1.3.5 Hyperbolic PDE’s (Wave)

Table 1.10: Hyperbolic PDE’s (Wave) breakdown of results. Time in seconds

#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time

335

Wave PDE in 1D

Finite length string

pict

General solution for both ends fixed. Domain is \(0\dots L\)

39.729

26.077

Yes

336

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital position zero (special case)

2.745

13.872

Yes

Yes

337

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital velocity zero (special case)

1.654

10.655

Yes

Yes

338

Wave PDE in 1D

Finite length string

pict

both ends fixed but domain is \(-\pi \dots \pi \). zero intial position, non zero initial velocity

6.485

60.958

Yes

Yes

339

Wave PDE in 1D

Finite length string

pict

both ends fixed but domain is \(-1 \dots 1\). intial position is an impulse, zero initial velocity

0.651

23.098

Yes

340

Wave PDE in 1D

Finite length string

pict

Logan book, page 28. Both ends fixed

1.011

2.843

341

Wave PDE in 1D

Finite length string

pict

non-zero initial velocity. Both ends fixed

102.57

8.794

342

Wave PDE in 1D

Finite length string

pict

Logan book page 149)

22.208

15.192

343

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (a)

25.092

54.316

344

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (b)

56.658

36.438

Yes

345

Wave PDE in 1D

Finite length string

pict

Both I.C. not zero

26.507

18.372

346

Wave PDE in 1D

Finite length string

pict

With constant source

42.579

30.72

347

Wave PDE in 1D

Finite length string

pict

Logan page 213

4.687

17.921

348

Wave PDE in 1D

Finite length string

pict

Telegraphy PDE

148.548

19.516

349

Wave PDE in 1D

Finite length string

pict

Dispersion term present (general case)


Due to adding dispersion term

66.087

25.109

Yes

350

Wave PDE in 1D

Finite length string

pict

Dispersion term present


Due to adding dispersion term

154.94

89.473

Yes

351

Wave PDE in 1D

Finite length string

pict

Dispersion term present (specific case)


Due to adding dispersion term

158.32

22.849

Yes

Yes

352

Wave PDE in 1D

Finite length string

pict

non-zero initial position


But sum should not include \(n=2\)

7.67


Handled \(n=2\) case correctly

75.615

353

Wave PDE in 1D

Finite length string

pict

With source

11.457

79.233

354

Wave PDE in 1D

Finite length string

pict

Right end free (general case)

26.182

34.789

Yes

355

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (general case)

7.292

25.612

Yes

356

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (special case)

90.868

15.144

Yes

Yes

357

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (general case)

53.247

33.813

Yes

358

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, underdamped)

110.821

29.971

Yes

Yes

359

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, critical damped)

110.575

36.941

Yes

Yes

360

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, over damped)

112.386

22.824

Yes

Yes

361

Wave PDE in 1D

Finite length string

pict

I.C. at different times, right end free, with source

39.847

85.616

Yes

362

Wave PDE in 1D

Finite length string

pict

Right end oscillates

73.611

62.784

363

Wave PDE in 1D

Finite length string

pict

Perioidic B.C.

2.343

4.542

364

Wave PDE in 1D

Finite length string

pict

Mixed B.C.

15.952

5.86

365

Wave PDE in 1D

Finite length string

pict

Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions

63.293

29.737

Yes

Yes

366

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed, (general case)

4.332

5.185

Yes

367

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed with specific initial position

16.204

13.644

Yes

368

Wave PDE in 1D

Semi-infinite domain

pict

Logan page 115, left end fixed with source

7.221

5.306

369

Wave PDE in 1D

Semi-infinite domain

pict

Left moving boundary condition

1.807

4.509

370

Wave PDE in 1D

Semi-infinite domain

pict

moving Left end

9.432

6.497

371

Wave PDE in 1D

Semi-infinite domain

pict

I.C. at \(t=1\)

9.345

1.518

372

Wave PDE in 1D

Semi-infinite domain

pict

B.C. at \(x=1\)

77.621

2.819

373

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (general solution)

10.585

1.703

374

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (Special solution)

4.216

6.313

Yes

Yes

375

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed. zero initial velocity (Special solution)

4.331

8.789

Yes

Yes

376

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial position (general solution)

13.593

1.712

377

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. Non zero initial position and velocity (general solution)

34.866

1.483

378

Wave PDE in 1D

Semi-infinite domain

pict

Left end free with source

1.029

1.232

379

Wave PDE in 1D

Infinite domain

pict

General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)

0.019

0.69

380

Wave PDE in 1D

Infinite domain

pict

General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)

0.005

0.55

381

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)

0.159

6.92

382

Wave PDE in 1D

Infinite domain

pict

No source. \(u_{tt} = u_{xx}\), with \(u(x,0) =e^{-x^2},u_t(x,0)=1\)

0.002

0.307

383

Wave PDE in 1D

Infinite domain

pict

With source term. \(u_{tt} = u_{xx} + m\)

0.01

0.444

384

Wave PDE in 1D

Infinite domain

pict

non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)

0.029

0.404

Yes

Yes

385

Wave PDE in 1D

Infinite domain

pict

Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)

0.004

0.319

386

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.032

0.365

387

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.002

0.29

388

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

8.986

0.491

389

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.002

0.306

Yes

Yes

390

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.004

4.527

Yes

Yes

391

Wave PDE in 1D

Infinite domain

pict

General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)

0.104

0.378

392

Wave PDE in 1D

Infinite domain

pict

General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position

0.003

0.347

Yes

Yes

393

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)

0.101

0.699

Yes

Yes

394

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)

63.765

0.695

Yes

395

Wave PDE in 1D

Infinite domain

system of 2 inhomogeneous linear hyperbolic system with constant coefficients

0.274

1.19

396

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, General solution

3.116

399.349

Yes

397

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, zero velocity. Specific example

0.471

32.39

Yes

Yes

398

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example

34.18

57.503

Yes

Yes

399

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example, delta in center

9.562

54.177

Yes

Yes

400

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed

4.864

32.94

401

Wave PDE in 2D

Cartesian coordinates

pict

All edges fixed (Haberman 8.5.5 (a)

29.711

3.128

402

Wave PDE in 2D

Cartesian coordinates

pict

2 edgs fixed, 2 free, zero initial velocity

6.093

63.49

403

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, general solution

1.518

(Timed out)

600.

Yes

404

Wave PDE in 2D

Cartesian coordinates

pict

With damping

7.018

88.935

405

Wave PDE in 2D

Cartesian coordinates

pict

On the whole plane

0.187

0.388

406

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency, fixed boundary, general case

2.339


Has unresolved Invlaplace calls

8.424

Yes

407

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency. Specific example. Both initial conditions not zero

2.425


Has unresolved Invlaplace calls

73.886

408

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency. Specific example. Both initial conditions not zero

2.25


Has unresolved Invlaplace calls. How to get series solution?

68.303

Yes

Yes

409

Wave PDE in 2D

Polar coordinates

no \(\theta \) dependency. Using integral transforms. Source present. Specific example

3.815

29.343

410

Wave PDE in 2D

Polar coordinates

no \(\theta \) dependency. Using integral transforms. Source present. Specific example

2.47

7.253

411

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, general solution

10.075

16.112

Yes

412

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial velocity, general solution

4.977

12.441

Yes

413

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial velocity, specific example

15.007

10.359

Yes

Yes

414

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial position, specific example

6.402

12.831

Yes

Yes

415

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial position with internal source (Haberman 8.5.5. (b)

0.043

4.539

Yes

416

Wave PDE in 3D

Spherical coordinates

pict

No I.C. no B.C.

0.042

6.306

417

Wave PDE in 3D

Cylindrical coordinates

pict

No I.C. no B.C.

0.006

0.668