x[0]=x/nch},eval(x(t),char_curve)),t,t=0..4], color=black,thickness=2),x=-4*nch..2*nch)],view=[-2..2,0..4]): display(char_plot);On the characteristics we need to solve the following ordinary differential equation:ode:=diff(w(t),t)=0;which coincides with our PDE upon the substitution NiMvLSUid0c2IyUidEctJSJ1RzYkLSUieEdGJkYn (restricted to a characteristic curve):eval(subs(w(t)=u(eval(x(t),char_curve),t),ode));The ODE is easy to solve:char_sol:=dsolve({ode,w(0)=f(x[0])});our solution of the PDE then is NiMvLSUidUc2JCUieEclInRHLSUid0c2JEYoLSZGJzYjIiIhRiY=:u:=unapply(subs(x[0]=solve(subs(x(t)=x,char_curve),x[0]),eval(w(t),char_sol)),x,t):'u(x,t)'=u(x,t);sol_plot:=plot3d(subs({c=0.5,f=f_ini},u(x,t)),x=-2..2,t=0..4,axes=boxed,scaling=constrained,numpoints=2000, shading=zhue): display(sol_plot,style=patchnogrid,view=-0.1..1,orientation=[-65,55]);shift:=transform((x,y)->[x,y,-0.1]):movie:=display([display(sol_plot,style=patchnogrid),display(sol_plot,style=wireframe)],insequence=true):display([movie,shift(char_plot)],orientation=[-90,0],view=[-2..2,0..4,-0.1..1],scaling=constrained);And this is how the initial profile evolves in time. animate(plot,[subs({c=0.5,f=f_ini},u(x,t)),x=-2..2,color=red,thickness=2],t=0..6,scaling=constrained);Clean-up:u:='u': f:='f': w:='w':

This example is a non-homogeneous equationwith non-constant coefficients, NiMvLCYtJSVkaWZmRzYkLSUidUc2JCUieEclInRHRiwiIiIqJilGLCIiI0YtLUYmNiRGKEYrRi0hIiJGLQ==pde:=diff(u(x,t),t)-t^2*diff(u(x,t),x)=1;The equation of the characteristics is char_eq:=diff(x(t),t)=-t^2;The characteristic curves are then given by the equationchar_curve:=dsolve({char_eq,x(0)=x[0]});And this is how the characteristic curves look like:nch:=5: char_plot:=display([seq(plot([subs({c=0.5,x[0]=x/nch},eval(x(t),char_curve)),t,t=0..4], color=black,thickness=2),x=-4*nch..8*nch)],view=[-2..2,0..4]): display(char_plot);On the characteristics we need to solve the following ordinary differential equation:ode:=diff(w(t),t)=1;which coincides with our PDE upon the substitution NiMvLSUid0c2IyUidEctJSJ1RzYkLSUieEdGJkYn (restricted to a characteristic curve):eval(subs(w(t)=u(eval(x(t),char_curve),t),ode));The ODE is easy to solve:char_sol:=dsolve({ode,w(0)=f(x[0])});our solution of the PDE then is NiMvLSUidUc2JCUieEclInRHLSUid0c2JEYoLSZGJzYjIiIhRiY=:u:=unapply(subs(x[0]=solve(subs(x(t)=x,char_curve),x[0]),eval(w(t),char_sol)),x,t):'u(x,t)'=u(x,t);In this case solution is not constant along the characteristics. Instead, it exibits the linear growth there:sol_plot:=plot3d(subs({c=0.5,f=f_ini},u(x,t)),x=-2..2,t=0..4,axes=boxed,scaling=constrained,numpoints=2000, shading=zhue): display(sol_plot,style=patchnogrid,view=-0.1..4,orientation=[-135,70]);Still, looking at the solution from the top we can see the characteristic curves:shift:=transform((x,y)->[x,y,-0.1]):movie:=display([display(sol_plot,style=patchnogrid),display(sol_plot,style=wireframe)],insequence=true):display([movie,shift(char_plot)],orientation=[-90,0],view=[-2..2,0..4,-0.1..4],scaling=constrained);And this is how the initial profile evolves in time. It is a combination of growth and accelerated motion to the left (with the speed of NiMqJCklInRHIiIjIiIi):animate(plot,[subs({c=0.5,f=f_ini},u(x,t)),x=-5..4,color=red,thickness=2],t=0..6,scaling=constrained);Clean-up:u:='u': f:='f': w:='w':

Walter A. Strauss, Partial Differential Equations: An Introduction, Wiley, 1992Richard Haberman, Elementary Applied Partial Differential Equations, 3rd edition, Prentice HallStanley J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover 1982

"While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material."