d Applying the convolution theorem find the solution of the following problem

D applying the convolution theorem find the solution

This preview shows page 15 - 20 out of 20 pages.

d). Applying the convolution theorem, find the solution of the following problem(9P.) 15
Image of page 15
Extra space for previous question Now consider Q L{ r } . We have q := L - 1 { Q } = L - 1 1 s - 3 - 1 s - 2 = e 3 t - e 2 t . So L - 1 { Q L{ r }} = q ? r = Z t 0 e 3( t - τ ) - e 2( t - τ ) 4 e τ ( H ( τ ) - H ( τ - 2)) When 0 t < 2 q ? r = Z t 0 e 3( t - τ ) - e 2( t - τ ) 4 e τ = 4 e 3 t Z t 0 e - 2 τ - 4 e 2 t Z t 0 e - τ = 4 e 3 t - 1 2 e - 2 τ τ = t τ =0 - 4 e 2 t ( - e - τ ) τ = t τ =0 = - 2 e 3 t ( e - 2 t - 1 ) + 4 e 2 t ( e - t - 1 ) = 2 e t + 2 e 3 t - 4 e 2 t . When 2 t q ? r = Z 2 0 e 3( t - τ ) - e 2( t - τ ) 4 e τ = - 2 e 3 t ( e 4 - 1 ) + 4 e 2 t ( e - 2 - 1 ) . So y = 2 e t - 2 e 3 t + e 2 t , if 0 t < 2 - 2 e 3 t ( e - 4 + 1 ) + e 2 t ( 4 e - 2 + 1 ) , if 2 t 16
Image of page 16
17
Image of page 17
Question: 3(Stability and Numerical Methods)9P.a). Consider the problemx0=x. If the initial condition isx(0) =c, the solution isx=cet.If a round off or measurement error ofoccurs in reading or assigning the value ofc(i.e.instead ofcthe initial condition is set toc+ ), what effect is there on the solution at thepointt= 10? Att= 20? Leave your answers in algebraic form. That is, you do not needto explicitly evaluate the exponentials.(4P.) 18
Image of page 18
Image of page 19
Image of page 20

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture