Convergence rate of Post-Widder approximate inversion of the ...

CONVERGENCE RATE OF POST-WIDDER APPROXIMATE INVERSION OF THE LAPLACE TRANSFORM Vu Kim Tuan Department of Mathematics and Computer Science Faculty of Science, Kuwait University P.O. Box 5969, Safat, Kuwait and Dinh Thanh Duc Department of Mathematics Quynhon Teacher’s Training College Quynhon, Vietnam Abstract. In this paper, error estimates for inverse Laplace transform by using Post-Widder approximation are obtained.

Keywords: Inverse Laplace transform, Post-Widder formula. AMS Mathematics Subject Classification: 65R20, 35R25. 1. Introduction Let f ∈ L2 (IR+ ). Then the integral F (p) =

Z

∞

e−pt f (t)dt

(1)

0

exists and is called the Laplace transform of f . The Laplace transform occurs frequently in the applications of mathematics, especially in those branches involving solution of differential equations and convolution integral equations. If the image F is known in the complex plane, the original f can be computed by the Bromwich contour integral [5] 1 Z d+i∞ F (p)ept dp, f (t) = 2πi d−i∞

d > 0.

(2)

However, if the image F is known only on the positive axis IR+ , one should use the Post-Widder formula instead (−1)n f (t) = lim n→∞ n!

 n+1

n t

F

(n)

n . t

 

(3)

Formula (3) itself is an approximation scheme for inverting the Laplace transform when the limit is dropped. Jagerman [3] applied this approximation scheme to invert the Laplace transform. However, so far the convergence rate of this approximation scheme has not been studied yet. In this short communication we obtain the convergence rate of the Post-Widder approximate inversion of the Laplace transform in some function spaces. 1

2. Convergence Rate Let f ∗ (s) =

Z

∞

ts−1 f (t)dt

(4)

0

be the Mellin transform of function f [4]. Applying the Parseval formula for the Mellin convolution [4] Z

∞

k(zy)f (y)dy =

0

1 Z 1/2+i∞ ∗ k (s)f ∗ (1 − s)z −s ds, 2πi 1/2−i∞

(5)

valid if k, f ∈ L2 (IR+ ), with k(y) = exp(−y), k ∗ (s) = Γ(s), we have F (p) =

Z

∞

exp(−pt)f (t)dt =

0

1 Z 1/2+i∞ Γ(s)f ∗ (1 − s)p−s ds. 2πi 1/2−i∞

(6)

From (6) one can prove that (−1)n n n! t

 n+1

F (n)

n 1 Z 1/2+i∞ Γ(1 − s + n) s ∗ = n f (s)t−s ds. t 2πi 1/2−i∞ Γ(1 + n)

 

(7)

Applying the Plancherel theorem for the Mellin transform [4] we obtain

   

(−1)n n n+1 (n) n

F − f (t)

n!

t

t

L2 (IR+ )





Γ(1 − s + n) s

= (2π)−1

n − 1 f ∗ (s)

Γ(1 + n)

.

(8)

L2 (1/2−i∞,1/2+i∞)

We have Γ(1 − s + n) s n = 1, n→∞ Γ(1 + n) lim

and

Γ(1 − s + n) s Γ(1 −