1.3.4 Elliptic PDE’s (Laplace, Poisson, Helmholtz)

Table 1.9: Elliptic PDE’s (Laplace, Poisson, Helmholtz) breakdown of results. Time in seconds

#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time

282

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge not

1.631

79.735

Yes

283

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, right edge not

1.743

8.797

Yes

284

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge has impulse

3.989

26.447

Yes

285

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (a)

0.935

72.504

286

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (b)

0.905

83.332

287

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (c)

4.763

58.768

288

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (d)

4.235

83.677

289

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (e)

20.264

73.192

Yes

290

Laplace in 2D

Cartesian coordinates

pict

Unit triangle B.C.

1.039

18.79

291

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

3.505

7.27

Yes

292

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.766

9.875

Yes

293

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

3.464

7.235

Yes

294

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

41.177

41.03

Yes

295

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.962

32.109

296

Laplace in 2D

Cartesian coordinates

Laplace PDE in 2D Cartessian with boundary condition as Dirac function

0.043

3.511

297

Laplace in 2D

Cartesian coordinates

pict

One side homogeneous

1.619

51.323

298

Laplace in 2D

Cartesian coordinates

pict

In right half plane

22.988

4.04

299

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

5.902

23.138

300

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem Upper half

1.282

6.671

301

Laplace in 2D

Cartesian coordinates

pict

Right half-plane

22.63

4.481

302

Laplace in 2D

Cartesian coordinates

pict

First quadrant

14.85

31.473

303

Laplace in 2D

Cartesian coordinates

pict

Neumann problem upper half-plane

4.935


used convert(sol,Int).

8.997

304

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

2.225

12.098

305

Laplace in 2D

Cartesian coordinates

pict

Strip in upper half

60.268

12.959

306

Laplace in 2D

Cartesian coordinates

pict

in Rectangle, right edge at infinity

5.899

19.588

307

Laplace in 2D

Polar coordinates

pict

Laplace PDE inside quarter disk, Neumann BC at edge

3.591

2.842

308

Laplace in 2D

Polar coordinates

pict

\(r=4\) and \(u=x^4\) at boundary of disk

253.737

22.021

Yes

309

Laplace in 2D

Polar coordinates

pict

\(r=1\) and \(u_r=x\) at boundary of disk

2.745

1.888

Yes

310

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. General solution

0.655

11.463

311

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. Specific boundary conditions

3.688

13.769

Yes

312

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.5 (c)

4.073

2.611

Yes

313

Laplace in 2D

Polar coordinates

pict

semi-circle

2.093

2.322

314

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.8 (b)

9.474

361.819

Yes

315

Laplace in 2D

Polar coordinates

pict

Circular annulus

0.547

9.499

316

Laplace in 2D

Polar coordinates

pict

Outside a disk

7.343

13.615

317

Laplace in 2D

Polar coordinates

pict

Outside a disk

4.194

15.946

Yes

318

Laplace in 3D

Spherical coordinates

In a sphere

0.036

3.401

319

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (a)

0.018

3.513

320

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (b)

0.015

3.773

321

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (c)

0.015

3.813

322

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (d)

0.015

3.881

323

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (e)

0.014

3.827

324

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (a)

0.016

4.865

325

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (b)

0.015

39.671

326

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (c)

0.02

53.644

327

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (d)

0.015

4.81

328

Poisson in 2D

Cartesian coordinates

pict

All boundaries at zero

0.02

82.75

Yes

329

Poisson in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

1.285

1.599

330

Poisson in 2D

Cartesian coordinates

pict

Poisson PDE in whole 2D plane

30.87

0.066

Yes

331

Helmholtz in 2D

Cartesian coordinates

pict

In rectangle

64.088

25.968

332

Helmholtz in 2D

Cartesian coordinates

pict

On whole plane


why? It solved earlier with BC?

0.004

0.289

333

Helmholtz in 2D

Cartesian coordinates

pict

Reduced Helmholtz Inside square

62.012

26.367

334

Helmholtz in 3D

Spherical coordinates

Chain reaction PDE

0.123


trivial solution

4.683

Yes