Question

Use equation (5.13) and the Fourier integral representation of to write a solution of the problem on the real line:

f(r

Also reformulate the solution using equation (5.18).

Problem:

Equation (5.13):

Equation (5.18):

Answer 1

E som given, Poe is, Up : klage for -21<x<0, 120.40 112,0)=3(9). , 100<<*, =0.--- wing the method of soperation of variable, we write U18+) = X(X). Tlt). substituting into a wegel- $ I = a (seperation constant) — => *"-2x = 0 and, T'-AKT = 0. so, The solutоо 2от Tavel, T= се 13.970, we have ß and, therefore I go u(xt) growing exponentially with time." from realistic physical considerations, it is reasonable to assume thaf 3(0) >0 as 1 0 , while fulgt) | <M as lod so But for Man to ucct) to remain bounded, a should be negative and thus we take d=-4?. Nowo, from equation , we save, x"-2x = 0 = x"+4? x = 0 So, x(x) = C, Cos uxe + C, 8in eix and, T1) = c ē uz kt Hence, u tru) = (A Cos uz 4 B Sinux) ē kwest - is the solution of 1? 0. wh pores A and B ame arbitrary constants. Since A and B ane ambitmany, we may coulidem them as functions of deleti A= A (de) , B=Bilde),

SU 80, 4(xt) = Jºula+nde) de ( wing principal of) superposition 3) uw t) = AW) Cos wx + Blue) Brin diz] e o from the initial condition ©, wehave u(3,0) = 3(2) = 1°C Alm) Cosuere + B(l) sin ur J du. The former integral theorem is given by, 31+)= 3(). Cod wlt->)dx Jdw. o. Thus, we may write, 3(-2) = + 1° 30°) Cas 'm (2-y) do Idee - + SUS 308) ( cos de Cos wy + Sin Sin uydzie = [Cor use bo 319) Co wy dy + Siosta Sin er jo 219) Seo ury day I due o hel, Alm) = +1° 3L9) cos ugdy B(u) = + 5*318) Se'n wy dy Thus, equation @ written in the form, f(x) = 4 [A() Cos we + B(6) Siour] du Comparing yze and -, we shallo write relation on ucao) = 3($) =L. 869) Cas ula-y) do Jdele Thus, from eya ©, we obfaris. (98, +) = To CS 313) Cos de 12-y) exp(-xx?t) do Jdece

Now, changing order of integration, wegel- u(x, t): ] 313) [ exp(-x ?t) Cos_ule-y)che Jay - weknou, sex cas (abs) ds every — getting, 8 = le diet and choosing. be to TRE so, eje @ becomes, ſ e Rest cas in (e-y) du = V to é: substituting ej? ☺'in az? ® , weget ulert) = hehet 19 769) exp [ by the Joy Replace, y by }, we obtain ulet) = de 33) e 2 kr Now, we have, 2100 = 2 Sinx for 12) sr Lo Zoo 1x) >r 80, 866) Esins 20m -228x'sr jo zor 121792 using ☺, wehave. ulet) = a dhe 8inzi e le net dz. . uilest). - ar kat T *Jatt (eot (in ) teny 93) -ezim erit (-2ext-2) dekt terrs (5+24*6+2)

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