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

Table 1.10: Elliptic PDE’s (Laplace, Poisson, Helmholtz) breakdown of results. Time in seconds

#	PDE	description	Mathematica		Maple		hand solved?	Animated?
			result	time	result	time
280	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge not	✓	1.441	✓	23.392	Yes
281	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, right edge not	✓	1.6	✓	1.217	Yes
282	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge has impulse	✓	3.91	✓	9.684	Yes
283	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (a)	✓	0.857	✓	22.092
284	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (b)	✓	0.893	✓	22.111
285	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (c)	✓	3.512	✓	22.17
286	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (d)	✓	3.17	✓	34.579
287	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (e)	✓	14.115	✓	21.461	Yes
288	Laplace in 2D Cartesian coordinates	Unit triangle B.C.	✓	0.728	✓	1.904
289	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	3.233	✓	1.632	Yes
290	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	2.288	✓	1.606	Yes
291	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	2.789	✓	1.651	Yes
292	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	89.049	✓	6.302	Yes
293	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	1.348	✓	3.834
294	Laplace in 2D Cartesian coordinates	Laplace PDE in 2D Cartessian with boundary condition as Dirac function	✓	0.047	✓	0.429
295	Laplace in 2D Cartesian coordinates	One side homogeneous	✓	5.1	✓	3.404
296	Laplace in 2D Cartesian coordinates	In right half plane	✓	4.281	✓	0.585
297	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	4.281	✓	2.222
298	Laplace in 2D Cartesian coordinates	Dirichlet problem Upper half	✓	1.507	✓	0.633
299	Laplace in 2D Cartesian coordinates	Right half-plane	✓	5.029	✓	0.705
300	Laplace in 2D Cartesian coordinates	First quadrant	✓	92.86	✗	2.852
301	Laplace in 2D Cartesian coordinates	Neumann problem upper half-plane	✓	5.277	✓ used convert(sol,Int).	0.894
302	Laplace in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	2.117	✓	1.228
303	Laplace in 2D Cartesian coordinates	Strip in upper half	✓	5.131	✓	1.376
304	Laplace in 2D Cartesian coordinates	in Rectangle, right edge at infinity	✓	4.392	✓	2.306
305	Laplace in 2D Polar coordinates	Laplace PDE inside quarter disk, Neumann BC at edge	✓	2.417	✓	0.649
306	Laplace in 2D Polar coordinates	\(r=4\) and \(u=x^4\) at boundary of disk	✓	2.707	✓	2.883	Yes
307	Laplace in 2D Polar coordinates	\(r=1\) and \(u_r=x\) at boundary of disk	✗	0.4	✗	0.4	Yes
308	Laplace in 2D Polar coordinates	Laplace inside disk. General solution	✓	0.356	✓	1.209
309	Laplace in 2D Polar coordinates	Laplace inside disk. Specific boundary conditions	✓	3.399	✓	1.845	Yes
310	Laplace in 2D Polar coordinates	Haberman 2.5.5 (c)	✗	3.091	✓	0.751	Yes
311	Laplace in 2D Polar coordinates	semi-circle	✓	1.397	✓	0.683
312	Laplace in 2D Polar coordinates	Haberman 2.5.8 (b)	✓	6.32	✓	70.728	Yes
313	Laplace in 2D Polar coordinates	Circular annulus	✓	1.168	✓	1.572
314	Laplace in 2D Polar coordinates	Outside a disk	✓	4.008	✓	1.229
315	Laplace in 2D Polar coordinates	Outside a disk	✓	3.021	✓	1.928	Yes
316	Laplace in 3D Spherical coordinates	In a sphere	✓	0.056	✓	0.693
317	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (a)	✗	0.014	✗	0.628
318	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (b)	✗	0.012	✗	0.642
319	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (c)	✗	0.013	✗	0.698
320	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (d)	✗	0.012	✗	0.692
321	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (e)	✗	0.016	✗	0.712
322	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (a)	✗	0.012	✗	0.802
323	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (b)	✗	0.013	✗	4.903
324	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (c)	✗	0.016	✗	4.587
325	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (d)	✗	0.011	✗	0.773
326	Poisson in 2D Cartesian coordinates	All boundaries at zero	✗	0.014	✓	6.977	Yes
327	Poisson in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	1.232	✓	0.258
328	Poisson in 2D Cartesian coordinates	Poisson PDE in whole 2D plane	✓	30.674	✓	0.107	Yes
329	Helmholtz in 2D Cartesian coordinates	In rectangle	✓	62.649	✓	2.399
330	Helmholtz in 2D Cartesian coordinates	On whole plane	✓ why? It solved earlier with BC?	0.005	✓	0.255
331	Helmholtz in 2D Cartesian coordinates	Reduced Helmholtz Inside square	✓	61.781	✓	2.916
332	Helmholtz in 3D Spherical coordinates	Chain reaction PDE	✗	0.093	✗ trivial solution	0.707	Yes