October-November 2009

UKSW - Faculty of Applied Mathematics by J.Tomasik, UdA Clermont 1 (France)

Problems on Fourier and Laplace Transforms

1

Laplace Transform

Exercice 1 Prove the equalities : 1. L(1)(s) =

1 s

n! sn+1 1 = s−a

2. L(tn )(s) = 3. L(eat )(s)

4. L(sin at)(s) =

5. L(cos at)(s) =

a s2 +a2 s s2 +a2

6.

e−as , where U (t − a) = L(U (t − a)(s) = s

(

1, t > a, a ∈ R, a > 0. 0, t < a,

Precise the domains of definition of these Laplace transforms. Exercice 2 Calculate Laplace transforms of functions below, where a, b, A, T are all positive real numbers.( A t, 0 6 t 6 a 1. f (t) = a A, t > a   u n ∈ N, n > 1, c.a.d nA, pour (n − 1)T 6 t < nT, o` 2. f (t) = A, 06tT  0, 06t 0, we put ( 1 , 06t6ε Fε (t) = ε . 0, t>ε 1. Find L(Fε ).

2. Prove that limε→0 L(Fε ) = 1. Exercice 4 Calculate the Laplace transform of the function F where a, b, ω, k ∈ R, a, b > 0 : 1.

F (t) = a sin ωt,

2.

F (t) = a(1 − e−bt ),

3.

F (t) = a cos(bt − k). N.B. Observe that the expression of the Laplace transform of the function ( F (t − a), t > a, G(t) = in terms of F is not applicable(3). Why not ? 0, t < a,

Exercice 5 Verify the following properties of the Laplace transform L : 1. L(c1 F1 + c2 F2 ) = c1 L(F1 ) + c2 L(F2 ).

2. L(eat F (t))(s) = (LF )(s − a). ( F (t − a), t > a, 3. For G(t) = we have (LG)(s) = e−as (LF )(s) (a > 0). 0, t 6 a, 4. For Fa (t) = F (at) we have (LFa )(s) = a1 (LF )( as ). Rt 5. For G(t) = 0 F (u)du we have (LG)(s) = (LFs)(s) .

6. L(tn F (t)) = (−1)n (LF )(n) 7. If the limt→0

F (t) t

exists, L( F (t) )(s) = t

R∞ s

(LF )(ζ)dζ.

8. If F is periodic, F (t + T ) = F (t), then

1 (LF )(s) = 1 − e−sT

Z

T

F (t)e−st dt 0

9. lims→+∞ (LF )(s) = 0. 10. lims→+∞ (LF )(s) = 0. 11. limt→0,t>0 F (t) = lims→∞ s(LF )(s) if the indicted limits exists (the Initial Value Theorem). 12. limt→∞ F (t) = lims→0 s(LF )(s) if the indicted limits exists (the Final Value Theorem).

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2

Fourier Series and Transforms

Exercice 6 Examine the convergence of Fourier series ∞ X cos nx

n2

1

,

∞ X sin nx

n2

1

,

∞ X einx 1

n!

.

Exercice 7 Calculate the Fourier series coefficients of the 2π−periodic function f , defined on the interval [−π, π] by the formula f (x) = π − |x|,

|x| 6 π.

Study the convergence (simple, uniforme) of the resulting Fourier series. Find out the Pabsolute, ∞ 1 value of the numerical series 1 n2 .

Exercice 8 1. Find the fourier coefficients in sin and cos of the periodic function F , defined on ] − 5, 5[\{0} by ( 0, −5 < x < 0 F (x) = . 3, 0 0.

2. Use the result of (1) to prove that, for p > 0 β > 0, the following holds Z ∞ cos pv π −pβ e . dv = v2 + β 2 2β 0 Exercice 13 Find a function f satisfying the integral equation : ( Z ∞ 1 − α, 0 6 α 6 1, f (x) sin αxdx = 0, α > 1. 0 3

Exercice 14 Verify that, for x > 0 holds Z ∞ cos λx π dλ = e−x . 2 λ +1 2 0 x2

Exercice 15 (Gauss function) Let f (x) = e− 2 . Verify that both f (x) and its Fourier transform ψ(ω) are solutions of the differential equation y 0 + xy = 0. Hint : Observe that the following equality holds Z ∞ Z ∞ d dK(ω, x) K(ω, x)dx = dx dω −∞ dω −∞ p . Prove out that ψ(ω) = (2)f (ω) and then calculate the Gauss integral Z ∞ x2 e− 2 dx −∞

.

Exercice 16 Evaluate, using the Parseval identity : R∞ , 1. 0 (x2dx +1)2 R ∞ 2 dx 2. 0 (xx2 +1) 2, .

3

Application of Fourier and Laplace Transforms

Exercice 17 Find the inverse Laplace transforms L−1 (f ) of rational fractions : 1. f (s) =

s (s+1)3 (s−1)2

2. f (s) =

1 (s2 +1)2

3. f (s) =

s+1 (s+2)(s+3)

4. f (s) =

s+1 (s+2)(s+3)2 (s2 +4s+5)

5. f (s) =

1 (s+3)(s+4)

6. f (s) =

s+2 (s+1)2 (s+3)

7. f (s) =

s+1 s2 +4s+16

Exercice 18 Let a, x be real numbers, 0 < x < a. We put f (s) =

shsx . s2 chsa

Determine L−1 (f ).

Exercice 19 Calculate the integral Z

∞

t2 cos 6te−4t dt.

0

Exercice 20 Find solutions y(t) of the differential equations 1. y 00 − 5y 0 + 6y = tet ,

y(0) = 0 = y 0 (0),

2. y 00 − y 0 = sin(2t),

y(0) = 1 and y 0 (0) = −1, 4

3. y 00 + 2y 0 + 5y = e−t sin(2t),

y(0) = 0 and y 0 (0) = 1,

Exercice 21 Find a function f (x) such that f (0) = 1 and Z x f (t) cos(x − t)dt = f 0 (t). 0

Exercice 22 Find all solutions y(t) of the differential equation ty 00 + 2y 0 + ty = 0,

y(0) = 1,

defined on the whole real line (i. e. Dy = R). Exercice 23 Solve the differential system dx = x + 5y dt dy = x − 3y dt x(0) = 1, y(0) = 2

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