1994 Mathematica Laplace Inv

Approximate Inversion of the Laplace Transform Alexander H-D. Cheng, Paston Sidauruk, University of Delaware Younane Abousleiman, University of 0 klahoma

The Laplace transform is often applied to linear partial differential equations to eliminate the time dimension. The analytical solutions thus obtained need paper provides Matbematica functions for approximate

The Laplace transform

F(s) = Sm f(t)e%t 0

is often used in the solution of linear partial differential equations to eliminate the time dimension. The resulting systems become easier to solve by analytical techniques [Detournay and Cheng 19881, or numerical methods such as finite element or boundary element methods [Cheng and Ou 1989; Cheng and Detournay 19881. The solutions thus obtained need to be inverted to the time domain. The exact inversion is normally difficult to carry out, so approximate inversion techniques are used.

There are many approximate Laplace inversion algorithms. (For comprehensive bibliographies, see [Piessens 197.5; Piessens and Dang 19761.) With issues of computa- tional efficiency and accuracy in mind, a handful of techniques have been tested and compared [Cost 1964; Davies and Martin 1979; Narayanan and Beskos 19821. Based on recommendations in the literature and our own experience in engineering applications, we present here the Mathematics package NLaplaceInversion .m, which implements five inversion algorithms, one in each of the five categories classified by Davies and Martin [ 19791.

It is important to emphasize that the inverse of the Laplace transform is not stable under reasonable perturba- tions [Bellman et al. 19661. This can be seen by considering the function f(t) = sin at, a > 0, and its Laplace transform F(s) = a/( a2 + s2). While F(s) can be made arbitrarily small by taking a sufficiently large, the inverse transform f(t) oscil- lates between -1 and 1. It is therefore impossible to devise a universal algorithm that performs well for all types of func-

Alexander H-D. Cheng is a professor of civil engineering at the University of Delaware. His research interests include groundwater, poroelasticity, boundary element methods, and chaotic dynamics.

Paston Sidauruk holds a Masters degree in civil engineering from the University of Delaware and is currently a Ph.D. candidate. He is on leave from the Hydrology Section of the Atomic Energy Agency in Jakarta, Indonesia.

Younane Abousleiman holds a Ph.D. from the University of Delaware. He is currently a Senior Research Scientist at the Rock Mechanics Research Center at the University of Oklahoma.

to be inverted to the time domain. inversion of the Laplace transform.

This

tions. In particular, functions of periodic nature, of spike shapes, and whose Laplace transforms contain poles (singu- larities), are normally difficult to handle. With these limita- tions in mind, we present below the five approximate Laplace inversion algorithms that are implemented in our package.

where has th

F(s) c

ln[3]:= F[

In[4]:= Nl

OLlt[4]= 0

The e:

ln[5]:= f I

ln[6]:= f 1

Out[6]= c

BY expec error

ln[7]:= xq

Stehfest Method ln[B]:=

The Stehfest method [Stehfest 19701 is a popular numerical inversion techn ique used in groundwater flow and petroleum reservoir engineering applications. The inversion formula is based on computing a sample of the time function using a delta-convergent series. The resultant series approximation takes the following form:

f(t) == yi7,cnF(*) n=l

where the coefficient c, is given by

c, = (-1) n+N/2 x

min( n, N/2)

x

/P/2(2@!

k=(n+1)/2 (N/2 - A$! Ic! (k - l)! (n - A$! (2k - n)!

The number of terms N in the series must be even. The inverse function NLInvSteh can be defined as

In[l]:= csteh[n_, i,] = (-l)^(i + n/2) Sum[ k^(n/2) (2k)! / ((n/2-k)! k! (k-l)! (i-k)! (2k-i)! >, {k, Floor[(i+l)/2], Min[i, n/2] 1 I;

ln[z]:= NLInvSteh[F_, s_, t,, n-1 := Log[2]/t Sum[ csteh[n,i] F /. s -> i Log[2l/t,

{i, 1, n) 1 // N

For example, the function

Th aboul event asym accur diver1 range

Fig (2) ( !?S for tl tions that enhai

76 THE MA THEMA TICA JOURNAL 0 1994 Miller Freeman Publications

F( ) 2 Kdm s =s (1) where K0 is the modified Bessel has the exact Laplace inversion

function of the second kind,

&f(t) = -E&-l/t) (2)

F(s) can be numerically inverted for the solution at t = 1:

ln[3]:= F[s_] = (2/s) BesselK[O, 2 Sqrt[s]];

ln[4]:= NLInvSteh[F[s], s, I, 61

Out[4]= 0.223359

The exact solution is

ln[5]:= f [t,] = - ExpIntegralEi[ -l/t ] // N;

ln[6]:= f [I]

Out[6]= 0.219384

By increasing the number of terms in the series, the error is expected to diminish. The following plot shows the relative error versus the number of terms N in log scale:

tn[7]:=

FIGURE 2. Comparison of Stehfest (dashed line) and Papoulis (dotted line) inversion FIGURE 4. Comparison of Stehfest (dashed results with exact solution (solid line), f(t) = -Ei(-1 /t). results with exact solution (solid line) f(t) = sin

We observe that it is inferior to the Stehfest result at both small and large times.

Figure 3 shows the inversion for the Laplace transform pair

using the Stehfest and the Papoulis methods. The results are excellent in both cases. It is not surprising that the Papoulis method works well for this problem, since its bases are exponential functions. For many diffusion problems, the solution is expected to contain multiple components of exponential decay. The exponential series based inversion techniques are highly efficient in those applications.

Next, we test the sine function:

W) = -& f(t) = sin t

(3 ) a

(3b)

Figure 4 displays the results for the Stehfest and the Papoulis inversion. They are both unsatisfactory.

Durbin-Crump Method

The Durbin method [Durbin 19761 approximates the time function f(t) by a periodic function of period T. The user is asked to provide the parameter t,,, , specifying that the inverse tranform will only be used in the interval [0,4 t,,,].

f(t) 1

0.5

0.1 0.05

0.01 0.005

0.001

FIGURE 3. Comparison results with exact solution

of Stehfest (dashed (solid line), f(t) = te-

line) and Papoulis (dotted line) inversion

line) and Papoulis (dotted line) inversion

The approximating function, with period T = 4 t,,, , is represented by its Fourier Series. The coefficients of the Laplace transform are equated to samples of the transform F(s):

f(t) = ${y + f)e[F(a + y)cos(y)] k=l

-Im[F(a + y)sin(F)]/

where i = 11-1. Th e parameter a can be chosen as

(4)

a =a 1nE -- 2T

where E is the error tolerance, for which the default value E = 1 x lo-* has been assigned, and a is the real part of the leading pole of the function F(s). For functions without poles, the default value a = 0 has been set. The following code implements the Durbin method:

adurb[tmax_, alpha_, tol_] := adurb[tmax, alpha, toll = alpha - Log[tol] / (8 tmax) // N

cdurb[F_, s_, tmax_, alpha_, tol_, k-1 := F /. s -> adurb[tmax, alpha, toll + I Pi k / (4 tmax) // N

cdurb[F_, s_, tmax_, alpha_, tol_, 01 := F/2 /. s -> adurb[tmax, alpha, toll // N

NLInvDurb[F_, s_, t_, j_, tmax_, alpha_:O, to1_:0.00000001] := NLInvDurb[F, s, t, j, tmax, alpha, toll =

Exp[ adurb[tmax, alpha, toll t I / (4 tmax) * Sum[ Re[ cdurb[F, s, tmax, alpha, tol, k] I

Cos[k Pit /(4 tmax)] - Im[ cdurb[F, s, tmax, alpha, tol, k] 1 Sin[k Pit /(4 tmax)],

{k, 0, j) 1 // N

The Durbin method has the same deficiency as a Fourier series: it converges slowly. Figure 5 shows the convergence of the Durbin inversion of the function (3a) to the exact solution with 20 and 50 terms. The SO-term result practically coincides with the exact solution.

FIGURE 5. terms. The e

Next,

We note s=l. -I NLInvDurt the dot demons and the

Despi that any tally in\ sion. Tfi ema tica method

The Durbin 19641. r

where t quantit note he cedure j

epsilon1 epsilon1 epsilo ( epsi

eP

epsilon1

epsilon[ NLInvD

NLInvCrm epsilo

The op Crump


FIGURE 5. Convergence of the Durbin inversion with 20 (dashed line) and 50 (dotted line) terms. The exact solution is f(t) = sin t (solid line).

Next, the test is extended to the Laplace transform pair:

(5 ) a

We note in particular that the function (5a) has a pole at s = 1. The proper inversion requires the use of a = 1 in NLInvDurb. The result using N = 100 is shown in Figure 6 as the dotted line. The result is reasonable. Although not demonstrated in the figure, we note that both the Stehfest and the Papoulis method badly fail in this situation.

Despite the reasonable approximation, it is recommended that any poles in a function should be removed and analyti- cally inverted prior to the execution of an approximate inversion. The proper tool for extracting the poles is again Muth- ematica. In those cases, even the Stehfest and the Papoulis method can perform a proper inversion.

The Crump method [Crump 19761 is an accelerated Durbin method that uses the epsilon-algorithm [MacDonald 19641. The algorithm is given as

where $7) = 0 and &brn) is the mth partial sum of (4). The quantity EN (O) hence represents the extrapolated result. We note here that N must be an even number. The Crump pro- cedure is implemented as:

epsilon[F_, s_, t_, tmax_, alpha_, Ml_, i_, m-1 := epsilon[F, s, t, tmax, alpha, tol, i, ml = epsilon[F, s, t, tmax, alpha, tol, i - 2, m + II + 1 / ( epsilon[F, s, t, tmax, alpha, tol, i - 1, m + 11 -

epsilon[ F, s, t, tmax, alpha, tol, i - 1, ml >

epsilon[F_, s_, t_, tmax_, alpha_, tol_, -1, m-1 := 0

epsilon[F_, s_, t_, tmax_, alpha_, toI_, 0, m-1 = NLInvDurb[F, s, t, m, tmax, alpha, toll

NLInvCrmp[F_, s_, t_, n_, tmax_, alpha_:O, tol_:O.OOOOOOO~l := epsilon[F, s, t, 0.2 tmax, alpha, tol, n, 01

The optimal value for T is fixed at 0.8 t,,, , as suggested by Crump [1976].

For the case of the sine function (3), the Crump method with N = 20 produces results comparable with the Durbin method using N = 50. However, the need to store a large

(m) number of li quantities in memory to avoid re-computation makes the Crump algorithm memory-inefficient.

f(t) t 150: ,

12s.

100:

75 -

SO-

2s :

1 2 3 4 St

FIGURE 6. Comparison of Durbin (dotted line) and Weeks (dashed line) inversion results with the exact solution f(t) = et (solid line).

Weeks Method

The Weeks method [Weeks 19661 is classified as a method that uses a bilinear transformation of s [Davies and Martin 19791. The approximate inversion formula is

where Lk are the Laguerre polynomials, and the parameters are

with a the leading pole of the function F(s), and H the Heav- iside unit step function. The coefficients ak are

N 1 a() = -

N+l h(0 ) j j=o N

2 al, = ~ N+l

h(8,) cos(ke,), N#O j=o

in which

n 2j+l (+---- 2 N+l

h(O) - - )I - cot $ Im

[i F c + & cot :

n )I The code for NLInvWks is given as:

I (6

VOLUME 4, ISSUE 2 79

talk, tmax_] := tn[n, tmax] = tmax / n

cuks[tmax_, alpha_] := cuks[tmax, alpha] = If[ (alpha + 1 / tmax) >= 0, alpha + 1 / tmax, 0 ]

thetab_, j-1 := (Pi/2) (2 j + I) / (n + 1)

h[F_, s_, n_, tmax_, alpha_, j-1 := h[F, s, n, tmax, alpha, j] = l/(2 tn[n, tmax]) (Re[F] - CotI theta[n, j]/2 1 Im[F]) /.

s -> cuks[tmax, alpha] + I / (2 tn[n, tmax]) Cot[ thetarn, j]/2 I // N

auks[F_, s_, n_, tmax_, alpha_, k_] := 2/(n + I) Sum[ h[F, s, n, tmax, alpha, j]

Cod k thetaIn, jl I, {j, 0, n) 1

auks[F_, s_, n_, tmax_, alpha_, 01 := l/(n + 1) SumI h[F, s, n, tmax, alpha, j], {j, 0, n)l

NLInvWks[F_, s_, t_, n_, tmax,, alpha_:01 := Exp[ ( cuks[tmax, alpha] - 1 / (2 tn[n, tmaxl) > t] * Sum[ auks[F, s, n, tmax, alpha, k]

LaguerreL[ k, t / tn[n, tmax] 1, {k, 0, n) 1 // N

The Weeks method received one of the highest ratings in [Davies and Martin 19791 for its overall accuracy tested against an assorted group of functions. Indeed, using merely 10 terms (and a = l), the method inverts (5a) quite accu- rately. The result is presented in dashed line in Figure 6, which virtually coincides with the exact solution.

Another tough test is provided by the Bessel function J. For the pair:

f(t) = Jo(t) VW

the inverted result (dotted line) is displayed against the exact solution in Figure 7.

Piessens Gaussian Quadrature Method

The Piessens method seeks to evaluate the Bromwich integral by numerical quadrature [Piessens 1971; 19731. The time function is expressed as a Gaussian quadrature sum:

fo=&wixiF(y i=l

where B is a positive real constant such that #F(s) is analytic and has no branch point at infinity. The abscissas Q, i = 1, . . . ) N, are the zeros of the orthogonal polynomials &(x). The polynomials Pi(z) are related to the generalized Bessel polynomials and are orthogonal with respect to the weight sp. They can be evaluated from the recursive relation:

PO(X) = 3

PI(X) = /3x - 1 Pi(X) = (U$ + bJ Pi-1 (x) + ci P&x), i 2 2

1

0.8

0.6

0.4

FIGURE 7. Comparison of Weeks (dotted line) and Piessens (dashed line) inversion results with the exact solution f(t) = Jo (t) (solid line).

in which

ai = (2i + j3 - 3)(2i + /3 - 2) i+/L--2

bi = (2i + j3 - 3)(2 - /5)

(i+B-2)(2i+/3-4)

Ci = (2i + /3 - 2)(i - 1)

(i+j3--2)(2i+/3--4)

The weights uti are given by

wi = (-y-1 (N-l)! [2N+B_2] T(N + B - I)Nx pN-l(l/Xi)

where I is the Gamma function. The complete code is the following:

apsns[i_, beta_] := apsns[i, beta] = (2 i + beta - 3) (2 i + beta - 2) / (i + beta - 2)

bpsns[i_, beta_] := bpsns[i, beta] = (2 i + beta - 3) (2 - beta) /

( (i + beta - 2) (2 i + beta - 4) >

cpsns[i_, beta_] := cpsns[i, beta] = (2 i + beta - 2) (i - 1) / ( (i + beta - 2) (2 i + beta - 4) >

p[i_, beta_, x-1 := (apsndi, beta] x + bpsndi, beta]) p[i - I, beta, x] +

cpsns[i, beta] p[i - 2, beta, x ]

p[O, beta,, x-1 := I

p[l, beta_, x,] := beta x - I

proot[n_, beta_] := proot[n, beta] = NSolveE p[n, beta, l/x] == 0, x ]

xi[n_, beta_, i,] := x /. proot[n, beta] [[i]]

ui[n_, beta_, i,] := ui[n, beta, i] = (-l>^(n - l)(n-I)!/ (n Gamma[n + beta - 11 xi[n, beta, ij-2 > ( (2 n + beta - 2) /

p[n - 1, beta, l/xi[n, beta, i] ] )-2 // N

NLInvPsn SumC F

We not To tc

using 1 line, WI the opti ber of t represe Nth or sB F(s) Maclau with sl( to the f recomn

As tl pan-:

Due point 2 branch The prc ure 8 s default in dasl- using I\ essenti; the fun

FIGURE (dotted lini


WLInvPsns[F_, s_, t_, n_, beta_:11 := (l/t) Rd Sum[ ui[n, beta, i.] xi[n, beta, i]^beta F /. s -> (xi[n, beta, ilk), {i, 1, d 1 1 // N

We note that the default value p = 1 has been set. To test the algorithm, the inversion of (7a) is performed

using 12 terms. The result is presented in Figure 7 in dashed line, which overlaps the exact solution. In fact, this result is the optimal one. The solution rapidly deteriorates if the number of terms is increased. The probable cause is the inefficient representation of (7a) by a polynomial series in l/s. (The Nth order Gaussian quadrature gives the exact result of #F(s) if it is a polynomial in l/s of degree 2N - 1.) The Maclaurin series expansion of (7a) gives an alternating series with slowly dying coefficients, which might have contributed to the failure of the method. For engineering purposes, it is recommended that N 2 10 be used for all functions.

As the final example, we choose the following transform pair:

F( ) 1 -l/s s=-e 21 S

f(t) - - LcosJa 4-t ?r Due to the presence of the factor l/G, there is a branch

point at infinity. However, the function s1i2 F(s) has no branch point and can be represented as a polynomial in l/s. The proper inversion should be performed with /3 = l/2. Fig- ure 8 shows the inversion by the Piessens method with the default (but wrong) b value in dotted line, and with /3 = l/2 in dashed line (indistinguishable from the exact solution), using N = 10. It is clear that for the quadrature method it is essential to learn about the existence of any branch point of the function before carrying out the inversion.

f(t)

FIGURE 8. Compar ison of the Piessens inversion using B = 0.5 (dashed line) and /3 = 1 (dotted line) with the exact solution f(t) = (1 /fi) cos N (solid line).

Conclusion

We have provided five approximate inversion algorithms (Stehfest, Papoulis, Durbin-Crump, Weeks, Piessens), one each in the five categories presented by Davies and Martin [ 19791. Based on a set of assorted test functions, some algorithms were found to be more accurate, while others are more efficient. The performance is dependent on the function type. It is difficult to recommend just one inversion method. The user should have some idea of the behavior of the function in order to choose the proper method. When in doubt, multiple algorithms should be used and their results compared. If used with care, the approximate Laplace inversion methods can broaden our range of techniques for finding (semi-) analytical solutions of partial differential equations.

References

Bellman, R., R.E. Kalaba, and J.A. Lockett. 1966. Numerical inversion of the Laplace transform: Applications to biol- ogy, economics, engineering and physics. American Else- vier, New York.

Cheng, A.H-D., and E. Detournay. 1988. A direct boundary element method for plane strain poroelasticity. ht. J. Num. Anal. Meth. Geomech. 12:551-572.

Cheng, A.H-D., and O.K. Morohunfola. 1993. Multilayered leaky aquifer systems: I. Pumping well solution. Water Resour. Res. 29:2787-2800.

Cheng, A.H-D., and K. Ou. 1989. An efficient Laplace transform solution for multiaquifer systems. Water Resour. Res. 25:742-748.

Cost, T.L. 1964. Approximate Laplace transform inversion in viscoelastic stress analysis. AIAA Journal 2:2157-2166.

Crump, K.S. 1976. Numerical inversion of Laplace transforms using a Fourier series approximation. J. Assoc. Computing Machinery 23~89-96.

Davies, B., and B. Martin. 1979. Numerical inversion of Laplace transform: a survey and comparison of methods. J. Camp. Phys. 33:1-32.

Detournay, E., and A.H-D. Cheng. 1988. Poroelastic response of a borehole in a non-hydrostatic stress field. ht. ]. Rock Me&. Min. Sci. Geomech. Abstr. 25:171-l 82.

Durbin, F. 1974. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abates method. Computer]. 17:371-376.

MacDonald, J.R. 1964. Accelerated convergence, divergence, iteration, extrapolation, and curve fitting. J. Appl. Phys. 10:3034-3041.

Narayanan, G.V., and D.E. Beskos. 1982. Numerical opera- tional methods for the time-dependent linear problems. ht. J. Numer. Meth. Engng. 18:1829-l 854.

Papoulis, A. 1957. A new method of inversion of the Laplace transform. Quart. Appl. Math. 14:405-414.

VOLUME4,lSSUEZ 81

Piessens, R. 1971. Gaussian quadrature formulas for the numerical integration of Bromwichs integral and the inversion of the Laplace transform. J. Eng. Math. 51-9.

Piessens, R. 1973. Gaussian quadrature formulas for Bromwichs integral. Comm. ACM 16:486-487.

Piessens, R. 1975. A bibliography on numerical inversion of the Laplace transform and applications. J. Camp. Appl. Math. 1:ll.S126.

Piessens, R., and N.D.P. Dang. 1976. A bibliography on numerical inversion of the Laplace transform and applications: a supplement. J. Comp. Appl. Math. 2:22.5-228.

Stehfest, H. 1970. Numerical inversion of Laplace transforms. Comm. ACM 13: 47-49 and 624.

Or 0

Prl Bernt

Weeks, W.T. 1966. Numerical inversion of Laplace transforms using Laguerre functions. J. ACM. 13:419-426.

The FOI waves c by its al analyze Oppenl WQI, phase s determi [1989] text.

The I graphic tudes a charact Oppen! [1982]. charactl these in of amp1 image r

The 1 charact book in input st

In[l]:= Sh

Bernt C. Californk devices.


1994 Mathematica Laplace Inv

Documents

Transcript of 1994 Mathematica Laplace Inv