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Unformatted text preview: MATH 220: DUHAMELS PRINCIPLE Although we have solved only the homogeneous heat equation on R n , the same method employed there also solves the inhomogeneous PDE. As an application of these methods, lets solve the heat equation on (0 , ) t R n x : (1) u t- k u = f, u (0 , x ) = ( x ) , with f C ([0 , ) t ; S ( R n x )), S ( R n ) (say) given. Namely, taking the partial Fourier transform in x , and writing F x u ( t, ) = u ( t, ), gives u t ( t, ) + k | | 2 u ( t, ) = f ( t, ) , u (0 , ) = ( F )( ) . We again solve the ODE for each fixed . To do so, we multiply through by e k | | 2 t : e k | | 2 t u t ( t, ) + e k | | 2 t k | | 2 u ( t, ) = e k | | 2 t f ( t, ) , and realize that the left hand side is d dt parenleftBig e k | | 2 t u ( t, ) parenrightBig . Thus, integrating from t = 0 and using the fundamental theorem of calculus yields e k | | 2 t u ( t, )- u (0 , ) = integraldisplay t e k | | 2 s f ( s, ) ds, so u ( t, ) = e k | | 2 t ( F )( ) + integraldisplay t e k | | 2 ( t s ) f ( s, ) ds. Finally, u ( t, x ) = F 1 parenleftbigg e k | | 2 t ( F )( ) + integraldisplay t e k | | 2 ( t s ) f ( s, ) ds parenrightbigg = F 1 parenleftBig e k | | 2 t ( F )( ) parenrightBig + integraldisplay t F 1 parenleftBig e k | | 2 ( t s ) f ( s, ) parenrightBig ds = (4 kt ) n/ 2 integraldisplay R n e | x y | 2 / 4 kt ( y ) dy + integraldisplay t (4 k ( t- s )) n/ 2 integraldisplay R n e | x y | 2 / 4 k ( t s ) f ( s, y ) dy ds. Note that the parts of the solution formula corresponding to the initial condition and the forcing term are very similar. In fact, let the solution operator of the homogeneous PDE u t- k u = 0 , u (0 , x ) = ( x ) , be denoted by S ( t ), so ( S ( t ) )( x ) = u ( t, x ) . Then (4 k ( t- s )) n/ 2 integraldisplay R n e | x y | 2 / 4 k ( t s ) f ( s, y ) dy = ( S ( t- s ) f s )( x ) , 1 2 where we let f s ( x ) = f ( s, x ) be the restriction of f to the time s slice. Thus, the solution formula for the inhomogeneous PDE, (1), takes the form (2) u ( t, x ) = ( S ( t ) )( x ) + integraldisplay t ( S ( t- s ) f s )( x ) ds. The fact that we can write the solution of the inhomogeneous PDE in terms of the solution of the Cauchy problem for the homogeneous PDE is called Duhamels principle . Physically one may think of (2) as follows. The expression S ( t- s ) f s is the solution of the heat equation at time t with initial condition f ( s, x ) imposed at time s . Thus, we think of the forcing as a superposition (namely, integral) of initial conditions given at times s between 0 (when the actual initial condition is imposed) and time t (when the solution is evaluated). Conversely, one could say that the initial condition amounts to a delta-distributional forcing,...
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- Fall '10