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AMath350.F12.A9.Sol

# y g y f clearly result of g y f further

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Unformatted text preview: the deﬁnition the Convolution x, y (b) have (u(!) y ) =(g (y )f (! ). Clearly )ˆ result) of g (y )f (! ). Further, that part(a) and to ˆ ˆ ,integral on theZ rightyconvergesy = Z the Fourier Transform any (x) uy (! , sinceto x does not exist for of f Z The ﬁrst the Fourier Transform of u(x, ) with respect Theorem we have x ˆ 4. Letvalue,yt) howeverx, t)} (x)g (y )e(x, t)dxi= dx. )(this f (x)e i x dx = g (y )fSince u(! (x, y F{u ˆ u ix F{u of =. )} =( the f = g (y exists, y second integralediverges Z was shown in class). (! ). i 1 Z Z Z (x z ) ( ˆ F( 1) f (! )e Transform xto integral obtain tx }} = ApplyyFourierthist } = Fi the PDE )}F{ thei left divergesffor any xz t) dz y and t). g { 6= 0, means {F{f (x to on (x ) value of = f (x (b) (f (x) + g (y ))e dx = f ( x) e dx + g (y ) e i dx. the Fourier Tranform of u(x, y ) does not exist. ˆ + ut + t ˆ ˆ The ﬁrst integral i! utheZright u = 0 x since the Fourier Transform of f (x) converges Z on 4. Let u(! , t) = F{u(x, t)} = ˆ u(x, t)e i dx. i 3. (a) exists, however the second)integral diverges,!usingwas shown in class). Since F{ (x a)} = (x a e ut x dx = (t + a (this the substitution property. e i i )ˆ ˆ= u Apply Fourier Transform to PDE to obtain g (y ) 6= 0, this means the integral on the left diverges for any value of y and ˆ (b) Let F{f (x)} = f (! ). Using the result of part(a) and the Convolution the Fourier Tranform+ uu+ tu) = 0 not exist. i! u ofˆt (x, y does ˆ ˆ1 Theorem we have Z i ˆ= u 3. (a) F{ 1 (x a)} =t (x a)e ut x dx = (t + ai,! )ˆ Z the substitution property. e i using ˆ F {f (! )e i } = F 1 {F{f (x)}F{ (x t)}} = f (x z ) (z t) dz = f (x t). ˆ(! ). Using the result of part(a) and the Convolution (b) Let F{f (x)} = f Z 1 4. LetTheorem we have t)} = u(! , t) = F{u(x, ˆ u(x, t)e i x dx. Z Apply 1 {f (! )e Transform 1to PDE )}F{ (x t)}} = Fourier i t } = F {F{f (x to obtain ˆ F f (x z ) (z t) dz = f (x t). Z 1 i! x AMATH Z t i! x Assignment i! u + Zˆt + tu = 0 ˆu ˆ x 4. Let u(! , t) = F{u(x, t)} = ˆ uux, t)e i (tdx...
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