The two-dimensional heat equation

1 Homog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionThe two-dimensional heat equationRyan C. DailedaTrinity UniversityPartial Differential EquationsLecture 12 DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionPhysical motivationGoal:Model heat flow in a two-dimensional object (thin plate).Set up:Represent the plate by a region in thexy-plane and letu(x,y,t) ={temperature of plate at position (x,y) a fixedt, the height of the surfacez=u(x,y,t) gives thetemperature of the plate at timetand position (x,y).}

2 Under ideal assumptions( uniform density, uniform specificheat, perfect insulation along faces, no internal heat sources etc.)one can show thatusatisfies thetwo dimensional heat equationut=c2 u=c2(uxx+uyy)DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionRectangular plates and boundary conditionsFor now we assume:The plate is rectangular, represented byR= [0,a] [0,b].yxbaThe plate is imparted with some initial temperature:u(x,y,0) =f(x,y),(x,y) edges of the plate are held at zero degrees:u(0,y,t) =u(a,y,t) = 0,0 y b,t>0,u(x,0,t) =u(x,b,t) = 0,0 x a,t> 2-D heat equationHomog.

3 Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionSeparation of variablesAssuming thatu(x,y,t) =X(x)Y(y)T(t),and proceeding as wedid with the 2-D wave equation , we find thatX BX= 0,X(0) =X(a) = 0,Y CY= 0,Y(0) =Y(b) = 0,T c2(B+C)T= have already solved the first two of these problems:X=Xm(x)= sin( mx), m=m a,B= 2mY=Yn(y)= sin( ny), n=n b,C= 2n,form,n then follows thatT=Tmn(t) =Amne 2mnt, mn=c 2m+ 2n=c m2a2+ 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionSuperpositionAssembling these results, we find that for anypairm,n 1 wehave the normal modeumn(x,y,t) =Xm(x)Yn(y)Tmn(t) =Amnsin( mx)sin( ny)e principle of superposition gives the general solutionu(x,y,t) = m=1 n=1 Amnsin( mx)sin( ny)e initial condition requires thatf(x,y) =u(x,y,0) = n=1 m=1 Amnsin(m ax)sin(n by),which is just the double fourier series forf(x,y).

4 DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionConclusionTheoremIff(x,y)is a sufficiently nice function on[0,a] [0,b], then thesolution to the heat equation with homogeneous Dirichletboundary conditions and initial conditionf(x,y)isu(x,y,t) = m=1 n=1 Amnsin( mx) sin( ny)e 2mnt,where m=m a, n=n b, mn=c 2m+ 2n, andAmn=4ab a0 b0f(x,y)sin( mx)sin( ny)dy 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionExampleA2 2square plate withc= 1/3is heated in such a way that thetemperature in the lower half is 50, while the temperature in theupper half is 0.

5 After that, it is insulated laterally, and thetemperature at its edges is held at 0. Find an expression that givesthe temperature in the plate fort> must solve the heat problem above witha=b= 2 andf(x,y) ={50ify 1,0ify> coefficients in the solution areAmn=42 2 20 20f(x,y)sin(m 2x)sin(n 2y)dy dx= 50 20sin(m 2x)dx 10sin(n 2y)dyDailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solution= 50(2(1 + ( 1)m+1) m)(2(1 cosn 2) n)=200 2(1 + ( 1)m+1)(1 cosn 2) mn= 3 m24+n24= 6 m2+n2, the solution isu(x,y,t) =200 2 m=1 n=1((1 + ( 1)m+1)(1 cosn 2)mnsin(m 2x) sin(n 2y)e 2(m2+n2)t/36).}

6 DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionInhomogeneous boundary conditionsSteady state solutions and Laplace s equation2-D heat problems with inhomogeneous Dirichlet boundaryconditions can be solved by the homogenizing procedure used inthe 1-D case:1. Find and subtract the steady state(ut 0);2. Solve the resulting homogeneous problem;3. Add the steady state to the result of Step will focus only on finding the steady state part of the 0 in the 2-D heat equation gives u=uxx+uyy= 0(Laplace s equation ),solutions of which are calledharmonic 2-D heat equationHomog.

7 Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionDirichlet problemsDefinition:TheDirichlet problemon a regionR R2is theboundary value problem u= 0 insideR,u(x,y) =f(x,y) on R. u=0u x,y() =f x,y()When the region is a rectangleR= [0,a] [0,b], the boundaryconditions will be given on each edge separately as:u(x,0)=f1(x),u(x,b)=f2(x),0<x<a,u(0,y )=g1(y),u(a,y)=g2(y),0<y< 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionSolving the Dirichlet problem on a rectangle Homogenization and superpositionStrategy:Reduce to four simpler problems and use (0,y)= g (y)u(x,0)=f (x)u(a,y)=g (y)u(x,b)=f (x)2121 u= 0( )* u= 0u(0,y)= 0u(x,0)=f (x)u(a,y)=0u(x,b)=01(A)u(0,y)=0u(x,0)=0u (a,y)=0u(x,b)=f (x)2 u= 0(B)= u(0,y)=g (y)u(x,0)=0u(a,y)=0u(x,b)=01 u= 0(C)u(0,y)= 0u(x,0)=0u(a,y)=g (y)u(x,b)=02 u= 0(D)DailedaThe 2-D heat equationHomog.

8 Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionRemarks:IfuA,uB,uCanduDsolve the Dirichlet problems (A), (B),(C) and (D), then the solution to ( ) isu=uA+uB+uC+ that the boundary conditions in (A) - (D) are allhomogeneous, with the exception of a single with inhomogeneous Neumann or Robin boundaryconditions (or combinations thereof) can be reduced in asimilar 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionSolution of the Dirichlet problem on a rectangleCase BGoal:Solve the boundary value problem (B): u= 0,0<x<a, 0<y<b,u(x,0)= 0,u(x,b) =f2(x),0<x<a,u(0,y)=u(a,y) = 0,0<y< (x,y) =X(x)Y(y) leads toX +kX= 0,Y kY= 0,X(0) =X(a) = 0,Y(0) = know the nontrivial solutions forXare given byX(x) =Xn(x) = sin( nx), n=n a,k= 2n(n N).

9 DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionInterludeThe hyperbolic trigonometric functionsThehyperbolic cosine and sine functionsarecoshy=ey+e y2,sinhy=ey e satisfy the following identities:cosh2y sinh2y= 1,ddycoshy= sinhy,ddysinhy= can show that the general solution to the ODEY 2Y= 0can (also) be written asY=Acosh( y) +Bsinh( y).DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionUsing = nandY(0) = 0, we findY(y)=Yn(y) =Ancosh( ny) +Bnsinh( ny)0=Yn(0) =Ancosh 0 +Bnsinh 0= yields the separated solutionsun(x,y) =Xn(x)Yn(y)=Bnsin( nx) sinh( ny),and superposition gives the general solutionu(x,y) = n=1 Bnsin( nx) sinh( ny).

10 Finally, the top edge boundary condition requires thatf2(x) =u(x,b)= n=1 Bnsinh ( nb) sin ( nx).DailedaThe 2-D heat equationHomog. Dirichlet BCsInhomog. Dirichlet BCsHomogenizingComplete solutionConclusionAppealing to the formulae for sine series coefficients, we can nowsummarize our (x)is piecewise smooth, the solution to the Dirichlet problem u= 0,0<x<a,0<y<b,u(x,0) = 0,u(x,b) =f2(x),0<x<a,u(0,y) =u(a,y) = 0,0<y< (x,y) = n=1 Bnsin( nx) sinh( ny),where n=n aandBn=2asinh( nb) a0f2(x) sin( nx) 2-D heat equationHomog. Dirichlet BCsInhomog.