Tutorial on Symbolic Computing with Mathematica - … on Symbolic Computing with Mathematica...

Tutorialon

Symbolic Computingwith Mathematica

Youngjoo Chung

School of Info. and Comm., [email protected]

2011. 7. 7

<< InitPackages`

Symbolic Computation

• Symbolic computation (algebraic manipulation)

– Mathematical computation with variables and constants according to the rules of algebra

Concept Formulation

CalculationVisualization

•Numerical Computing

•Graphics

•Symbolic Computing

•Science•Engineering

• Commercial software

MathematicaMapleMathCadMatlabReduceMacsymaScratchpadDerive

Basic Concepts of Mathematica

• Mathematica is the world's only fully integrated environment for formulation, calculationand visualization for technical computing.

• The concept of Mathematica is to create once and for all a single system that could handleall the various aspects of technical computing in a coherent and unified way.

2 Mathematica Tutorial.nb

• The concept of Mathematica is to create once and for all a single system that could handleall the various aspects of technical computing in a coherent and unified way.

• Manipulation of the very wide range of objects involved in technical computing using onlya fairly small number of basic primitives (over 3,000 built in the kernel of version 7).

• The functionality can be easily extended through user-defined functions and externalprograms.

• Platform-independent interactive documents known as notebooks (.nb files)

• System-provided macros and user-defined macros in packages (.m files)

• Interactive and non-interactive sessions

• Communication with external programs using MathLink

• Mathematica computing environment

TCP/IP

mle.exeMathLink

Standard Packages

MathematicaKernel

Windows/UnixPlatform

Windows/UnixPlatform

Add-On Packages

User Packages

Other Packages

User Frontend

Numerical Analysis

Control &Instrumentation

– External applications are written in C/C++. – Mathematica is used for both data input and output. – External applications and Mathematica communicate via MathLink. – The functionality can be extended through user-defined functions and external programs.

• More information available at http://www.wolfram.com

Mathematica Tutorial.nb 3

Algebraic Calculation

Ÿ Algebraic Calculations (Mathematica Tutorial)

Ÿ Symbolic Calculations (Mathematica Tutorial)

Ÿ Expand

Ÿ Factor

Ÿ Apart

Ÿ Together

Ÿ Simplify

Basic Algebra

Ÿ Polynomial Algebra (Mathematica Guide)

Ÿ Polynomial Systems (Mathematica Guide)

Elementary Functions

Ÿ Elementary Functions (Mathematica Guide)

Trigonometric Functions

Ÿ Trigonometric Functions (Mathematica Guide)

Ÿ Trigonometric Expressions (Mathematica Tutorial)

Complex Variables

Ÿ Complex Numbers (Mathematica Guide)

Ÿ Functions of Complex Variables (Mathematica Guide)

Flow Control

Ÿ Conditionals (Mathematica Guide)

Ÿ Loops and Control Structures (Mathematica Tutorial)

Ÿ Flow Control (Mathematica Guide)

Formula Manipulation


Formula Manipulation

Ÿ Formula Manipulation (Mathematica Guide)

Ÿ Manipulating Equations (Mathematica Guide)

User Package Functions

• Mathematica provides a solid foundation for symbolic computing and a number of support-ing functions. – Built-in kernel functions – User-defined functions to complement the built-in kernel functions

• A mechanism is needed to facilitate the algebraic manipulation of mathematical expressionswith the following properties: – Seamless integration with the computing environment of Mathematica – Selective targeting of the object to apply functions – Improved handling of subscripts (a1), tildes (ã), hats (â), etc. – On-line setting and clearing of attributes – Addition of comments – Improved handling of derivatives, integrals and summations – Minimal use of variables

• Algebraic manipulation of formulas using symbolic computing – Mathematica is not a word processor or an equation editor. – Application of functions – Substitution using mathematical identities – Allows focusing on the principles instead of time-consuming and error-prone calculations. – Good readability – Minimization of human errors during calculations – Expressions that closely resemble the traditional mathematical style, e.g., subscripts andvector notations

• A large collection of user-defined functions – Basic Algebra – Differentiation – Integration – Summation – Trigonometric Functions – Complex Variables – Vectors and Matrices – Polynomials and Series – Functional Analysis – Equations – Operator Analysis – Plotting – Etc.


• A large collection of user-defined functions – Basic Algebra – Differentiation – Integration – Summation – Trigonometric Functions – Complex Variables – Vectors and Matrices – Polynomials and Series – Functional Analysis – Equations – Operator Analysis – Plotting – Etc.

• MPMAF – An abbreviation of MPMapApplyFunc – The platform for algebraic manipulation of all or parts of an expression

• User-defined functions – The function names start with the prefix “MP”. – Algebraic manipulation of expressions – Mathematical identities – Can be used separately independent of the macro MPMAF, in which case the featuresprovided by the options of MPMAF cannot be used.

Basic Algebra

• Expansion

ExpandAHx + aL2Ea2 + 2 a x + x2

The power exponent must be an integer. Otherwise, use MPExpandBinomial.

MPExpandBinomial@Hx + aLn, kDMPEvalSum@%D

âk=0

n ak x-k+n n!k! H-k + nL!Ha + xLn


MPExpandBinomial@Ha + b + cLn, k, lDMPEvalSum@%DPowerExpand@%D

âk=0

n âl=0

-k+n ak bl c-k-l+n n!k! l! H-k - l + nL!

cnb + cc

n a + b + cb + c

n

Ha + b + cLn• Factoring

FactorAx2 + Ha + bL x + a bEHa + xL Hb + xLMPFactor

MPFactor@a + b, aDa 1 +

ba

• Reduction of fractions

x2x + a ®

Hx + aL2 - a2 - 2 a Hx + aL + 2 a2x + a

x2

a + x®

-a2 - 2 a x + Ha + xL2a + x

ApartB x2x + a , xF

-a + x +a2

a + x

MPApart

MPApartB sx2 exn2 e0 - ex

+sy2 ey

n2 e0 - ey+

sz2 ezn2 e0 - ez

, 9ex, ey, ez=F% �. -sx2 - sy2 - sz2 ® -1

-sx2 - sy2 - sz2 -n2 sx2 e0

-n2 e0 + ex-

n2 sy2 e0-n2 e0 + ey

-n2 sz2 e0

-n2 e0 + ez

-1 -n2 sx2 e0

-n2 e0 + ex-

n2 sy2 e0-n2 e0 + ey

-n2 sz2 e0

-n2 e0 + ez

• Separation of variables for solving differential equations


âyâx Š 2 ãx y

MPMAFB%, MPSepVars, 8All, x, Side ® Right<,MPDiffToInt, :All, x, y, Constant ®

12 Log@c1D>,

MPEvalInt, All,MPSolve, 8All, y<F

âyâx

Š 2 ãx y

Log@yD2

Š ãx +Log@c1D

2y Š ã2 ã

x c1

Direct solution using MPDSolveâyâx Š 2 ãx yMPMAF@%, MPDSolve, 8All, y, x, RCN ® c1<D

âyâx

Š 2 ãx y

y Š ã2 ãx c1

Trigonometric Functions

• Adding sin and cos functions

MPTrigAddSinCos@a Cos@xD + b Sin@xD, SinDa2 + b2 Sin@x + ArcTan@b, aDD

• Conversion to other functions

MPTrigConvert@Tan@xD, CosD1 - Cos@xD2 Sec@xD

• Conversion to half angle

MPTrigToHalf@Cos@2 xD, SinD1 - 2 Sin@xD2

• Conversion to double angle


MPTrigToDoubleACos@xD2, CosE12

H1 + Cos@2 xDLComplex Variables

• Asterisk (*) is interpreted as the complex conjugate

MPComplexExpandAHãu+vL*, u, TargetFunctions ® ConjugateEãv+u

*

The built-in function ComplexExpand gives a rather different (even though equivalent)result.

ComplexExpand@Conjugate@ãu+vD, u, TargetFunctions ® ConjugateDãv+

12 Hu+Conjugate@uDL CoshB 1

2Hu - Conjugate@uDLF - ãv+ 1

2 Hu+Conjugate@uDL SinhB 12

Hu - Conjugate@uDLF• The exponential form of a complex number

MPComplexToExp@x + ä yDãä ArcTanA y

x E x2 + y2

Polynomials and Series

• Eliminate the second highest order term

MPMergePolyAx3 + a x2 + c, xE-a3

27+ c -

a2 x3

+a3+ x

3

• Transform a polynomial by making a replacement of the variable

MPTransPolyAx3 + a x2 + c, x, a + xEc + a2 Ha + xL - 2 a Ha + xL2 + Ha + xL3

A case of two variables

MPTransPolyAx2 Hy + aL, 8x, y<, 8x - 1, y + 1<Ea + H-2 + 2 aL H-1 + xL + H-1 + aL H-1 + xL2 + y + 2 H-1 + xL H1 + yL + H-1 + xL2 H1 + yL

• Taylor series


MPTaylorB 1 + x , x, 3F1 +

x2-x2

8+x3

16

MPTaylorB 1 + Hx + DxL2 , Dx, 1F1 + x2 +

x Dx

1 + x2

MPAFE@y@xD, MPTaylor, 8All, x - x0, 2, Variables ® x<, CS ® x0, Head ® TildeTildeDy@xD » y@x0D + Hx - x0L y¢@x0D + 1

2Hx - x0L2 y¢¢@x0D

Equations

• Adding equations

MPAddEqs@8a Š b, c Š d, e Š f<Da + c + e Š b + d + f

• Subtracting equations

MPSubtEqs@8a Š b, c Š d, e Š f<Da - c - e Š b - d - f

• Move a subpart of an equation to left- or right-hand side

MPMoveEq@x + y Š z, y, RightD-x + z Š y

• Eliminating variables from a set of equations

MPEli@8x + y Š 1, x - y + 2 z Š 1, y - 3 z Š 3<, 8y, z<D �� Simplify

2 x Š 5

• Solve equations and write

MPSolve@8x + y Š 1, x - y + 2 z Š 1, y - 3 z Š 3<, 8x, y, z<D:x Š

52, y Š -

32, z Š -

32>


Differentiation

• The built-in derivative operator D, or ¶, evaluates the expression immediately.

D@f@xD, xDf¢@xD

The total derivative operator Dt assumes all symbols are dependent variables.

Dt@a f@xD, xDDt@a, xD f@xD + a f¢@xD

• The partial differential operator ¶¶x delays the evaluation until the command MPEvalD or

MPExpandD is encountered.

y Š¶3

¶x2 ¶yHa f + b gL

MPMAF@%, MPExpandD, 8At@2D, 8f, g<<Dy Š

¶3Ha f + b gL¶x2 ¶y

y Š a¶3f

¶x2 ¶y+ b

¶3g¶x2 ¶y

The function MPExpandD is used to expand an expression using the differential operators.

MPExpandDB ¶

¶xf + Sin@gD

h , 8f, g, h<F1h

¶f¶x

+ Cos@gD ¶g¶x

-f + Sin@gD

h2¶h¶x

• Integration by parts using the differential operator

MPIntegrateByPartsDBHFžà f ¶g¶x âxF

f g - à g ¶f¶x

âx

• Merging the differential operator

MPMergeDBa ¶f¶x + b ¶g

¶x F¶Ha f + b gL

¶x

• Transformation of variables


• Transformation of variables

MPTransDB ¶2f¶w2

, VT ® :w, l, w Š2 p cl

>Fl3

2 c2 p2¶f

¶l+

l4

4 c2 p2¶2f

¶l2

• Evaluation of vector operators

MPAFEAÑ2y, MPEvalVecOps, 8All, y Š y@r, jD, Cylindrical@r, j, zD<, Apply ® ExpandEMPMAFB%, MPSimpD, : 1

r

¶y

¶r+¶2y

¶r2, Mult ® r>F

Ñ2y Š1r

¶y

¶r+¶2y

¶r2+

1r2

¶2y

¶j2

Ñ2y Š1r2

¶2y

¶j2+1r

¶

¶rr¶y

¶r

MPAFEAÑ2y, MPEvalVecOps, 8All, y Š y@x, y, zD, Cartesian@x, y, zD<EMPMAFB%, MPTransD, 8At@2D,

VT ® 88x, y, z<, 8r, q, j<, 8x Š r Sin@qD Cos@jD, y Š r Sin@qD Sin@jD, z Š r Cos@qD<<,Apply ® HExpandžPowerExpand žSimplifyžð &L<,

MPSimpD, :: 2r ¶y

¶r +¶2y

¶r2>, : Cot@qD

r2¶y

¶q+

1r2

¶2y

¶q2>>F

Ñ2y Š¶2y

¶x2+¶2y

¶y2+¶2y

¶z2

Ñ2y Š2r

¶y

¶r+Cot@qD

r2¶y

¶q+¶2y

¶r2+

1r2

¶2y

¶q2+Csc@qD2

r2¶2y

¶j2

Ñ2y ŠCsc@qD2

r2¶2y

¶j2+1r

¶2Hr yL¶r2

+Csc@qD

r2¶

¶qSin@qD ¶y

¶q

Integration

• Wrap the integral with HoldForm (HF) to prevent evaluation.

– Without the wrap

F Š à f@xD âxF Š à f@xD âx

– With the wrap


F Š HFžà f âx

F Š à f âx• Combining multiple integrals

KHFžà f@xD âxO KHFžà g@xD âxOMPCombInt@%, RV ® 8x, x¢<D

Kà f@xD âxO à g@xD âxà à f@xD g@x¢D âx¢ âx

• Convolution in Fourier transform

:f@xD Š HFžà F@kD ãä k x âk, g@xD Š HFžà G@kD ãä k x âk>:f@xD Š à F@kD ãä k x âk, g@xD Š à G@kD ãä k x âk>HFžà F@kD G@kD ãä k x âkMPMAFB%, RA, :All, :F@kD Š

12 p à f@xD ã-ä k x âx, G@kD Š

12 p à g@xD ã-ä k x âx>>,

MPCombInt, 8All, RV ® 8x, x¢, x¢¢<<,MPFourierToDiracDelta, 8All, k<,MPEvalIntDelta, 8All, x¢¢<F

à F@kD G@kD ãä k x âk12 p à Kà f@x¢¢D g@x¢D d@x - x¢ - x¢¢D âx¢O âx¢¢

12 p à f@x - x¢D g@x¢D âx¢

• Transformation of the integration variables

MPTransIntBà f âx , VT ® 9x, y, y Š x2=Fà f2 y

ây

MPAFEBHFžà-¥

¥ 1x2 + a2

âx, MPEvalInt, Apply ® PowerExpandFà-¥

¥ 1x2 + a2

âx Šp

a


MPAFEBHFžà-¥

¥ 1x2 + a2

âx, MPTransInt,8All, VT ® 8x, q, x Š a Tan@qD<<, RA ® Sign@aD Š 1, PostApply ® MPEvalIntFà-¥

¥ 1x2 + a2

âx Šp

a

HFžIntegrate@f, x, y, zDMPMAF@%, MPTransInt, 8All,

VT ® 88x, y, z<, 8r, q, f<, 8x Š r Sin@qD Cos@fD, y Š r Sin@qD Sin@fD, z Š r Cos@qD<<<,Apply ® PowerExpandD

à à à f âz ây âxà à à f r2 Sin@qD âf âq âr

• Replacement of the integration variable with others

MPTransIntBà f âx3 , Weight ® r2 Sin@qD, RV ® 9x3, 8r, 0, ¥<, 8q, 0, p<, 8f, 0, 2 p<=Fà0

¥à0

pà0

2 pf r2 Sin@qD âf âq âr

• Merging multiple integrals

MPMergeIntBa à f âx + b à g âx + c à h âxFà Ha f + b g + c hL âx

• Merging integration intervals

MPMergeIntIntervalBà0

af@tD ât + à

a

bf@tD âtF

à0

bf@tD ât

• Integration by parts

MPIntegrateByPartsBHFžà-¥

xy¢@yD y2 ây, y¢@yDF

-à-¥

x2 y y@yD ây - Iy2 y@yDMyŠ-¥ + x2 y@xD

Summation

• Changing the summation intervals


MPChSumIntervalBHFž ân=-¥

¥

an, 8n, 81, 2<, 8n3, n4<<Fân=1

2an + â

n=n3

n4an

• Combining multiple summations. The indexes are replaced to avoid duplication.

MPCombSumBân=1

¥

fn ân=1

¥

gn, RV ® 8n, m<Fân=1

¥ âm=1

¥

fn gm

• Merging multiple summations

MPMergeSumBâk=0

¥

A@kD +âk=0

¥

B@kDFâk=0

¥ HA@kD + B@kDL• Merging summation intervals

MPMergeSumIntervalBân=0

k-1an + â

n=k

NanF

ân=0

Nan

Vectors and Matrices

• Expansion of vector expressions

MPExpandVecBA ‰Ja BÓ‰CÓNFa AÓ.CÓ BÓ - a AÓ.BÓ CÓMPExpandVecAÑ × Ie EÓM, 9e, E

Ó=Ee Ñ × E

Ó+ HÑeL.EÓ

• Matrix form of equations

MPEqsMatForm@8a x + b y Š z1, c x + d y Š z2<, 8x, y<DK a bc d O × K x

y O Š J z1z2 N

• Expansion of Ñ ´Ñ ´ÑA


• Expansion of Ñ ´Ñ ´ÑA

MPExpandVecAÑ ‰Ñ ‰A, AEÑIÑ × AÓM - Ñ2AÓ

• Deleting a column from a matrix

Table@i j, 8i, 3<, 8j, 3<D �� MatrixFormMPDeleteColumns@%, 2D �� MatrixForm

1 2 32 4 63 6 91 32 63 9

• Inserting a column in a matrix

Table@i j, 8i, 3<, 8j, 3<D �� MatrixFormMPInsertColumns@%, 80, 0, 0<, -1D �� MatrixForm

1 2 32 4 63 6 91 2 3 02 4 6 03 6 9 0

• Diagonalization of a matrix

1 4 94 4 09 0 4

;

MPDiagMatrix@%D �� MatrixForm

12 J5 + 397 N 0 0

0 12 J5 - 397 N 0

0 0 4

• Transform a vector from Cartesian to spherical coordinates

MPVecFromCart@80, 0, 1<, Spherical@r, q, jDD8Cos@qD, -Sin@qD, 0<The result is z= r cosq -q

`sinq.

• Transform a vector from spherical to Cartesian coordinates


MPVecToCart@80, 0, 1<, Spherical@r, q, jDD8-Sin@jD, Cos@jD, 0<The result is j= - x sinj + y cosj.

Functional Analysis

• Variable transformation

y¢@xD + y@xDMPTransEqB%, y@xD Š g@sD, VT ® :x, s, x Š s2, s Š x >F

y@xD + y¢@xDg@sD + g¢@sD

2 s

A case of two variables

¶x,xf@x, yD - ¶y,yf@x, yDMPTransEqB%, f@x, yD Š g@x, hD,VT ® :8x, y<, 8x, h<, :x Š

x + y2 , h Š

x - y2 >>, Apply ® SimplifyF

-fH0,2L@x, yD + fH2,0L@x, yDgH1,1L@x, hD

• Abbreviation of derivatives

MPAbbrevFAä pH0,1L@x, tD - 2 a pH1,0L@x, tD qH1,0L@x, tD, Deriv ® TrueEä pt - 2 a px qx

The subscripts denote the derivatives. The function arguments can be restored using MPRe-storeFunctions.

MPRestoreF@ä pt - 2 a px qx, 8p, q<, Deriv ® TrueD-2 a pH0,1L@t, xD qH0,1L@t, xD + ä pH1,0L@t, xD

• Taking a factor out of a function

MPFactorFunc@f@a xD + Ñ ‰Ha AL, CurlH, aDa Ñ ‰A + f@a xD

• Putting a factor inside a function


MPInsideFunc@f@a xD + a Ñ ‰A, CurlH, aDÑ ‰Ha AL + f@a xD

• Merging arguments of a linear function

MPMergeFunc@a f@x, zD + b f@y, zD, fDf@a x + b y, zD

Operator Analysis

• Expansion of operator expressions

MPExpandBraket@HXa + Xb LëAëH a\ + b\L, ADXa A a\ + Xa A b\ + Xb A a\ + Xb A b\MPCommutatorB Q

l0 2+ ä

l0

Ñ 2P, Q

l0 2- ä

l0

Ñ 2PF

MPExpandOp@%, 8Q, P<D-

Q

2 l0-ä P l02 Ñ

ëQ

2 l0+ä P l02 Ñ

+Q

2 l0+ä P l02 Ñ

ëQ

2 l0-ä P l02 Ñ

ä

ÑPëQ -

ä

ÑQëP

• Simplify a commutator using a commutation rule @Q, PD º QP - PQ= ä Ñ

MPCommutatorAQ2, P2EMPSimpOp@%, 8Q, P, ä Ñ<D

-P2ëQ2 + Q2ëP2

2 Ñ2 + 4 ä Ñ QëP

Examples

– Ref. Chap 9, Arfken, Mathematical Methods for Physicists

Ÿ Parachutist

• The equation of motion is

m v Š m g - b v2

where -b v2 is the drag force and m g is the force of the gravitational attraction. The initialvelocity when the parachute opens at time t = 0 is vH0L = vi.• The terminal velocity, v0, can be found from the equation of motion as t ®¥; when there isno acceleration, v = 0, so


• The terminal velocity, v0, can be found from the equation of motion as t ®¥; when there isno acceleration, v = 0, so

m v Š m g - b v2MPMAFA%, RA, 9All, 9v Š 0, v Š v0==,MPSolve, 8All, v0, -1<, MergeP ® TrueE

m v Š g m - b v2

0 Š g m - b v02

v0 Šg mb

• The variables t and v separate.

m v Š m g - b v2

MPMAFB%, RA, :At@1D, v Šâvât >,

MPSepVars, 8All, t, Times, Side ® Right<,MPDivDenom, 8At@1D, m<F

m v Š g m - b v2

m âvg m - b v2

Š ât

âv

g - b v2m

Š ât

which can be integrated to yield the velocity at time t.

âvg - b v2

m

Š ât

MPMAFB%, MPDiffToInt, 8All, 8v, vi, v<, t<, ,MPEvalInt, All, , ,MPSolve, 8All, v<, Hold ® vi, MergeP ® True, ,TrigExpand, At@2D, Apply ® Simplify,

MPDivFrac, :At@2D, bg m CoshB b g

m tF>, MergeP ® True, ,

MPEli, :At@2D, v0 Šg mb , b>, Apply ® PowerExpand, , ,

MPEli, :At@2D, v0g Š T, g>F

âv

g - b v2m

Š ât


àvi

v 1

g - b v2m

âv Š à 1 âtv Š

g mb

TanhB b gm

t + ArcTanhB bg m

viFFv Š

g mb

vi + g mb TanhB b g

m tFg mb + vi TanhB b g

m tFv Š

v0 Ivi + v0 TanhA tT EM

v0 + vi TanhA tT E

where

:v0 Š g mb , T Š

v0g >

• Verify that our solution satisfies the originial equation of motion:

MPAFEBv , MPExpandD, :All, v Šv0 Ivi + v0 TanhA t

T EMv0 + vi TanhA t

T E , OverDot ® t>, Apply ® SimplifyFMPMAFB%, MPTrigConvert, 8At@2D, Tanh<,MPEli, :At@2D, v Š

v0 Ivi + v0 TanhA tT EM

v0 + vi TanhA tT E , TanhBtT F>, Apply ® Simplify,

RR, :At@2D, :v0 Š g mb , T Š

v0g >>, Apply ® ExpandF

v Šv0 Iv02 - vi2M

T Iv0 CoshA tT E + vi SinhA t

T EM2v Š

-v2 + v02

T v0

v Š g -b v2

m

that is, Newton's equation of motion.

• Substituting the numerical data, v0 and T are


:v0 Š g mb , T Š

v0g >

MPMAFB%, MPEli, 8All, v0, 1, Keep ® True<, ExpandMergeP ® True,

RA, :All, :g Š 9.8 MeterSecond2

, b Š 700 Kilo GramMeter , m Š 70 Kilo Gram>>, Apply ® PowerExpand,

Convert, :At@1, 2D, MileHour >F

:v0 Š g mb

, T Šv0g

>:v0 Š 0.989949 Meter

Second, T Š 0.101015 Second>

:v0 Š 2.21445 MileHour

, T Š 0.101015 Second>Plot vHtL.


v Šg mb

vi + g mb TanhB b g


m tFMPMAFB%, RA, :At@2D, :g Š 9.8 Meter

Second2, b Š 700 Kilo Gram

Meter ,

m Š 70 Kilo Gram, vi Š 60 MileHour , t Š t Second>>, Apply ® PowerExpand,

MPDivFrac, :At@2D, MileHour >,

Convert, :: Hour MeterMile Second , 1>, :At@2D, Mile

Hour >>,MPDivEq, :All, Mile

Hour >,Plot, :At@2D, 8t, 0, 0.1<, PlotRange ® 80, 60<,AxesLabel ® :t@SecondD, vB MileHour F>, ImageSize ® 500>, Take ® LastF

v Šg mb

vi + g mb TanhB b g


m tFHour vMile

Š2.21445 H60 + 2.21445 [email protected] tDL

2.21445 + 60 [email protected] tD

0.00 0.02 0.04 0.06 0.08 0.10tHSecondL

10

20

30

40

50

60

vMile

Hour


Ÿ Series Solution of Differential Equations

• We apply the method of series soution to the linear (classical) oscillator equation.(9.84)

â2yâx2

+ w2 y Š 0

with known solutions y = sinHw xL, cosHw xL.– We try the series solution of the form(9.85)

y@xD Š xk Ia0 + a1 x + a2 x2 + a3 x3 +ºM Š HFžâl=0

¥

al xk+l

y@xD Š xk Iº + a0 + x a1 + x2 a2 + x3 a3M Š âl=0

¥

al xk+l

with a0 ¹ 0 and the exponent k and all the coefficients al still undetermined.

– By substituting (9.85) into Eq. (9.84), we have(9.86)

â2yâx2

+ w2 y Š 0

MPMAFB%, RA, :At@1D, y Š âl=0

¥

al xk+l>, Apply ® MPEvalDFy w2 +

â2yâx2

Š 0

w2 âl=0

¥

xk+l al + âl=0

¥

x-2+k+l H-1 + k + lL Hk + lL al Š 0

which gives


w2 âl=0

¥

xk+l al +âl=0

¥


MPMAFA%, MPShiftSum, 9AtA1, x-2+k+lE, 8l, 2<=,MPChSumLimits, 8At@1D, 8l, 0, ¥<<,MPMergeSum, At@1D,MPFactor, 9At@1D, xk+l=, Base ® _SumE

w2 âl=0

¥

xk+l al + âl=0

¥


âl=0

¥ Ixk+l w2 al + xk+l H1 + k + lL H2 + k + lL a2+lM + H-1 + kL k x-2+k a0 + k H1 + kL x-1+k a1 Š 0

âl=0

¥

xk+l Iw2 al + H1 + k + lL H2 + k + lL a2+lM + H-1 + kL k x-2+k a0 + k H1 + kL x-1+k a1 Š 0

– From the lowest power of x, xk-2, vanishing of the coefficient yieldsH-1 + kL k a0 Š 0

Since a0 was chosen to be nonzero, we have(9.87)

H-1 + kL k Š 0

This equation is called the indicial equation and we have k = 0 or k = 1.

– Since the coefficient of xk+ jH j ³ 0L vanishes, this results in(9.88)

w2 al + H1 + k + lL H2 + k + lL a2+l Š 0MPMAFA%, RA, 8At@1D, l Š j<,MPSolve, 9All, a2+j=E

w2 al + H1 + k + lL H2 + k + lL a2+l Š 0

w2 aj + H1 + j + kL H2 + j + kL a2+j Š 0

a2+j Š -w2 ajH1 + j + kL H2 + j + kL

• Depending on whether k = 0 or k = 1, we have different recurrence relations. First withk = 0, the recurrence relation (Eq. (9.88)) becomes(9.89)


a2+j Š -w2 ajH1 + j + kL H2 + j + kL

MPMAF@%, RA, 8At@2D, k Š 0<Da2+j Š -

w2 ajH1 + j + kL H2 + j + kLa2+j Š -

w2 ajH1 + jL H2 + jLwhose solution is(9.90)

a2+j Š -w2 ajH1 + jL H2 + jL

MPMAFA%, MPRSolve, 9All, aj, 8j, 0<=,RA, 8All, j Š 2 n<, SimpOne ® True, Apply ® PowerExpand,MPToFactorial, At@2DE

a2+j Š -w2 ajH1 + jL H2 + jL

a2 n ŠH-1Ln w2 n a0Gamma@1 + 2 nDa2 n Š

H-1Ln w2 n a0H2 nL!and our solution of the differential equation is(9.91)

y Š âl=0

¥

al xk+l

MPMAFB%, RA, 8At@2D, k Š 0<,MPChSumInterval, 8At@2D, 8l, 8n, 0, ¥, l Š 2 n<<<,RA, :At@2D, a2 n Š

H-1Ln w2 n a0H2 nL! >, Apply ® MPFactorSum,

MPEvalSum, At@2DFy Š â

l=0

¥

al xk+l

y Š a0 ân=0

¥ H-1Ln x2 n w2 nH2 nL!y Š Cos@x wD a0

• Similarly, if we choose the indicial equation root k = 1 (Eq. (9.88)), the solution becomes(9.94)


y ŠSin@x wD a0

w

• From (9.91) and (9.94), the general solution can be put

y Š c1 Cos@x wD + c2 Sin@x wD


Mathematica Language

Ÿ Language Overview (Mathematica Guide)

Structure of Mathematica Expressions

Ÿ Expressions (Mathematica Tutorial)

Constants

Ÿ Numbers (Mathematica Overview)

Ÿ Types of Numbers (Mathematica Tutorial)

Ÿ Integer (Built-in Mathematica Symbol)

Ÿ Rational (Built-in Mathematica Symbol)

Ÿ Real (Built-in Mathematica Symbol)

Ÿ Complex (Built-in Mathematica Symbol)

Ÿ Mathematical Constants (Mathematica Guide)

Variables

Ÿ Defining Variables (Mathematica Tutorial)

Ÿ Eliminating Variables (Mathematica Tutorial)

Patterns

Ÿ Patterns (Mathematica Guide)

Ÿ Introduction to Patterns

Ÿ Verbatim Patterns

Ÿ Example

• In four-wave mixing, two waves of frequencies w1 and w2 interact and generate additionalfrequencies through nonlinear mixing. Suppose the two waves are given by


Ei Š12 IEŽi ã-ä wi t + EŽi* ãä wi tM

for i = 1, 2, and the nonlinear process is due to the third-order Kerr effect:

PNL Š e0 c3 E3

Find out what frequencies are present in the output wave near w1 » w2.

• The field is given by

E Š HFžâi=1

2Ei

MPMAFB%, RA, :At@2D, Ei Š12 IEŽi ã-ä wi t + EŽi* ãä wi tM>,

MPEvalSum, At@2DFE Š â

i=1

2Ei

E Š12

âi=1

2 Iã-ä t wi EŽi + ãä t wi HEŽiL*ME Š

12

Iã-ä t w1 EŽ1 + ã-ä t w2 EŽ2 + ãä t w1 HEŽ1L* + ãä t w2 HEŽ2L*Mand the nonlinear mixing produces

PNL Š e0 c3 E3

MPMAFB%, RA, :At@2D, E Š12 Iã-ä t w1 EŽ1 + ã-ä t w2 EŽ2 + ãä t w1 HEŽ1L* + ãä t w2 HEŽ2L*M>,

Expand, At@2, ED, FactorExp ® ä tFPNL Š E3 e0 c3

PNL Š18e0 c3 Iã-ä t w1 EŽ1 + ã-ä t w2 EŽ2 + ãä t w1 HEŽ1L* + ãä t w2 HEŽ2L*M3

PNL Š18e0 c3 Jã-3 ä t w1 EŽ13 + 3 ãä t H-2 w1-w2L EŽ12 EŽ2 + 3 ãä t H-w1-2 w2L EŽ1 EŽ22 + ã-3 ä t w2 EŽ23 + 3 ã-ä t w1 EŽ12 HEŽ1L* +

6 ã-ä t w2 EŽ1 EŽ2 HEŽ1L* + 3 ãä t Hw1-2 w2L EŽ22 HEŽ1L* + 3 ãä t w1 EŽ1 IHEŽ1L*M2 + 3 ãä t H2 w1-w2L EŽ2 IHEŽ1L*M2 +ã3 ä t w1 IHEŽ1L*M3 + 3 ãä t H-2 w1+w2L EŽ12 HEŽ2L* + 6 ã-ä t w1 EŽ1 EŽ2 HEŽ2L* + 3 ã-ä t w2 EŽ22 HEŽ2L* +6 ãä t w2 EŽ1 HEŽ1L* HEŽ2L* + 6 ãä t w1 EŽ2 HEŽ1L* HEŽ2L* + 3 ãä t H2 w1+w2L IHEŽ1L*M2 HEŽ2L* +3 ãä t H-w1+2 w2L EŽ1 IHEŽ2L*M2 + 3 ãä t w2 EŽ2 IHEŽ2L*M2 + 3 ãä t Hw1+2 w2L HEŽ1L* IHEŽ2L*M2 + ã3 ä t w2 IHEŽ2L*M3N

The frequencies are


PNL Š18 e0 c3 Jã-3 ä t w1 EŽ13 + 3 ãä t H-2 w1-w2L EŽ12 EŽ2 + 3 ãä t H-w1-2 w2L EŽ1 EŽ22 + ã-3 ä t w2 EŽ23 + 3 ã-ä t w1 EŽ12 HEŽ1L* +6 ã-ä t w2 EŽ1 EŽ2 HEŽ1L* + 3 ãä t Hw1-2 w2L EŽ22 HEŽ1L* + 3 ãä t w1 EŽ1 IHEŽ1L*M2 + 3 ãä t H2 w1-w2L EŽ2 IHEŽ1L*M2 +ã3 ä t w1 IHEŽ1L*M3 + 3 ãä t H-2 w1+w2L EŽ12 HEŽ2L* + 6 ã-ä t w1 EŽ1 EŽ2 HEŽ2L* + 3 ã-ä t w2 EŽ22 HEŽ2L* +6 ãä t w2 EŽ1 HEŽ1L* HEŽ2L* + 6 ãä t w1 EŽ2 HEŽ1L* HEŽ2L* + 3 ãä t H2 w1+w2L IHEŽ1L*M2 HEŽ2L* +3 ãä t H-w1+2 w2L EŽ1 IHEŽ2L*M2 + 3 ãä t w2 EŽ2 IHEŽ2L*M2 + 3 ãä t Hw1+2 w2L HEŽ1L* IHEŽ2L*M2 + ã3 ä t w2 IHEŽ2L*M3N

MPMAFA%, frequencies Š Cases@ð, ãa_, 80, ¥<D &, At@2D, Take ® Last,Part, 8At@2D, 82<<, Level ® 1,MPDiv, 8At@2D, ä t<,Cases, 9At@2D, wi_ 2 wi_ - wj_=, Apply ® UnionE

PNL Š18e0 c3 Jã-3 ä t w1 EŽ13 + 3 ãä t H-2 w1-w2L EŽ12 EŽ2 + 3 ãä t H-w1-2 w2L EŽ1 EŽ22 + ã-3 ä t w2 EŽ23 + 3 ã-ä t w1 EŽ12 HEŽ1L* +

6 ã-ä t w2 EŽ1 EŽ2 HEŽ1L* + 3 ãä t Hw1-2 w2L EŽ22 HEŽ1L* + 3 ãä t w1 EŽ1 IHEŽ1L*M2 + 3 ãä t H2 w1-w2L EŽ2 IHEŽ1L*M2 +ã3 ä t w1 IHEŽ1L*M3 + 3 ãä t H-2 w1+w2L EŽ12 HEŽ2L* + 6 ã-ä t w1 EŽ1 EŽ2 HEŽ2L* + 3 ã-ä t w2 EŽ22 HEŽ2L* +6 ãä t w2 EŽ1 HEŽ1L* HEŽ2L* + 6 ãä t w1 EŽ2 HEŽ1L* HEŽ2L* + 3 ãä t H2 w1+w2L IHEŽ1L*M2 HEŽ2L* +3 ãä t H-w1+2 w2L EŽ1 IHEŽ2L*M2 + 3 ãä t w2 EŽ2 IHEŽ2L*M2 + 3 ãä t Hw1+2 w2L HEŽ1L* IHEŽ2L*M2 + ã3 ä t w2 IHEŽ2L*M3N

frequencies Š 8-3 w1, -2 w1 - w2, -w1 - 2 w2, -3 w2, -w1, -w2, w1 - 2 w2, w1,2 w1 - w2, 3 w1, -2 w1 + w2, -w1, -w2, w2, w1, 2 w1 + w2, -w1 + 2 w2, w2, w1 + 2 w2, 3 w2<

frequencies Š 8w1, 2 w1 - w2, w2, -w1 + 2 w2<Four frequencies are present in the output wave and they are in the order

frequencies Š 82 w1 - w2, w1, w2, -w1 + 2 w2<if w1 < w2.

Functions

Ÿ Mathematical Functions (Mathematica Guide)

Programming

Ÿ Procedural Programming (Mathematica Guide)

Ÿ Defining Functions (Mathematica Tutorial)

• Functions can be called recursively.

f@n_D := n f@n - 1D; f@0D = 1;


f@100D93 326 215 443 944 152 681 699 238 856 266 700 490 715 968 264 381 621 468 592 963 895 217 599 993 229 „915 608 941 463 976 156 518 286 253 697 920 827 223 758 251 185 210 916 864 000 000 000 000 000 000 „000 000

100!

93 326 215 443 944 152 681 699 238 856 266 700 490 715 968 264 381 621 468 592 963 895 217 599 993 229 „915 608 941 463 976 156 518 286 253 697 920 827 223 758 251 185 210 916 864 000 000 000 000 000 000 „000 000

Clear@fDŸ Functional Programming (Mathematica Guide)

Ÿ Pure Functions (Mathematica Tutorial)

Ÿ Functional Operations (Mathematica Overview)

Ÿ Example: data manipulation

– Generate the data

data = RandomReal@8-10, 10<, 100D89.2069, 4.38196, 7.36707, 2.44153, -2.71293, -7.84716, -2.51371, -7.20865, -0.35902,0.547298, -5.73894, 6.72848, 0.891682, 6.47749, 2.32973, -1.13296, -9.28741, 2.6097,-3.3001, 1.31403, 3.28091, -6.56867, -4.45742, -3.50691, -5.22019, -7.31909,8.72281, 2.20909, -1.75738, 8.50959, 4.91354, 7.45721, 5.48025, -1.42342, 9.46615,4.72104, 7.33425, -1.08151, 5.55214, 2.43006, 8.78305, -8.43983, -1.63389,-5.64446, -6.33726, 6.14862, -5.05503, 7.83732, 9.89825, -6.70792, -1.25488,5.78948, -6.64355, -4.31986, 5.54391, 5.09748, 4.97521, -7.33067, -5.96476,0.273494, 4.95501, -9.22198, -4.24909, -6.01423, 1.07997, -0.97207, -0.425487,1.10524, 4.61603, -1.08979, -4.19288, -4.14596, 0.206045, -3.44104, -5.51746,-3.39021, 4.92202, -8.64063, -2.70173, 2.66111, 7.56427, 3.89614, 4.80096,-7.25148, -9.66792, 2.19267, -8.06609, 2.25114, 5.52372, 9.20761, -7.88423,4.23404, -6.21864, 6.96702, 5.57343, 7.28515, 9.36244, -4.81503, 6.72125, -0.399189<

– Number of data

Length@dataD100

– Addition of all data

Plus žž data

34.7723

– Select positive numbers


Select@data, NegativeD8-2.71293, -7.84716, -2.51371, -7.20865, -0.35902, -5.73894, -1.13296, -9.28741,-3.3001, -6.56867, -4.45742, -3.50691, -5.22019, -7.31909, -1.75738, -1.42342,-1.08151, -8.43983, -1.63389, -5.64446, -6.33726, -5.05503, -6.70792, -1.25488,-6.64355, -4.31986, -7.33067, -5.96476, -9.22198, -4.24909, -6.01423, -0.97207,-0.425487, -1.08979, -4.19288, -4.14596, -3.44104, -5.51746, -3.39021, -8.64063,-2.70173, -7.25148, -9.66792, -8.06609, -7.88423, -6.21864, -4.81503, -0.399189<Cases@data, _?NegativeD8-2.71293, -7.84716, -2.51371, -7.20865, -0.35902, -5.73894, -1.13296, -9.28741,-3.3001, -6.56867, -4.45742, -3.50691, -5.22019, -7.31909, -1.75738, -1.42342,-1.08151, -8.43983, -1.63389, -5.64446, -6.33726, -5.05503, -6.70792, -1.25488,-6.64355, -4.31986, -7.33067, -5.96476, -9.22198, -4.24909, -6.01423, -0.97207,-0.425487, -1.08979, -4.19288, -4.14596, -3.44104, -5.51746, -3.39021, -8.64063,-2.70173, -7.25148, -9.66792, -8.06609, -7.88423, -6.21864, -4.81503, -0.399189<

– Select data larger than 5

Select@data, ð > 5 &D87.97756, 7.06071, 6.25687, 7.77989, 8.95827, 9.383, 6.92007, 6.09351,9.35384, 8.79465, 6.81701, 9.68147, 5.23082, 6.14421, 8.4387, 6.12878,7.46329, 6.51098, 9.32175, 8.12827, 9.10668, 9.73478, 8.44398, 6.23945<Cases@data, _?Hð > 5 &LD87.97756, 7.06071, 6.25687, 7.77989, 8.95827, 9.383, 6.92007, 6.09351,9.35384, 8.79465, 6.81701, 9.68147, 5.23082, 6.14421, 8.4387, 6.12878,7.46329, 6.51098, 9.32175, 8.12827, 9.10668, 9.73478, 8.44398, 6.23945<

– Group two numbers and add the numbers in each group.

Partition@ð, 2D &ždataPlus žžž %889.2069, 4.38196<, 87.36707, 2.44153<, 8-2.71293, -7.84716<, 8-2.51371, -7.20865<,8-0.35902, 0.547298<, 8-5.73894, 6.72848<, 80.891682, 6.47749<, 82.32973, -1.13296<,8-9.28741, 2.6097<, 8-3.3001, 1.31403<, 83.28091, -6.56867<, 8-4.45742, -3.50691<,8-5.22019, -7.31909<, 88.72281, 2.20909<, 8-1.75738, 8.50959<, 84.91354, 7.45721<,85.48025, -1.42342<, 89.46615, 4.72104<, 87.33425, -1.08151<, 85.55214, 2.43006<,88.78305, -8.43983<, 8-1.63389, -5.64446<, 8-6.33726, 6.14862<,8-5.05503, 7.83732<, 89.89825, -6.70792<, 8-1.25488, 5.78948<, 8-6.64355, -4.31986<,85.54391, 5.09748<, 84.97521, -7.33067<, 8-5.96476, 0.273494<, 84.95501, -9.22198<,8-4.24909, -6.01423<, 81.07997, -0.97207<, 8-0.425487, 1.10524<, 84.61603, -1.08979<,8-4.19288, -4.14596<, 80.206045, -3.44104<, 8-5.51746, -3.39021<,84.92202, -8.64063<, 8-2.70173, 2.66111<, 87.56427, 3.89614<, 84.80096, -7.25148<,8-9.66792, 2.19267<, 8-8.06609, 2.25114<, 85.52372, 9.20761<, 8-7.88423, 4.23404<,8-6.21864, 6.96702<, 85.57343, 7.28515<, 89.36244, -4.81503<, 86.72125, -0.399189<<813.5889, 9.8086, -10.5601, -9.72236, 0.188278, 0.989546, 7.36917, 1.19678, -6.67771,-1.98607, -3.28776, -7.96432, -12.5393, 10.9319, 6.75221, 12.3708, 4.05682,14.1872, 6.25274, 7.9822, 0.343213, -7.27836, -0.188635, 2.78229, 3.19033, 4.53461,-10.9634, 10.6414, -2.35546, -5.69127, -4.26698, -10.2633, 0.107901, 0.679749,3.52624, -8.33884, -3.235, -8.90766, -3.7186, -0.040626, 11.4604, -2.45052,-7.47525, -5.81495, 14.7313, -3.65019, 0.748378, 12.8586, 4.54741, 6.32206<

Packages


Packages

Ÿ Mathematica Packages (Mathematica Tutorial)

Ÿ Package Development (Mathematica Guide)

Ÿ A sample package

Sample.m

<< Sample`

Context@AddTwoDSample`

AddTwo@a, bDAddTwo@a, bDRemove@Sample`AddTwoDRemove@Global`AddTwoD

Communication with External Programs

Ÿ Introduction to MathLink (Mathematica Tutorial)

Ÿ MathLink C API Developer Guide (Windows)

Ÿ MathLink API (Mathematica Guide)

Ÿ Example (addtwo.exe)

Ÿ addtwo.c

#include "mathlink.h"extern int addtwo( int i, int j);int addtwo( int i, int j){

return i+j;}


Ÿ addtwo.tm

int addtwo P(( int, int));:Begin::Function: addtwo:Pattern: AddTwo[i_Integer, j_Integer]:Arguments: { i, j }:ArgumentTypes: { Integer, Integer }:ReturnType: Integer:End::Evaluate: AddTwo::usage = "AddTwo[x, y] gives the sum of two machine integers x and y."

Ÿ Compile and link

SET CL=/nologo /c /DWIN32 /D_WINDOWS /W3 /O2 /DNDEBUG /I\IncludeSET LINK=/NOLOGO /SUBSYSTEM:windows /INCREMENTAL:no kernel32.libuser32.lib gdi32.lib /LIBPATH:\LibMPREP addtwo.tm -o addtwotm.cCL addtwo.c addtwotm.cLINK addtwo.obj addtwotm.obj ml32i2m.lib /OUT:addtwo.exe

Ÿ Install and run

Install

mlelink = Install@"I:�Mathematica�MathLink�MathLinkExamples�addtwo�addtwo.exe"D;The two arguments must be integers.

AddTwo@1, 2D3

[email protected], 2.0DAddTwo@1., 2.D

Uninstall.

Uninstall@mlelinkD"I:\Mathematica\MathLink\MathLinkExamples\addtwo\addtwo.exe"


Ÿ Example (Modeling of Optical Fiber Modes)

Ÿ Initialization

• Open the MathLink connection.

– Windows

mlelink = Install@"mle.exe"D;– Unix

mlelink = Install@LinkConnect@"aaa"DD;– Uninstall

[email protected]

• Set the default directory.

SetDirectory@"D:�Temp"DD:\Temp

Directory@DD:\Temp

• Turn off Sellmeier formula by default.

DspnSetSellmeier@FalseD;Ÿ Refractive Index Profiles

• Step index

The data format for the step index profile is{Step, ncore, {acore-clad, nclad}}or{Step, Dcore, {acore-clad, Dclad}}.D is given in percent (%).


profile = 8Step, 1, 84, 0<<[email protected], profile, Scale ® Delta,AxesLabel ® 8r@mmD, "D@%D"<, ImageSize ® 500D8Step, 1, 84, 0<<

2 4 6 8rHmmL

0.2

0.4

0.6

0.8

1.0

D@%D

• a-index

The data format for the a index profile is {Alpha, a, {{0, ncore}, {acore, n1}, {r2, n2}, ... , {rN , nN}, {acore-clad, nclad}}or{Alpha, a, {{0, Dcore}, {acore, D1}, {r2, D2}, ... , {rN , DN}, {acore-clad, Dclad}}.


profile = 8Alpha, 2, 880, 1<, 84, -0.5<, 86, -0.5<, 88, 0<<<[email protected], profile, Scale ® Delta,AxesLabel ® 8r@mmD, "D@%D"<, ImageSize ® 500D8Alpha, 2, 880, 1<, 84, -0.5<, 86, -0.5<, 88, 0<<<

5 10 15rHmmL

-0.4

-0.2

0.2

0.4

0.6

0.8

1.0

D@%D

• General index

The data format for the general index profile is{General, {0, ncore}, {r1, n1}, {r2, n2}, ... , {rN , nN}, {acore-clad, nclad}}or{General, {0, Dcore}, {r1, D1}, {r2, D2}, ... , {rN , DN}, {acore-clad, Dclad}}.


profile = 8General, 880, 1<, 84, -0.5<, 86, -0.5<, 86, 0<<<[email protected], profile, Scale ® Delta,AxesLabel ® 8r@mmD, "D@%D"<, ImageSize ® 500D8General, 880, 1<, 84, -0.5<, 86, -0.5<, 86, 0<<<

2 4 6 8 10 12rHmmL

-0.4

-0.2

0.2

0.4

0.6

0.8

1.0

D@%D

Ÿ Characteristic Equation

• Plot the characteristic equation.

profile = 8Step, 1.46, 84, 1.45<<;DspnPlotCharEquation@1, 1.55, profile,AxesLabel ® 8"neff", "Log10ÈGÈ"<, ImageSize ® 500D

1.452 1.454 1.456 1.458 1.460neff

-6

-5

-4

-3

Log10 ÈGÈ

Ÿ Effective Index, Group Index, and Dispersion


Ÿ

Effective Index, Group Index, and Dispersion

• Calculate the effective indexes, group indexes and dispersion of the guided modes.

profile = 8Step, 1.46, 84, 1.45<<;sols = DspnNeffNgroupDspn@1, 81.2, 1.6, 0.02<, profile, NumModes ® 1D8Mode ® 8HE1,1<, Lambda ® 81.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34,

1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6<,Neff ® 881.4573, 1.45723, 1.45716, 1.45709, 1.45702, 1.45696, 1.45689,

1.45682, 1.45675, 1.45668, 1.45661, 1.45654, 1.45647, 1.45641,1.45634, 1.45627, 1.4562, 1.45613, 1.45606, 1.45599, 1.45592<<,

Ngroup ® 881.46137, 1.46138, 1.46139, 1.4614, 1.46141, 1.46142, 1.46143,1.46143, 1.46144, 1.46144, 1.46144, 1.46144, 1.46144, 1.46144,1.46144, 1.46143, 1.46143, 1.46142, 1.46142, 1.46141, 1.4614<<,

Dspn ® 882.10049, 1.93054, 1.75769, 1.58207, 1.4038, 1.22301, 1.03981, 0.854336,0.666699, 0.477018, 0.285407, 0.0919793, -0.103155, -0.299888, -0.498114,-0.697731, -0.898639, -1.10074, -1.30395, -1.50817, -1.71333<<<

• Plot the effective indexNeff

ListPlot@Transpose@8Lambda �. sols, HNeff �. solsL@@1DD<D,Joined ® True, AxesLabel ® 8l@mmD, Neff<, ImageSize ® 500D

1.3 1.4 1.5 1.6lHmmL

1.4562

1.4564

1.4566

1.4568

1.4570

1.4572

Neff

• Plot the group indexNg


ListPlotATranspose@8Lambda �. sols, HNgroup �. solsL@@1DD<D,Joined ® True, AxesLabel ® 9l@mmD, Ng=, ImageSize ® 500E

1.3 1.4 1.5 1.6lHmmL

1.46138

1.46139

1.46140

1.46141

1.46142

1.46143

1.46144

Ng

• Plot the dispersion D

ListPlot@Transpose@8Lambda �. sols, HDspn �. solsL@@1DD<D,Joined ® True, AxesLabel ® 8l@mmD, "D@ps�nm�kmD"<, ImageSize ® 500D

1.3 1.4 1.5 1.6lHmmL

-1

1

2

D@ps�nm�kmD

Ÿ Poynting Vector and Field Profile

• Calculate the Poynting vector.


profile = 8Step, 1.46, 84, 1.45<<;sols = DspnNeff@1, 1.55, profileDSetCoordinates@Cartesian@x, y, zDD;ssol = DspnPoyntingVector@1, 1.55, 1, profile, Phase ® p � 2, MaxRadius ® 20D;8Mode ® 8HE1,1<, Lambda ® 1.55, Neff ® 81.4561<<

• Plot the Poynting vector

Plot@HPoyntingVector@@3DD �. ssolL@x, 0D, 8x, -10, 10<, AxesLabel ® 8r@mmD, Sz<D

-10 -5 5 10rHmmL

0.005

0.010

0.015

0.020

Sz

DensityPlot@HPoyntingVector@@3DD �. ssolL@x, yD,8x, -10, 10<, 8y, -10, 10<, PlotPoints ® 50, PlotRange ® AllD


Plot3D@HPoyntingVector@@3DD �. ssolL@x, yD, 8x, -10, 10<, 8y, -10, 10<,AxesLabel ® 8x, y, Sz<, BoxRatios ® 81, 1, 1<, PlotPoints ® 50, PlotRange ® AllD

Ÿ Example (Hardware Control via GPIB)

Ÿ Initialization

• Open the MathLink connection.

mlelink = Install@"mle.exe"D;mlelink = Install@LinkConnect@"aaa"DD;Uninstall@mlelinkD

mle.exe

• Initialization of the GPIB interface for data acquisition and control.

GPIBInitialize@8"GPIB0", 0, IsBoard ® True<,8"tds360", 1, TimeOut ® 11<, 8"hp3245a", 9, TimeOut ® 11<Dboard = "GPIB0", address = 0, eos = 10, timeout = 10device = "tds360", address = 1, board = "GPIB0", eos = 10, timeout = 11device = "hp3245a", address = 9, board = "GPIB0", eos = 10, timeout = 11


GPIBIfClear@"GPIB0"DInterface cleared: board "GPIB0".

• Set the default directory.

SetDirectory@"D:�Temp"DD:\Temp

Directory@DD:\Temp

Ÿ Communication Examples

• Identfication of the equipment

GPIBClear@"tds360"D;GPIBWrite@"tds360", "*IDN?"D;GPIBRead@"tds360"D

TEKTRONIX,TDS 360,0,CF:91.1CT FV:v1.09

• Retrieve the data from the equipment and store the data to a file.

GPIBWrite@"tds360", "CURVe?"D;GPIBReadFile@"tds360", "data.txt", TimeOut ® 11, Delete ® TrueD

Read to file: device "tds360" to file "D:\Temp\data.txt".

• Switch the data source to Channel 1.

GPIBWrite@"tds360", "DATa:SOUrce CH1"D;• Verify the data source.

GPIBWrite@"tds360", "DATa:SOUrce?"D;GPIBRead@"tds360"D

:DAT:SOU CH1

Ÿ Square Pulses and the Fourier Spectrum

• Set the amplitude and frequency of the signal source.

GPIBWrite@"hp3245a", "RST"D;GPIBWrite@"hp3245a", "APPLY SQV 1.0"D;GPIBWrite@"hp3245a", "FREQ 1E3"D;

• Read the data from the oscilloscope and the store the data in the variable "data".


GPIBWrite@"tds360", "CURVe?"D;data = GPIBRead@"tds360", TimeOut ® 11D;

• Parse the data and put the t and y data in the variable " pdata". The format of "pdata" is88t1, y1<, 8t2, y2<, ...<. The time ti is in the unit of second and the data yi is in the unit of volt.

GPIBClear@"tds360"D;GPIBWrite@"tds360", "WFMPre:CH1:XINcr?"D;xinc = ToExpression@

StringReplace@Part@StringSplit@GPIBRead@"tds360"D, " "D, 2D, "E" ® "*^"DD;GPIBWrite@"tds360", "WFMPre:CH1:YMUlt?"D;ymu = ToExpression@StringReplace@

Part@StringSplit@GPIBRead@"tds360", TimeOut ® 12D, " "D, 2D, "E" ® "*^"DD;adata = StringDrop@data, 6D;pdata = ToExpression �ž StringSplit@adata, ","D;pdata = Table@8xinc * i, ymu * pdata@@iDD<, 8i, Length@pdataD<D;Length@pdataD

1000

• Plot the data.

ListPlot@pdata, Joined ® True, ImageSize ® 500,PlotRange ® 8-0.8, 0.8<, AxesLabel ® 8"sec", "V"<D

0.001 0.002 0.003 0.004 0.005sec

-0.75

-0.5

-0.25

0.25

0.5

0.75

V

… Graphics …

• Fourier transform. First separate the time and signal data. Put the time data in the variable"t" and the signal data in the variable "y".8t, y< = Transpose@pdataD;The frequency data


f =1

Length@tD Ht@@2DD - t@@1DDL HRange@Length@tDD - 1L;The Fourier transform of the signal data

fy = Fourier@yD;Plot the spectrum data.

ListPlotATransposež9f * 0.001, HAbs �ž fyL2=, PlotRange ® 880, 10<, All<,ImageSize ® 500, AxesLabel ® 9"f@kHzD", "ÈF@yD 2"=E

2 4 6 8 10f@kHzD

20

40

60

80

100

ÈF@yDÈ2

… Graphics …

• Clear the data.

f =.; t =.;

Ÿ Sinusoidal Waves and Curve Fitting

• Set the amplitude and frequency of the signal source.

GPIBWrite@"hp3245a", "RST"D;GPIBWrite@"hp3245a", "APPLY ACV 1.0"D;GPIBWrite@"hp3245a", "FREQ 1E3"D;

• Read the data from the oscilloscope and the store the data in the variable "data".

GPIBWrite@"tds360", "CURVe?"D;data = GPIBRead@"tds360", TimeOut ® 11D;




GPIBClear@"tds360"D;GPIBWrite@"tds360", "WFMPre:CH1:XINcr?"D;xinc = ToExpression@

StringReplace@Part@StringSplit@GPIBRead@"tds360"D, " "D, 2D, "E" ® "*^"DD;GPIBWrite@"tds360", "WFMPre:CH1:YMUlt?"D;ymu = ToExpression@StringReplace@

Part@StringSplit@GPIBRead@"tds360", TimeOut ® 12D, " "D, 2D, "E" ® "*^"DD;adata = StringDrop@data, 6D;pdata = ToExpression �ž StringSplit@adata, ","D;pdata = Table@8xinc * i, ymu * pdata@@iDD<, 8i, Length@pdataD<D;Length@pdataD

1000

• Plot the data.

pl1 = ListPlot@pdata, Joined ® True, ImageSize ® 500, AxesLabel ® 8"sec", "V"<D

0.001 0.002 0.003 0.004 0.005sec

-0.4

-0.2

0.2

0.4

V

… Graphics …

• Curve fitting. Fit the data to the sine curve and find the fitting parameters.

f = 1000;fitparams = FindFit@pdata, A + B Sin@2 p f t + fD, 8A, B, f<, tD8A ® -0.009912, B ® -0.507288, f ® 0.159094<

• Plot the sine curve using the fitting parameters.


pl2 = Plot@Evaluate@A + B Sin@2 p f t + fD �. fitparamsD, 8t, 0, 0.005<,ImageSize ® 500, PlotStyle ® 8RGBColor@1, 0, 0D<, AxesLabel ® 8"t@secD", "V"<D

0.001 0.002 0.003 0.004 0.005t@secD

-0.4

-0.2

0.2

0.4

V

… Graphics …

• Comparison of the original data and the fitting curve.

Show@8pl1, pl2<D

0.001 0.002 0.003 0.004 0.005sec

-0.4

-0.2

0.2

0.4

V

… Graphics …


If you want to know more about symbolic computing,

please visit:

http://ontl.gist.ac.kr® Lectures ® Symbolic Computing Methods and

Applications


Tutorial on Symbolic Computing with Mathematica - … on Symbolic Computing with Mathematica...

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