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

Table 1.8: 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	✓	0.293	✓	78.897	Yes
281	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, right edge not	✓	0.674	✓	7.527	Yes
282	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge has impulse	✓	2.443	✓	25.953	Yes
283	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (a)	✓	0.287	✓	58.794
284	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (b)	✓	0.277	✓	72.414
285	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (c)	✓	3.84	✓	50.321
286	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (d)	✓	3.146	✓	76.114
287	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (e)	✓	17.265	✓	70.724	Yes
288	Laplace in 2D Cartesian coordinates	Unit triangle B.C.	✓	0.948	✓	19.002
289	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	2.19	✓	7.02	Yes
290	Laplace in 2D Cartesian coordinates	Top edge at infinity	✓	1.523	✓	7.72	Yes
291	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	2.248	✓	9.542	Yes
292	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	40.198	✓	46.18	Yes
293	Laplace in 2D Cartesian coordinates	Right edge at infinity	✓	1.681	✓	53.657
294	Laplace in 2D Cartesian coordinates	Laplace PDE in 2D Cartessian with boundary condition as Dirac function	✓	0.043	✓	3.918
295	Laplace in 2D Cartesian coordinates	One side homogeneous	✓	0.566	✓	67.987
296	Laplace in 2D Cartesian coordinates	In right half plane	✓	21.874	✓	4.062
297	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	0.861	✓	35.773
298	Laplace in 2D Cartesian coordinates	Dirichlet problem Upper half	✓	1.339	✓	5.267
299	Laplace in 2D Cartesian coordinates	Right half-plane	✓	22.702	✓	4.731
300	Laplace in 2D Cartesian coordinates	First quadrant	✓	14.532	✗	32.221
301	Laplace in 2D Cartesian coordinates	Neumann problem upper half-plane	✓	4.536	✓ used convert(sol,Int).	9.638
302	Laplace in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	1.167	✓	10.724
303	Laplace in 2D Cartesian coordinates	Strip in upper half	✓	60.25	✓	12.8
304	Laplace in 2D Cartesian coordinates	in Rectangle, right edge at infinity	✗	0.859	✓	35.71
305	Laplace in 2D Polar coordinates	Laplace PDE inside quarter disk, Neumann BC at edge	✓	2.708	✓	2.815
306	Laplace in 2D Polar coordinates	\(r=4\) and \(u=x^4\) at boundary of disk	✓	143.09	✓	13.167	Yes
307	Laplace in 2D Polar coordinates	\(r=1\) and \(u_r=x\) at boundary of disk	✗	0.01	✗	1.92	Yes
308	Laplace in 2D Polar coordinates	Laplace inside disk. General solution	✓	0.593	✓	9.977
309	Laplace in 2D Polar coordinates	Laplace inside disk. Specific boundary conditions	✓	3.703	✓	7.162	Yes
310	Laplace in 2D Polar coordinates	Haberman 2.5.5 (c)	✗	2.03	✓	2.379	Yes
311	Laplace in 2D Polar coordinates	semi-circle	✓	1.289	✓	2.211
312	Laplace in 2D Polar coordinates	Haberman 2.5.8 (b)	✓	8.536	✓ But has unresolved Invfourier and Fourier calls	36.41	Yes
313	Laplace in 2D Polar coordinates	Circular annulus	✓	2.22	✓	1.924
314	Laplace in 2D Polar coordinates	Outside a disk	✓	6.559	✓	1.35
315	Laplace in 2D Polar coordinates	Outside a disk	✓	4.247	✓	1.549	Yes
316	Laplace in 3D Spherical coordinates	In a sphere	✓	0.028	✓	0.63
317	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (a)	✗	0.015	✗	0.796
318	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (b)	✗	0.013	✗	0.722
319	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (c)	✗	0.013	✗	0.76
320	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (d)	✗	0.013	✗	0.818
321	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (e)	✗	0.013	✗	0.878
322	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (a)	✗	0.014	✗	1.071
323	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (b)	✗	0.013	✗	5.71
324	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (c)	✗	0.018	✗	6.572
325	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (d)	✗	0.013	✗	1.244
326	Poisson in 2D Cartesian coordinates	All boundaries at zero	✗	0.013	✗	20.204	Yes
327	Poisson in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	0.22	✓	0.308
328	Poisson in 2D Cartesian coordinates	Poisson PDE in whole 2D plane	✓	0.009	✓	0.012	Yes
329	Helmholtz in 2D Cartesian coordinates	In rectangle	✓	62.821	✓	4.592
330	Helmholtz in 2D Cartesian coordinates	On whole plane	✓ why? It solved earlier with BC?	0.004	✓	0.062
331	Helmholtz in 2D Cartesian coordinates	Reduced Helmholtz Inside square	✓	60.777	✓	15.635
332	Helmholtz in 3D Spherical coordinates	Chain reaction PDE	✗	0.107	✗ trivial solution	1.06	Yes