Abstract
Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = ∫A 0 tα exp(-xtβ)F(t)dt In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 485-500 |
| Number of pages | 16 |
| Journal | Chemical Engineering Communications |
| Volume | 204 |
| Issue number | 4 |
| DOIs | |
| State | Published - Apr 3 2017 |
Keywords
- Asymptotics
- Expansion-point improvement
- Exponential improvement
- Laplace Integrals with Finite Upper Limits (FUL)
- Watson’s lemma (WL)
Cite this
General Laplace Integral Problems : Accuracy Improvement and Extension to Finite Upper Limits. / Hanna, Owen T.; Davis, Richard A.
In: Chemical Engineering Communications, Vol. 204, No. 4, 03.04.2017, p. 485-500.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - General Laplace Integral Problems
T2 - Accuracy Improvement and Extension to Finite Upper Limits
AU - Hanna, Owen T.
AU - Davis, Richard A.
PY - 2017/4/3
Y1 - 2017/4/3
N2 - Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = ∫A 0 tα exp(-xtβ)F(t)dt In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.
AB - Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = ∫A 0 tα exp(-xtβ)F(t)dt In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.
KW - Asymptotics
KW - Expansion-point improvement
KW - Exponential improvement
KW - Laplace Integrals with Finite Upper Limits (FUL)
KW - Watson’s lemma (WL)
UR - http://www.scopus.com/inward/record.url?scp=85014694321&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85014694321&partnerID=8YFLogxK
U2 - 10.1080/00986445.2016.1277524
DO - 10.1080/00986445.2016.1277524
M3 - Article
VL - 204
SP - 485
EP - 500
JO - Chemical Engineering Communications
JF - Chemical Engineering Communications
SN - 0098-6445
IS - 4
ER -