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
281	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge not	✓	0.246	✓	20.454	Yes
282	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, right edge not	✓	0.713	✓	1.493	Yes
283	Laplace in 2D Cartesian coordinates	Rectangle, 3 edges zero, buttom edge has impulse	✓	2.681	✓	13.09	Yes
284	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (a)	✓	0.198	✓	15.391
285	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (b)	✓	0.192	✓	18.329
286	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (c)	✗	0.405	✓	22.601
287	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (d)	✗	0.403	✓	16.85
288	Laplace in 2D Cartesian coordinates	Haberman 2.5.1 (e)	✗	0.796	✓	22.019	Yes
289	Laplace in 2D Cartesian coordinates	Unit triangle B.C.	✓	2.267	✓	3.069
290	Laplace in 2D Cartesian coordinates	Top edge at infinity	✗	2.308	✓	1.489	Yes
291	Laplace in 2D Cartesian coordinates	Top edge at infinity	✗	2.32	✓	1.616	Yes
292	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	2.313	✓	1.476	Yes
293	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	0.783	✓	9.34	Yes
294	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	2.258	✓	11.718
295	Laplace in 2D Cartesian coordinates	Laplace PDE in 2D Cartessian with boundary condition as Dirac function	✓	0.045	✓	1.641
296	Laplace in 2D Cartesian coordinates	One side homogeneous	✓	0.585	✓	18.027
297	Laplace in 2D Cartesian coordinates	In right half plane	✓	19.451	✓	1.513
298	Laplace in 2D Cartesian coordinates	Right edge at infinity	✗	0.829	✓	6.872
299	Laplace in 2D Cartesian coordinates	Dirichlet problem Upper half	✓	0.94	✓	1.91
300	Laplace in 2D Cartesian coordinates	Right half-plane	✓	20.57	✓	1.968
301	Laplace in 2D Cartesian coordinates	First quadrant	✓	15.696	✗	18.134
302	Laplace in 2D Cartesian coordinates	Neumann problem upper half-plane	✓	5.249	✓ used convert(sol,Int).	3.617
303	Laplace in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	1.257	✓	1.789
304	Laplace in 2D Cartesian coordinates	Strip in upper half	✗	0.369	✓	3.974
305	Laplace in 2D Cartesian coordinates	in Rectangle, right edge at infinity	✗	0.83	✓	7.553
306	Laplace in 2D Polar coordinates	Laplace PDE inside quarter disk, Neumann BC at edge	✗	2.091	✓	0.992
307	Laplace in 2D Polar coordinates	\(r=4\) and \(u=x^4\) at boundary of disk	✓	149.331	✓	5.08	Yes
308	Laplace in 2D Polar coordinates	\(r=1\) and \(u_r=x\) at boundary of disk	✗	0.016	✗	0.339	Yes
309	Laplace in 2D Polar coordinates	Laplace inside disk. General solution	✓	0.134	✓	1.492
310	Laplace in 2D Polar coordinates	Laplace inside disk. Specific boundary conditions	✗	0.041	✓	1.51	Yes
311	Laplace in 2D Polar coordinates	Haberman 2.5.5 (c)	✗	2.194	✓	0.695	Yes
312	Laplace in 2D Polar coordinates	semi-circle	✗	2.064	✓	0.67
313	Laplace in 2D Polar coordinates	Haberman 2.5.8 (b)	✗	0.011	✓ But has unresolved Invfourier and Fourier calls	35.961	Yes
314	Laplace in 2D Polar coordinates	Circular annulus	✓	2.316	✓	3.298
315	Laplace in 2D Polar coordinates	Outside a disk	✗	0.008	✓	2.042
316	Laplace in 2D Polar coordinates	Outside a disk	✗	0.038	✓	1.556	Yes
317	Laplace in 3D Spherical coordinates	In a sphere	✗	0.021	✓	1.758
318	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (a)	✗	0.003	✗	1.592
319	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (b)	✗	0.003	✗	1.779
320	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (c)	✗	0.003	✗	1.819
321	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (d)	✗	0.003	✗	1.919
322	Laplace in 3D Cylinderical coordinates	Haberman 7.9.1 (e)	✗	0.003	✗	1.862
323	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (a)	✗	0.003	✗	2.144
324	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (b)	✗	0.003	✗	7.48
325	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (c)	✗	0.003	✗	9.012
326	Laplace in 3D Cylinderical coordinates	Haberman 7.9.2 (d)	✗	0.003	✗	1.993
327	Poisson in 2D Cartesian coordinates	All boundaries at zero	✗	0.01	✗	23.384	Yes
328	Poisson in 2D Cartesian coordinates	Dirichlet problem in a rectangle	✓	0.211	✓	0.745
329	Poisson in 2D Cartesian coordinates	Poisson PDE in whole 2D plane	✗	0.02	✓	0.02	Yes
330	Helmholtz in 2D Cartesian coordinates	In rectangle	✓	2.593	✓	4.516
331	Helmholtz in 2D Cartesian coordinates	On whole plane	✗ why? It solved earlier with BC?	0.004	✓	0.039
332	Helmholtz in 2D Cartesian coordinates	Reduced Helmholtz Inside square	✗	1.134	✓	20.8
333	Helmholtz in 3D Spherical coordinates	Chain reaction PDE	✗	0.036	✗ trivial solution	2.122	Yes