Partial Differential Equations I: Basics and Separable ...

1 18 Partial Differential Equations I: Basics and Separable SolutionsWe now turn our attention to Differential Equations in whichthe unknown function to be deter-mined which we will usually denote byu depends on two or more variables. Hence thederivatives are Partial derivatives with respect to the various variables.(By the way, it may be a good idea to quickly review theA Brief Review of ElementaryOrdinary Differential Equations , Appendex A of these notes. We will be using some of thematerial discussed there.) Intro and ExamplesSimple ExamplesIf we have a horizontally stretched string vibrating up and down, letu(x,t)=the vertical position at timetof the bit of string at horizontal positionx,and make some almost reasonable assumptions regarding the string, the universe and the laws ofphysics, then we can show thatu(x,t)satisfies 2u t2 c2 2u x2=0wherecis some positive constant dependent on the physical properties of the stretched equation is called theone-dimensional wave equation(with no external forces).

2 If, instead, we have a uniform one-dimensional heat conducting rod along theX axis andletu(x,t)=the temperature at timetof the bit of rod at horizontal positionx,then, after applying suitable assumptions about heat flow, etc., we get theone-dimensional heatequation u t 2u x2=f(x,t).Here, is a positive constant dependent on the material propertiesof the rod, andf(x,t)describes the thermal contributions due to heat sources and/or sinks in the & Page: 18 2 PDEs I: Basics and Separable SolutionsThe physicists in the class, of course, are also well acquainted withSchr dinger s equationi h u t+ h22m 2u x2=V(x)uwhere handmare positive constants andV(x)is some potential energy problems involving two- and three-dimensional objects lead to similar Partial differ-ential Equations with 2replacing 2u/ , we have themulti-dimensional wave equation, 2u t2 c2 2u=0 ,and themulti-dimensional heat equation, u t 2u=f(x,t).If, in either of these, we reach an equilibrium or steady-state ( , u/ t=0 ), then theseequations reduce to eitherLaplace s equation 2u=0orPoisson s equation 2u=f(x).

3 Basic TerminologyTheorderof a Partial Differential equation is the order of the highest derivative explicitly appear-ing. In practice, most Partial Differential Equations of interest are second order (a few are firstorder and a very few are fourth order). We will concentrate onsecond-order linear second-order Partial Differential equation (in variablesx1,x2, ..,xn) is said to belinearif it can be written asXjkajk 2u xk xj+Xlbl u xl+cu=f.( )wheref,cand theajk s andbl s are constants or are functions of the variables (but notu).This equation is said to behomogeneousif (and only if)f 0 . We will concentrate our initialattention on homogeneous Equations because they are a little easier to deal with and because wemust solve a homogeneous equation anyway to get the completesolution to any nonhomogeneouslinear convenience, we ll occasionally letLdenote the Differential operator defined by theleft side of equation ( ),L=Xjkajk 2 xk xj+Xlbl xl+ : 2uis the divergence of the gradient ofu.

4 In Cartesian coordinates, 2u= 2u x2+ 2u y2+ .2 Take MA 506 or MA 526 to learn how to reduce first-order, linearpartial Differential Equations to first-order, linearordinary Differential and ExamplesChapter & Page: 18 3 That is, for any sufficiently differentiable functionw,L[w] =Xjkajk 2w xk xj+Xlbl w xl+ this, equation ( ) can be written more succinctly asL[u] =f,and, if it is homogeneous, asL[u] =0 .You should be aware that second-order linear Partial Differential Equations are often classifiedas being hyperbolic , parabolic or elliptic . Crudelyspeaking:Hyperbolic Partial Differential Equations are Partial Differential Equations like the wave equation, 2u t2 c2 2u=0 .Parabolic Partial Differential Equations are Partial Differential Equations like the heat equation, u t 2u=0 .Elliptic Partial Differential Equations are Partial Differential Equations like Laplace s equation, 2u=0 .For an intelligent discussion of the classification of second-order Partial Differential Equations ,take a true Partial Differential equation course (MA 506 or MA 526-626).

5 Linearity/Principle of SuperpositionLettingL=Xjkajk 2 xk xj+Xlbl xl+c,it is easily verified that, given a bunch of constants c1,c2,c3,.. and a correspondingbunch of sufficiently differentiable functions u1,u2,u3,.. thenL[c1u1+c2u2+c3u3+ ] =c1L[u1] +c2L[u2] +c3L[u3] + .Thus,Lis alineardifferential particular, ifu1,u2,u3,..are all solutions to the homogeneous equationL[u] =0 ,thenL[c1u1+c2u2+c3u3+ ] =c1L[u1] +c2L[u2] +c3L[u3] + =c1 0+c2 0+c3 0+ =0 .version: 3/8/2014 Chapter & Page: 18 4 PDEs I: Basics and Separable SolutionsThis gives usTheorem (principle of superposition)Any linear combination of solutions to a homogeneous linearpartial Differential equation is alsoa solution to that homogeneous Partial Differential will use this often, even with linear combinations involving infinitely many terms (and,at times, slop over issues of the convergence of the resulting infinite series).At this point we should spend a few seconds to observe thatL[0] =Xjkajk 20 xk xj+Xlbl 0 xl+c 0=0.

6 So the constant functionu=0 is a solution to every homogeneous linear Partial differentialequation. This not-so-exciting solution is often called thetrivial solution. Our main interest, ofcourse, will be in thenontrivial SolutionsIn general, we cannot find general solutions ( , relatively simple formulas describing allpossible solutions) to second-order Partial one notable exception iswith the one-dimensional wave equation 2u t2 c2 2u x2=0 .Using a clever change of variables, it can be shown that this has the general solutionu(x,t)=f(x ct)+g(x+ct)( )wherefandgare arbitrary sufficiently differentiable functions of a single Exercise :Verify that, iff(s)andg(s)are any two twice-differentiable functions ofone variable, thenu(x,t)=f(x ct)+g(x+ct)satisfies 2u t2 c2 2u x2= that, at any given timet, the graph ofy=f(x ct)3 General solutions to first-order linear Partial Differential Equations can often be =x+ctand =x ctthe wave equation simplifies to 2u =0.

7 Integrating twice then gives youu=f( )+g( ), which is formula ( ) after the change of of Variables for Partial Differential Equations (Part I)Chapter & Page: 18 5is just the graph ofy=f(x)shifted to the right byct. Thus, thef(x+ct)part of formula( ) can be viewed as a fixed shape traveling to the right with speedc. Likewise, theg(x+ct)part of formula ( ) can be viewed as a fixed shape traveling to the left with speedc. Unsurprisingly, these are generally known astraveling Exercise :Illustrate these last few statements by sketchingu(x,t)=f(x ct)as a function ofxatt=0,t=1,t=2andt=3, using, say,c=2and eitherf(s)=arctan(s)orf(s)=(1if 1<s< Exercise :Letf(s)be any twice-differentiable function of one variable and letnbeany unit vector. Verify thatu(x,t)=f(n x ct)is a solution to the three-dimensional wave equation 2u t2 c2 2u=0.(This is aplane wavesolution f(n x ct)remains constant on planes perpendicular tonand traveling with speedcin the direction ofn.))

8 Separation of Variables for Partial DifferentialEquations (Part I) Separable FunctionsA function ofNvariablesu(x1,x2, .. ,xN)isseparableif and only if it can be written as a product of two functions ofdifferent variables,u(x1,x2, .. ,xN)=g(x1, .. ,xk)h(xk+1, .. ,xN).It iscompletely separableif and only if it can be written as a product ofNfunctions, each ofwhich is a function of just one variable,u(x1,x2, .. ,xN)=g1(x1)g2(x2)g(x3) gN(xN).! Example :The following functions are all Separable :u(x,t)=e 6tsin(x)version: 3/8/2014 Chapter & Page: 18 6 PDEs I: Basics and Separable Solutionsu(x,y,t)=px2+y2 yx sin(3t)u(r, ,t)=rtan( )sin(3t)with the first and the last being completely applications, our functions are often really scalar fields, and, to simplify our work, weoften try to find coordinate systems under which these fields are given by Separable Separable SolutionsA first step towards solving many Partial Differential equation problems is to find all possibleseparable solutions to a given homogeneous linear Partial Differential equation ( , all solutionsto the Partial Differential equation given by Separable functions).

9 Here is a procedure for findingall such solutions. For simplicity, we will explicitly discuss Equations involving just two variablesxandt, though as will be noted, the method is easily extended to Equations involving morevariables. To illustrate its use, we ll go ahead and find all Separable solutions to the simpleone-dimensional heat equation u t 6 2u x2=0 .( ) the solutionu(x,t)can be written asu(x,t)=g(x)h(t),plug this into the Partial Differential equation, and compute the derivatives .Lettingu(x,t)=g(x)h(t)in u t 6 2u x2=0.( )yields t[g(x)h(t)] 6 2 x2[g(x)h(t)] = , after differentiating, isg(x)h (t) 6g (x)h(t)= good algebra, rearrange the last equation obtained togetformula oftonly=formula ofxonly( )Some side notes:(a)Dividing through byg(x)h(t)is usually a good idea.(b)For reasons that won t be clear for a while, it is usually a good idea to move any floating constants to the side with thetvariable.(c)If you can get the equation into form ( ), then your Partial Differential equation issaid to beseparable.

10 Otherwise, the Partial Differential equation is not Separable and this approach leads to a disappointing dead end. For the rest of this discussion,we will assume the of Variables for Partial Differential Equations (Part I)Chapter & Page: 18 7In our example:g(x)h (t) 6g (x)h(t)=0H g(x)h (t) 6g (x)h(t)g(x)h(t)=0g(x)h(t)H h (t)h(t) 6g (x)g(x)=0H h (t)h(t)=6g (x)g(x)H h (t)6h(t)=g (x)g(x).3. Observe that the only way we can haveformula oftonly=formula ofxonlyis for both sides to be equal to a single constant (see exercise on page 18 10).Because it will later simplify our work slightly, we will denote this constant by .This is theseparation constant(and is totally unknown at this point).After making this observation:(a)Write this fact down using the separation constant.(b)Observe further that this gives us a pair ofordinarydifferential Equations , eachinvolving .(c)Write out that pair of ordinary Differential Equations in assimplified a form as our example, we haveh (t)6h(t)=g (x)g(x)=.