Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V

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708 Chapter 10

10.44. Consider transverse wave motion in a taut string over the semi-inﬁnite interval

I ={x |0 <x<∞}. The left end of the string is held secure, and the initial

displacement distribution f(x) and initial speed distribution g(x) are given following.

There is an external source term h(x, t) (dimension of force per unit mass) acting and

c = 1/2.

Boundary conditions:

u(0,t)= 0 and u(∞,t)<∞

Initial conditions:

f(x) = e

−x

and g(x) = x e

−x

External force:

h(x, t) = e

−t

10.45. Consider transverse wave motion in a taut string over the semi-inﬁnite interval

I ={x |0 <x<∞}. The left end of the string is held secure, and the initial

displacement distribution f(x) and initial speed distribution g(x) are given following.

There is an external source term h(x, t) (dimension of force per unit mass) acting and

c = 1/2.

Boundary conditions:

u(0,t)= 0 and u(∞,t)<∞

Initial conditions:

f(x) = e

−x

and g(x) = x e

−x

External force:

h(x, t) = e

−x

−t

10.46. Consider transverse wave motion in a taut string over the semi-inﬁnite interval

I ={x |0 <x<∞}. The left end of the string is held secure, and the initial

displacement distribution f(x) and initial speed distribution g(x) are given following.

There is an external source term h(x, t) (dimension of force per unit mass) acting and

c = 1/2.

Boundary conditions:

u(0,t)= 0 and u(∞,t)<∞

Initial conditions:

f(x) = e

−x

and g(x) = x e

−x

External force:

h(x, t) = sin(x) e

−t

References

Abell, M., Braselton, J., Differential Equations with Maple V, Academic Press, 1994.

Abell, M., Braselton, J., Maple V by Example, Academic Press, 1994.

Abell, M., Braselton, J., The Maple V Handbook, Academic Press, 1994.

Artino, C., Johnson, J., Kolod, J., Exploring Calculus with Maple, John Wiley and Sons, 1994.

Bauldry, W., Johnson, J., Linear Algebra with Maple, John Wiley and Sons, 1995.

Bauldry, W.C., Fielder, J.R., Calculus Laboratories with Maple, Brooks/Cole, 1995.

Burbulla, D.C.M., Dodson, C.T.J., Self-Tutor for Computer Calculus Using Maple, Prentice

Hall Canada, 1993.

Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., Watt, S.M., First

Leaves: A Tutorial Introduction to Maple V, Springer-Verlag, 1992.

Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., Watt, S.M., Maple V

Language Reference Manual, Springer-Verlag, 1991.

Char, B.W., Geddes, K.O., Gonnet, G.H., Leong, B.L., Monagan, M.B., Watt, S.M., Maple V

Library Reference Manual, Springer-Verlag, 1991.

Cheung, C., Harer, J., A Guide to Multivariable Calculus with Maple V, John Wiley and Sons,

1994.

Decker, R., Calculus and Maple V, Prentice Hall, 1994.

Devitt, J.S., Calculus with Maple V, Brooks/Cole, 1993.

Ellis, W., Lodi, E., Maple for the Calculus Student: A Tutorial, Brooks/Cole, 1989.

Ellis, W., Lodi, E., Johnson, E., and Schwalbe, D., The Maple V Flight Manual, Brooks/Cole,

1992.

Fattahi, A., Maple V Calculus Labs, Brooks/Cole, 1995.

Harris, K., Lopez, R., Discovering Calculus with Maple, John Wiley and Sons, 1995.

709

710 References

Heck, A., Introduction to Maple-A Computer Algebra System, Springer-Verlag, 1993.

Holmes, M.H., Ecker, J.G., Boyce, W.E., Siegmann, W.L., Exploring Calculus with Maple,

Addison-Wesley, 1993.

Johnson, E., Linear Algebra with Maple V, Brooks/Cole, 1993.

Kreyszig, H.E., Maple Computer Manual for Advanced Engineering Mathematics, John Wiley

and Sons, 1994.

Redfern, D., The Maple Handbook, Springer-Verlag, 1995.

Index

acceleration, 218

angular frequency

damped, 222, 246, 417, 440, 472

multiplicity, 472–473

natural, 522, 554–555

undamped, 443, 473, 516

areal density, 274, 472

basis vector, 5, 6, 9, 13–14, 32

Bessel function

ﬁrst kind, 60

second kind, 62

boundary condition

Dirichlet, 78

mixed, 129

Neumann, 78

nonmixed, 73, 74, 77

periodic, 129

regular, 74, 77

Robin, 78

characteristic equation, 23

concavity, 162, 163, 218

convergence

of derivative, 187

pointwise, 82, 83

uniform, 82, 83, 187

convolution integral, 568–570

coordinate radius, 190, 244, 365, 437

damping coefﬁcient, 217, 414, 460

decay time, 406

mode, 406

differential equation, 4

Bessel, 56–63, 65

Cauchy-Euler, 29–32, 64, 115, 119,

124

Euler, 9, 24–28, 51–55, 64

ﬁrst-order, 19, 166, 371

ﬁrst-order homogeneous, 14

ﬁrst-order linear nonhomogeneous,

14, 37, 63

ﬁrst-order nonhomogeneous, 63

homogeneous, 13

linear operator, 10

normal linear second-order

nonhomogeneous, 13

ordinary, 8–10

ordinary linear homogeneous, 8

711

712 Index

differential equation (continued)

partial, 10–11

second-order, 8, 45–49

second-order constant coefﬁcients,

23–28

second-order linear, 23–28, 147

second-order linear

nonhomogeneous, 63

second-order nonhomogeneous, 37

second-order variable coefﬁcients,

28–32

differential operator, 73

second-order linear, 75

diffusion equation, homogeneous in one

spatial dimension

basis vectors, 166

dimension of solution space, 166

eigenfunctions, 165, 195

eigenfunction expansion solution,

182

eigenvalues, 165, 192

Fourier coefﬁcients, 205, 206

Fourier-Bessel series expansion, 194

homogeneous, 163

homogeneous boundary conditions,

163

initial conditions, 168, 194

method of separation of variables,

163, 191, 205, 206

partial differential in cylindrical

coordinate system, 190, 205

partial differential in rectangular

coordinate system, 161, 205

separation constant, 164, 191

spatial boundary conditions, 165

spatial differential equation, 167,

168, 193

Sturm-Liouville problem, 165

time-dependent differential equation,

166, 194

diffusion equation, homogeneous in two

spatial dimensions

basis vector, 345

boundary condition, 341

coupling equation, 342

double Fourier coefﬁcients, 348, 372,

395

double Fourier series expansion, 348,

372

eigenfunctions, 343, 368, 395

eigenvalues, 343, 344, 368

homogeneous boundary condition,

341, 366

initial condition, 347, 351, 371, 377

method of separation of variables,

341, 366, 394, 395

partial differential in cylindrical

coordinates, 365, 402

partial differential in rectangular

coordinates, 339, 396

r-dependent equation, 368, 370

separation constant, 342, 367

theta-dependent equation, 370

time-dependent equation, 344, 347,

359, 364, 369

x-dependent equation, 342

y-dependent equation, 343

diffusion equation, nonhomogeneous in

one spatial dimension, 477, 546

eigenfunction, 481

eigenvalue, 481

Index 713

Fourier coefﬁcient, 482, 488

Fourier series expansion, 488

homogeneous boundary condition,

477, 478

initial condition, 478, 479, 488

linear (steady-state) solution, 478,

480, 546, 547

method of separation of variables,

481

nonhomogeneous boundary

condition, 477, 479, 546, 547

nonhomogeneous source term, 478

substitution of condition, 478

time-dependent equation, 484

variable (transient) solution, 478,

484, 489, 546

x-dependent equation, 539

diffusion equation, nonhomogeneous

over inﬁnte domain

convolution integral, 568–570

Fourier integral, 564, 565

Fourier transform, 565

initial condition, 564

method of separation of variables,

564

nonhomogeneous partial differential

equation, 564

singular eigenfunctions, 564

time-dependent equation, 565

x-dependent equation, 564

diffusion equation, nonhomogeneous

over semi-inﬁnite domain,

570–573

Fourier transform, 571, 572

Fourier integral, 571, 572

homogeneous boundary condition,

570

diffusion operator, in one spatial

dimension

cylindrical coordinates, 190–194, 205

rectangular coordinates, 161–166,

205

diffusion operator, in two spatial

dimensions

cylindrical coordinates, 365–371, 402

rectangular coordinates, 339–341,

394

Dirac delta function, 557, 560

displacement distribution, 219, 223, 226

dot product, 6

driving (source) function, 13, 16

dynamic equation of motion, 46

eigenfunction

regular, 558, 621

singular, 560, 606

eigenvalue equation, 80

Euler’s constant, 62

Euler formula, 9, 558, 561

external applied forces, 217, 409, 437,

523, 604

Fourier coefﬁcient, 82, 147

complex, 558, 559

regular, 620

singular, 620

714 Index

Fourier integral, 557–560, 606

cosine, 562

sine, 563

Fourier integral theorem, 559

Fourier series, 82, 146, 619

complex form, 558

convergence of, 82

cosine, 562

real form, 558

sine, 563

Fourier transform, 559, 560

convergence of, 82

cosine, 562

pair, 560

sine, 561–563

Frobenius method of series solution,

49–50, 56–63

function

even, 561, 562

odd, 561–563

fundamental frequency, 473

Green’s formula, 75–81

Green’s function

ﬁrst-order, 16, 63, 484, 546

initial value, 569

second-order, 36–44, 63, 516, 547

source value, 568–570

heat loss, 400

heat loss coefﬁcient, 400

heat phenomena, 162, 190, 340

heat sources, 162, 163

Heaviside function, 539, 581

Hermite equation, 148

initial amplitude, 225, 249, 511

indicial equation, 50, 52, 58, 59

initial condition, 19

initial displacement, 410, 415, 417, 443

initial speed, 410, 415, 443, 511

initial value problem, 45–49

inner product, 5

integrating factor, 70, 138

internal heat source, 477

Klein-Gordon equation, 270, 271

Kronecker delta function, 8, 75

Laguerre equation, 148

Laplace equation, over non-inﬁnite

domain, 10, 276

as diffusion equation, 276

as wave equation, 276

basis vectors, 279, 280, 302

boundary conditions, 276

eigenfunctions, 278, 280, 301, 302

eigenvalues, 278, 280, 301

Fourier coefﬁcients, 283, 306

Fourier series expansion, 283, 306

homogeneous boundary condition,

277–279, 301, 302, 325, 326

Index 715

method of separation of variables,

276, 300

partial differential in cylindrical

coordinates, 301–306, 326

partial differential in rectangular

coordinates, 276–277, 325

r-dependent equation, 303, 326

separation constant, 277

steady-state, 275, 276

theta-dependent equation, 301, 307

x-dependent equation, 278

y-dependent equation, 278, 280

Laplace equation, homogeneous over

inﬁnite/semi-inﬁnte domain,

606–611

convolution integral, 611

convolution method, 609

Fourier coefﬁcient, 607, 609

Fourier integral, 607, 608

Fourier integral method, 606

Fourier transform, 607, 608

method of images, 609

method of separation of variables,

606

with respect to x variable, 606

with respect to y variable, 607

x-dependent equation, 606

y-dependent equation, 607, 608

Laplacian operator, 275, 339

linear density, 554

linear independence, 5, 14, 50

magnitude of temperature, 477

maple commands, 2, 4, 51, 65

> (prompt), 2

:= (command equality statement), 2

; (semi-colon), 2

: (colon), 2

+ (addition), 2

* (product), 2

/ (quotient), 2

abs (absolute value), 613

animate, 22

animate3d, 200

assume, 574

BesselJ, 60, 136

BesselY, 62, 136

coeff, 54

collect, 54, 60

combine, 47, 118

convert, 62

cylinderplot, 310

diff (derivative), 3

eval, 22

evalc, 25

exp (exponential), 17

expand, 44

factor, 3

for ...do (loop), 54

fourier, 615

fsolve, 103, 182

Int (integral), 3

invfourier, 595

invlaplace, 648

laplace, 648

limit, 323, 544

plot, 4, 22

plot3d, 287

radsimp, 88

restart, 4, 17

716 Index

maple commands (continued)

series, 54

simplify, 18

solve, 25

sqrt, 28

subs, 3

Sum (summation), 3

trig, 47

unprotect, 88

value (%), 4

with (inttrans), 4, 594

with (plots), 4

mass density, 161, 215, 274

Maclaurin series, 54

method of reduction of order, 32–36

method of undetermined coefﬁcients,

17, 23, 28, 40

method of variation of parameters,

36–44, 73

natural angular oscillation frequency,

46, 554–555

Newton’s law of cooling, 20, 67

non-collinear, 5

non-linear superposition, 37

norm, 6, 77

normal, 8, 9

normalization, 7, 88, 92, 96, 125

operators, 81, 348, 444, 565

linear, 10, 161, 190, 217, 437

partial differential, 11, 161, 217, 345

orthogonal, 7, 77, 159

orthonormal, 7, 77, 92, 133, 288, 361,

432

partial differential equation, 10, 29, 73,

161

diffusion equation, 10, 190, 339, 564

homogeneous, 11, 163, 219, 340

Laplace, 275, 606

wave equation, 10, 217, 275, 409,

639

particular solution, 14, 16, 36

polar angle, 365, 437

Rayleigh quotient, 158–159

recursion formula, 50, 52, 58

residual term, 52, 58

resistance-capacitance circuit, 67

resistance-inductance circuit, 67

resonance, 539, 544–545, 554–555

restoring force, 267, 269

self-adjoint, 76, 158

series RLC circuit, 70

simple spring mass system, 69

slope, 162

solution space, 10

dimension of, 13

span, 5, 14

Index 717

spatial domains

inﬁnite, 557

semi-inﬁnite, 557

spatially invariant, 162, 190, 339

spatial-time dependent temperature,

161, 339

spatial-time dependent wave amplitude,

217, 409

spatial variable, 162, 221, 301, 339, 409

speciﬁc heat, 161, 215, 339

speed, 162, 273

speed distribution, 219, 223, 226

statement of orthogonality, 75, 77, 159

statement of orthonormality, 75–81, 137

Sturm-Liouville operator, 75, 86, 146

Sturm-Liouville problem, 74, 165, 221,

278, 301, 342

Bessel differential equation, 136, 140

Cauchy-Euler differential equation,

115, 119, 124

convergence, 82

differential operator, 73, 75

eigenfunction, 74

eigenvalue, 73–75, 86

Euler differential equation, 78, 86,

90, 94, 99, 104, 109, 119

Fourier coefﬁcient, 74, 98, 138, 348

Fourier series, 73, 134

homogeneous differential equation,

73, 221

homogeneous nonmixed boundary

condition, 73

nonregular, 129

periodic boundary condition, 129

regular, 86

regular boundary condition, 74, 77,

159

superposition of solutions, 331–332

cylindrical coordinates, 336

rectangular coordinates, 327

Taylor series, 49

Tchebycheff equation, 148

telegraph equation, 267, 269

tension, 472, 554

thermal conductivity, 161, 215

thermal diffusivity, 162, 339, 477

magnitude of, 215

signiﬁcance of, 215

time-rate of input, 339

time variable, 162

vector space

dimension of, 5

linear, 4

two-dimensional, 4

veriﬁcation of solution (three-step)

procedure, 186

viscous damping, 67, 69

wave amplitude, 409, 510

initial, 679

wave equation, 10

basis vectors, 222

damped angular frequency of

oscillation, 222, 246