[PDF] [PDF] TRANSFORMS Methods of finding inverse Laplace

View - Download [PDF] TRANSFORMS

Previous PDF

Next PDF

LAPLACE

TRANSFORMS

MURRAY R. SPIEGEL, Ph. D.

Laplace transforms applications completely

explainedv

Works with all major texts

450 fully solved problems

Perfect for brushup

or exam prep

Use with these courses:

Operational Calculus 9 Electrical Engineering

9 Mechanics

RT College Ikthematics

SCHAUM'S OUTLINE OF

THEORY AND PROBLEMS

OF

LAPLACETRANSFORMS

MURRAY R. SPIEGEL, Ph.D.

Former Professor and Chairman.

Mathematics Department

Rensselaer Polytechnic Institute

Hartford Graduate Center

SCHAUM'S OUTLINE SERIES

McGRAW-HILL

New York

San FranciscoWashingtun. D.C. Auckland Rogoid Caracas Lisbon

London

MadridAlexien City,MilanAluntrcal New Delhi

San Juan

SingaporeSydneylbkva"lomnro

Copyright © 1965 by McGraw-Hill. Inc. All Rights Reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval -system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. 60231

12131415S11SH754321069

Preface

The theory of Laplace transforms or Laplace transformation, also referred to as

operational calculus, has in recent years become an essential part of the mathematicalbackground required of engineers, physicists, mathematicians and other scientists.Thisis because, in addition to being of great theoretical interest in itself, Laplace transformmethods provide easy and effective means for the solution of many problems arising invarious fields of science and engineering.

The subject originated in attempts to justify rigorously certain "operational rules"used by Heaviside in the latter part of the 19th century for solving equations in electro-

magnetic theory.These attempts finally proved successful in the early part of the 20thcentury through the efforts of Bromwich, Carson, van der Pol and other mathematicianswho employed complex variable theory.

This book is designed for use as a supplement to all current standard texts or as a

textbook for a formal course in Laplace transform theory and applications.It should alsobe of considerable value to those taking courses in mathematics, physics, electrical engi-neering, mechanics, heat flow or any of the numerous other fields in which Laplacetransform methods are employed.

Each chapter begins with a clear statement of pertinent definitions, principles andtheorems together with illustrative and other descriptive material.This is followed by

graded sets of solved and supplementary problems. The solved problems serve to illustrateand amplify the theory, bring into sharp focus those fine points without which the studentcontinually feels himself on unsafe ground, and provide the repetition of basic principlesso vital to effective learning.Numerous proofs of theorems and derivations of formulasare included among the solved problems.The large number of supplementary problemswith answers serve as a complete review of the material in each chapter.

Topics covered include the properties of Laplace transforms and inverse Laplace

transforms together with applications to ordinary and partial differential equations, integralequations, difference equations and boundary-value problems.The theory using complexvariables is not treated until the last half of the book.This is done, first, so that thestudent may comprehend and appreciate more fully the theory, and the power, of thecomplex inversion formula and, second, to meet the needs of those who wish only anintroduction to the subject.Chapters on complex variable theory and Fourier series andintegrals, which are important in a discussion of the complex inversion formula, havebeen included for the benefit of those unfamiliar with these topics.

Considerably more material has been included here than can be covered in most first

courses.This has been done to make the book more flexible, to provide a more usefulbook of reference and to stimulate further interest in the topics.

I wish to take this opportunity to thank the staff of the Schaum Publishing Companyfor their splendid cooperation.

M. R. SPIEGEL

Rensselaer Polytechnic Institute

January, 1965

CONTENTS

Page

Chapter

THE LAPLACE TRANSFORM................................1

Definition of the Laplace transform. Notation. Laplace transforms of someelementary functions.Sectional or piecewise continuity.Functions of ex-ponential order.Sufficient conditions for existence of Laplace transforms.Some important properties of Laplace transforms. Linearity property. Firsttranslation or shifting property.Second translation or shifting property.Change of scale property. Laplace transform of derivatives.Laplace trans-form of integrals.Multiplication by tn.Division by t.Periodic functions.Behavior o (s)as s--.Initial-value theorem.Final-value theorem.Generalization of initial-value theorem. Generalization of final-value theorem.Methods of finding Laplace transforms.Direct method.Series method.

Method of differential equations.Differentiation with respect to a parameter.Miscellaneous methods. Use of Tables. Evaluation of integrals. Some special

functions.The gamma function.Bessel functions.The error function.The complementary error function.Sine and cosine integrals.Exponentialintegral.Unit step function.Unit impulse or Dirac delta function.Null

functions.Laplace transforms of special functions.1

Chapter

2

Chapter

3THE INVERSE LAPLACE TRANSFORM ....................

42
Definition of inverse Laplace transform. Uniqueness of inverse Laplace trans-

forms.Lerch's theorem. Some inverse Laplace transforms. Some importantproperties of inverse Laplace transforms.Linearity property.First transla-tion or shifting property.Second translation or shifting property.Changeof scale property.Inverse Laplace transform of derivatives.Inverse Laplacetransform of integrals.Multiplication by sn.Division by s.The convolutionproperty.Methods of finding inverse Laplace transforms.Partial fractions

method.Series methods.Method of differential equations.Differentiationwith respect to a parameter.Miscellaneous methods.Use of Tables.Thecomplex inversion formula.The Heaviside expansion formula.The betafunction.Evaluation of integrals.

APPLICATIONS TO DIFFERENTIAL EQUATIONS..........

78

Ordinary differential equations with constant coefficients.Ordinary differen-tial equations with variable coefficients.Simultaneous ordinary differentialequations.Applications to mechanics.Applications to electrical circuits.Applications to beams. Partial differential equations.

Chapter

4APPLICATIONS TO INTEGRAL ANDDIFFERENCE EQUATIONS..................................

Integral equations.Integral equations of convolution type.Abel's integralequation. The tautochrone problem. Integro-differential equations. Differenceequations.Differential-difference equations.

Chapter

5COMPLEX VARIABLE THEORY.............................

The complex number system.Polar form of complex numbers.Operations inpolar form. De Moivre's theorem.Roots of complex numbers.Functions.Limits and continuity.

Derivatives.Cauchy-Riemann equations.Line in-tegrals. Green's theorem in the plane. Integrals. Cauchy's theorem. Cauchy'sintegral formulas. Taylor's series. Singular points.

Poles.Laurent's series.

Residues.

Residue theorem.Evaluation of definite integrals.112 136

CONTENTS

Chapter

6FOURIER SERIES AND INTEGRALS........................Fourier series. Odd and even functions. Half range Fourier sine and cosineseries. Complex form of Fourier series. Parseval's identity for Fourier series.Finite Fourier transforms. The Fourier integral. Complex form of Fourierintegrals.Fourier transforms.Fourier sine and cosine transforms.Theconvolution theorem. Parseval's identity for Fourier integrals.Relationshipof Fourier and Laplace transforms.Page

173

Chapter

7THE COMPLEX INVERSION FORMULA...................201The complex inversion formula.The Bromwich contour.Use of residuetheorem in finding inverse Laplace transforms. A sufficient condition forthe integral around r to approach zero.Modification of Bromwich contourin case of branch points. Case of infinitely many singularities.

Chapter

8APPLICATIONS TO BOUNDARY-VALUE PROBLEMS.......219Boundary-value problems involving partial differential equations. Some im-portant partial differential equations. One dimensional heat conduction equa-tion.One dimensional wave equation.Longitudinal vibrations of a beam.Transverse vibrations of a beam. Heat conduction in a cylinder. Transmissionlines.Two and three dimensional problems.Solution of boundary-valueproblems by Laplace transforms.

APPENDIX A. TABLE OF GENERAL PROPERTIES OF

LAPLACE TRANSFORMS.................................243 APPENDIX B. TABLE OF SPECIAL LAPLACE TRANSFORMS.......... 245

APPENDIX C. TABLE OF SPECIAL FUNCTIONS

.......................255

Chapter 1

The Laplace Transform

DEFINITION OF THE LAPLACE TRANSFORM

Let F(t) be a function of t specified for t> 0.

Then the Laplace transform of F(t),

denoted by 4 (F(t)), is defined by {F(t))=f(s)=f e-StF(t) dt(1) 0

where we assume at present that'the parameter s is real.Later it will be found usefulto consider s complex.

The Laplace transform of F(t) is said to exist if the integral (1) converges for somevalue of s; otherwise it does not exist.For sufficient conditions under which the Laplacetransform does exist, see Page 2.

NOTATION

If a function of t. is indicated in terms of a capital letter, such as F(t), G(t), Y(t), etc.,the Laplace transform of the function is denoted by the corresponding lower case letter,

i.e. f (s), g(s), y(s), etc.In other cases, a tilde (-) can be used to denote the Laplace trans-form. Thus, for example, the Laplace transform of u(t) is is (s).

LAPLACE TRANSFORMS OF SOME ELEMENTARY FUNCTIONS

The adjacent table shows

Laplace transforms of variouselementary functions.For de-tails of evaluation using defini-tion (1), see Problems-1 and 2.For a more extensive table seeAppendix B, Pages 245 to 254.F(t)

-C {F(t)} = f(8) 1.

118>0s

2.ts2s>0

3.to!8 > 0sn

n = 0, 1, 2, ... Note.Factorial n = n! = 12 nAlso, by definition 0! = 1.

4.eat1s > as-a

5. sin ata> 0s_82+a2 6. cos at88 > 082a2 7. sinh ata8 > jai82 - a2 8. cosh at8 > lat82a2 1 2

THE LAPLACE TRANSFORM[CHAP. 1

SECTIONAL OR PIECEWISE CONTINUITY

A function is called sectionally continuous or piecewise continuous in an intervalc< t-< a if the interval can be subdivided into a finite number of intervals in each ofwhich the function is continuous and has finite right and left hand limits.

F(t) I/ j i tatit21t3R

Fig. 1-1

An example of a function which is sectionally continuous is shown graphically in

Fig. 1-1 above.This function has discontinuities at ti, t2 and t3.Note that the right andleft hand limits at t2, for example, are represented bylim F(t2 + E) = F(t2 + 0) = F(t2+)

e

0andlim F(t2 - E) = F(t2- 0) = F(t2-)respectively, where cis positive.E-+0

FUNCTIONS OF EXPONENTIAL ORDER

If real constants M > 0 and y exist such that for all t > N

I e-It F(t) I

< MorI F(t) 1< Melt we say that F(t) is a function of exponential order y as t- - or, briefly, is of exponential order. Example 1.F(t) = t2is of exponential order 3 (for example), since;t2j = t2 < eat for all t > 0.

Example 2.

F(t) =et3is not of exponential order sinceIe-vt et' 1 =et3-yt can be made larger thanany given constant by increasing t.

Intuitively, functions of exponential order cannot "grow" in absolute value more rapidly

than Me"' as t increases.In practice, however, this is no restriction since M and y can beas large as desired.

Bounded functions, such as sin at or cos at, are of exponential order. SUFFICIENT CONDITIONS FOR EXISTENCE OF LAPLACE TRANSFORMS

Theorem 1-1.

If F(t) is sectionally continuous in every finite interval 0 < t< N andof exponential order y for t > N, then its Laplace transform f (s) exists for all s > y.

For a proof of this see Problem 47.It must be emphasized that the stated conditionsare sufficient to guarantee the existence of the Laplace transform.If the conditions arenot satisfied, however, the Laplace transform may or may not exist [see Problem 32].Thus the conditions are not necessary for the existence of the Laplace transform.

For other sufficient conditions, see Problem 145.

CHAP. 11

THE LAPLACE TRANSFORM3

SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORMS

In the following list of theorems we assume, unless otherwise stated, that all functions satisfy the conditions of Theorem 1-1 so that their Laplace transforms exist. 1.

Linearity property.

Theorem 1-2.

If c1 and C2 are any constants while F1(t) and F2(t) are functions with Laplace transforms f i (s) and f2 (s) respectively, then .({C1F1(t)+C2F2(t)I=C14 {Fi(t)} + c2a({F2(t)}=clfl(s) + C2f2(s)(2) The result is easily extended to more than two functions. Example.C (4t2 - 3 cos 2t + 5e-1}=4.C {t2} - 3.C {cos 2t} + 5.4 {e-t}

4(83) --3(s2+4)+5Cs+1)

83s+5
s3_ s2+4s+1 The symbol C, which transforms F(t) into f (s), is often called the Laplace trans-

formation operator.Because of the property of t expressed in this theorem, we saythat e( is a linear operator or that it has the linearity property.

2.First translation or shifting property.

Theorem 1-3.

IfaC {F(t)} = f(s)then

a( {eal F(t)}=f(s - a)(3)

Example.Sincee {cos 2t} =S

.32+ 4 ,we have .C{e-tcos2t} = s+1=s+1(s+1)2+4s2+2s+5 3.

Second translation or shifting property.

Theorem 1-4.

If(F(t))sandG(t) = td(t - a)t .({G(t))=e-as f(s)(4) Example.Sincei {t3} =s4= s4 ,the Laplace transform of the function G(t) =j(t - 2)3 t>2 10t<2 is 6e-2s/g4

4.Change of scale property.

Theorem 1-5.

If.( {F(t)} = f(s),then

.t {F(at)}

Example.

Since.( {sin t} =s2 + 1we have(5)

{sin 3t} = 1 1_3

3 (s/3)2 + 1

s2 + 9

4THE LAPLACE TRANSFORM

5.Laplace transform of derivatives.

Theorem 1-6.

Ift {F(t)} = f(s),then

i (F'(t)) =s f(s) - F(O)[CHAP. 1 (6)

if F(t) is continuous for 0 < t < N and of exponential order for t > N while F'(t) issectionally continuous for 0 < t < N.

Example.

If F(t) = cos 3t, thenr {F(t)} =82 + 9and we have

{F'(t)} {-3 sin 3t}=s (82+9 )- 182+ `9

The method is useful in finding Laplace

transforms without integration [see

Problem 151.

Theorem 1-7.If in Theorem 1-6,F(t)fails to be continuous att = 0 butlim F(t) = F(0+)exists [but is not equal to F(0), which may or may not exist], thent-.o

.C {F'(t)) =sf(s) - F(0+)(7)

Theorem 1-8.

If in Theorem 1-6, F(t) fails to be continuous at t = a, then

J (F'(t))

=s f (s) - F(O) - e -°s { F(a+) - F(a-) }(8) where F(a+) - F(a-) is sometimes called the jump at the discontinuity t = a.For more than one discontinuity, appropriate modifications can be made.

Theorem 1-9.

If{F(t) } = f (s),then

{F"(t)} =s2 f(s) - sF(0) - F'(0)(9) if F(t) and F'(t) are continuous for 0 = t < N and of exponential order for t > N while F"(t) is sectionally continuous for 0 t-:5 N. If. F(t) and F'(t) have discontinuities, appropriate modification of (9) can be made as in Theorems 1-7 and 1-8.

Theorem 1-10.

If(' {F(t)} = f(s),then

4 (Fcn)(t) }

=sn f(s) -sn-' F(0) - sn2 F'(0) -sFcn-2,(0)- Fcn 1)(0)(10) if F(t), F'(t),...,F`n-"(t) are continuous for 0tN and of exponential order for t > N while F(n)(t) is sectionally continuous for 0 < t < N. 6.

Laplace transform of integrals.

Theorem 1-11.

IfC {F(t)} = f(s),then

of if foF(u)du1

Example.Since t {sin 2t} =-+4 ,we have

J sin 2u du-2(J r

1US(82+4)

as can be verified directly.

CHAP. 1)

THE LAPLACE TRANSFORM5

7.Multiplication by tn.

Theorem 1-12.

If.{F(t)} = f(s),then

-(' {tn F(t)} _(-1)n dsnf(s)

Example.SinceC {e2t}=_1_s - 2 'we have

.e {te2t} .e {t2e2t}= (-1)n f(n)(s) d111 ds(s-2(s-2)2d( 1) _22d82 s-2(s-2) 8.

Division by t.

Theorem 1-13.

If.{F(t)} = f(s),then

provided lim F(t)/t exists. t-.of f (u) du(13) s

Exam le.

Since1sin tp({sin t} -82 + 1and Eo= 1,we havet

sin t l __f xdu t jJu2 + 1-tan' 1(11s)

9.Periodic functions.

Theorem 1-14.

Let F(t) have period T > 0 so that F(t + T) = F(t) [see Fig. 1-2]. Then F(t)T f° e-" F(t) dt {F'(t)}_(14)1 - e-ST

Fig. 1-2

10.Behavior o (s) as s -* co.

Theorem 1-15.

Ife (F(t)} = f(s),then

lim f (s)=0

S -. 'Jo

11.

Initial-value theorem.

Theorem 1-16.

If the indicated limits exist, then

lim F(t) =lim s f (s)t--0s-(15) (16)

6THE LAPLACE TRANSFORM

12.

Final-value theorem.

Theorem 1-17.

If the indicated limits exist, then

lim F(t)=lim s f (s)t-. os-.o[CHAP. 1 (17) 13.

Generalization of initial-value theorem.

If urn F(t)/G(t) = 1,

then we say that for values of t near t = 0 [small t], F(t) ist-.oclose to G(t) and we write F(t) - G(t) as t- 0.

Similarly if

lim f(s)/g(s) = 1,then we say that for large values of s, f (s) is -4 00close to g(s) and we write f (s) - g(s) as s With this notation we have the following generalization of Theorem 1-16.

Theorem 1-18.

If F(t) - G(t) as t - 0, then f(s) - g(s) as s -> oo where f (s).C (F(t)) and g(s) _ .( {G(t)}. 14.

Generalization of final-value theorem.

If lim F(t)/G(t) = 1, we write F(t) - G(t) as t

Similarly if lim f(s)/g(s) = 1,tWSowe write f (s) -- g(s) as s - 0.Then we have the following generalization of Theorem

1-17. _ Theorem 1-19.If F(t) ~ G(t) as t - oc, the (s) - g(s) as s - 0 where f (s) .C {F(t)} andg(s) _ .({G(t)}.

METHODS OF FINDING LAPLACE TRANSFORMS

Various means are available for determining Laplace transforms as indicated in the following list.

1.Direct method.This involves direct use of definition (1).

2. Series method.If F(t) has a power series expansion given by F(t) =ao + ait + a2t2 +=I ante(18)n-o its Laplace transform can be obtained by taking the sum of the Laplace transforms of each term in the series. Thus ao a,2!-a2n! an.C {F(t)}=+++_(19)SS-s'3n-U$n+1 A condition under which the result is valid is that the series (19) be convergent for s > y.

See Problems 34, 36, 39 and 48.

3. Method of differential equations.This involves finding a differential equation satis- fied by F(t) and then using the above theorems.See Problems 34 and 48. 4. Differentiation with respect to a parameter.See Problem 20. 5. Miscellaneous methods involving special devices such as indicated in the above theo- rems, for example Theorem 1-13.

6.Use of Tables (see Appendix).

CHAP. 1]

THE LAPLACE TRANSFORM

EVALUATION OF INTEGRALS

Iff (s) = C {F(t) },then

Taking the limit as s -> 0, we have7

f0 e-StF(t) dt=f(s)(20)f

F(t) dt=f(0)(21)

0assuming the integral to be convergent.

The results (20) and (21) are often useful in evaluating various integrals.

See Problems

45 and 46.

SOME SPECIAL FUNCTIONS

1.

The Gamma function.

I > 0, we define the gamma function by

r(n) =fun- ' e-u du(22) 0 The following are some important properties of the gamma function. 1. r(n + 1)=n r(n),n > 0 Thus since r(1)1, we have I'(2) = 1, r(3) = 2 ! = 2, or(4) = 3! and in general r(n + 1) = n!,i is a positive integer.For this reason the function is some- times called the factorial function. 2. r(:)= N5 3.

4.For large n,r(n+1)-27rnn11 e-n

[Here - means "approximately equal to for large n". More exactly, we writeF(n) ~ G(n) iflim F(n)/G(n) = 1.]This is called Stirling's formula.

n < 0 we can define r(n) by r(n) =r(n + 1) n II.

Bessel functions.

We define a Bessel function of order n by

_t2(t)=2nr(n+1)12(2n+2)

Some important properties are0

_ ...(23)2.4(2n + 2)(2n + 4) 1.

J-n (t)(-1)n Jn (t)i is a positive integer

2.

Jn+Z (t) =2tJ, (t) -(t)

3. dtto Jn-i (t).I = 0, we haveJo(t) = -J1(t).

4.e 'h"-1/u) =1'Jn(t)unr(p) r(1- p)

-sin per '0 < p < 1 9!=-a This is called the generating function for the Bessel functions.

8THE. LAPLACE TRANSFORM

5.J. (t) satisfies Bessel's differential . equation.

t2 Y11(t) + t Y'(t) + (t2 - n2) Y(t) _

It is convenient to define

J0(it) = i-nIn(t) where I0(t)Bessel function of order n. III.

The Error function is defined as

erf (t) =2e-u2 du 0 IV. The Complementary Error function is defined as erfc (t) =1erf (t).1 -2fte-u2du 0 V.

The Sine and Cosine integrals are defined by

t

Si (t)= fsin udu

0u

Ci (t)

VI.

The Exponential integral is defined asf

Cos udutu

00 eudu5Ei (t)=(26) (27) (28) VII.The Unit Step function, also called Heaviside's unit function, is defined as 'U(t - a) _{0t < a(29)

See Fig. 1-3.1t > a

u(t - a) i a t0 [CHAP. 1 is called the modified (24)

5e_u2du

(25) I 1/E

Fig. 1-3

Fig. 1-4

VIII.The Unit Impulse function or Dirac delta function...

Consider the function

FE (t)

quotesdbs_dbs13.pdfusesText_19

[PDF] laplace transform calculator

[PDF] qqs

[PDF] niveau bright

[PDF] abréviation exemple

[PDF] test bright gratuit

[PDF] formule du demi-perimetre du rectangle

[PDF] mensuration homme musculation

[PDF] mesurer tour de cuisse homme

[PDF] orientation scolaire belgique

[PDF] conseiller d'orientation bruxelles

[PDF] vocabulaire géométrie collège

[PDF] test tricam

[PDF] exemple programme larp

[PDF] test paramétrique exercice corrigé

[PDF] test non paramétrique pdf