1.3.2 Parabolic PDE’s (Diffusion)

Table 1.8: Parabolic PDE’s (Diffusion) breakdown of results. Time in seconds

#	PDE	description	Mathematica		Maple		hand solved?	Animated?
			result	time	result	time
150	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	General initial conditions	✓	92.535	✓	1.555	Yes
151	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Specific initial condition	✓	25.196	✓	0.827	Yes	Yes
152	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC \(u=x(1-x)\)	✓	29.404	✓	0.912	Yes	Yes
153	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (a)	✓	0.055	✓	2.379	Yes
154	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC hat function	✓	44.575	✓	1.102	Yes	Yes
155	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (b)	✓	0.802	✓	2.872	Yes
156	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (c)	✓ but \(n=3\) should be special case	93.233	✓ handled \(n=3\) case correctly.	3.454	Yes
157	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (d)	✗ (Timed out)	600.	✓	4.566	Yes
158	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	46.186	✓	0.837	Yes
159	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	92.384	✓	0.848	Yes	Yes
160	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (general case)	✓	98.212	✓	1.603	Yes
161	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (special case)	✓	85.342	✓	1.725	Yes	Yes
162	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (general case)	✓	49.174	✓	2.31	Yes
163	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (special case)	✓	95.91	✓	2.87	Yes	Yes
164	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Math 4567 Exam	✓	26.572	✓	1.604	Yes
165	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	With source	✓	61.128	✓	4.249
166	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	special initial condition	✓	2.824	✗ (Timed out)	600.	Yes
167	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction and Euler-Cauchy Sturm-Liouville	✓	25.661	✓	2.084
168	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Using growth form reaction term	✓	152.625	✓	0.882	Yes
169	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using logistic form for reaction term	✓	3.143	✗	2.02
170	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using Aleee form for reaction term	✓	5.429	✗	2.654
171	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Haberman 2.3.8	✓	93.624	✓	1.432	Yes
172	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	93.128	✓	1.382	Yes
173	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	0.438	✓	1.442
174	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✗	40.745	✓	1.184
175	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (general case)	✓	32.546	✓	1.909	Yes
176	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (special case)	✓	92.464	✓	1.702	Yes	Yes
177	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.2 (general case)	✓	31.546	✓	1.676	Yes
178	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Left end zero, right end insulated, no source	✓	26.574	✓	1.663	Yes
179	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	One end insulated	✓	32.074	✓	1.793
180	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.7 (general case)	✓	31.743	✓	1.363	Yes
181	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	specific case	✓	23.593	✓	1.061	Yes	Yes
182	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (a)	✓	21.49	✓	4.548
183	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Both ends insulated, no source	✓	32.954	✓	1.324	Yes
184	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (b) (special case)	✓	32.144	✓	3.016	Yes
185	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (c) (special case)	✓	32.008	✓	3.414	Yes
186	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (d)	✓	31.543	✓	2.665
187	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	both ends insulated	✓	61.651	✓	2.144
188	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	2.423	✓	0.936
189	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.25	✓	58.454	✓	3.591
190	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	external source	✓	43.316	✓	9.488
191	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction (general case)	✓	91.889	✓	0.86	Yes	Yes
192	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.23	✓	95.055	✓	3.445
193	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.21	✓	93.818	✓	1.96
194	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Euler-Cauchy Sturm-Liouville	✓	32.677	✗ (Timed out)	600.	Yes
195	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	2.286	✓	1.516	Yes
196	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Mixed BC	✓	3.138	✓	2.919
197	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 8.2.1 (f)	✓	93.293	✓	2.519
198	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (general case)	✓	33.038	✓	1.873	Yes
199	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (special case)	✓	20.289	✓	1.226	Yes	Yes
200	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (general case)	✓	95.03	✓	1.502	Yes
201	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (special case)	✓	0.009	✓	1.289	Yes
202	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC, special case	✓	14.619	✓	0.94	Yes	Yes
203	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	convection heat loss	✓	7.353	✗ (Timed out)	600.	Yes
204	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	1.798	✓	2.14
205	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	62.389	✓	0.99
206	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	54.29	✓	3.592	Yes
207	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (general case)	✓	65.521	✓	3.611	Yes
208	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	64.923	✓	2.507	Yes	Yes
209	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	65.993	✓	2.413	Yes	Yes
210	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	Haberman 8.3.6 (special case)	✓	154.843	✓	2.947	Yes	Yes
211	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	BC depends on time (special case)	✓	24.685	✓	1.381
212	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). IC given	✓	4.048	✓	4.277
213	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). No IC given	✓	0.011	✗	0.62	Yes
214	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). Damped heat PDE. No IC given	✗	0.022	✗	0.801	Yes
215	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends non-homogeneous BC (general case)	✓	93.33	✓	1.871	Yes
216	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	non-homogeneous BC (special case)	✓	27.836	✓	1.088	Yes
217	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.1 (special case)	✓	92.982	✓	1.029	Yes
218	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (general case)	✓	103.13	✓	1.961	Yes
219	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (special case)	✓	154.771	✓	1.155	Yes	Yes
220	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space and time only (general case)	✓	48.597	✓	2.457	Yes
221	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (general case)	✓	86.706	✓	3.557	Yes
222	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (special case)	✓	96.834	✓	2.968	Yes	Yes
223	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends nonhomogeneous	✓	64.258	✓	2.66	Yes
224	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (a) (general case)	✓	34.899	✓	2.049	Yes
225	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (d) (general solution)	✓	103.	✓	1.771	Yes
226	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source that depends on space only (general solution)	✓	128.677	✗ (Timed out)	600.	Yes
227	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source that depends on space only (special case)	✓	163.229	✓	3.857	Yes	Yes
228	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source that depends on time and space (general solution)	✓	69.729	✓	3.34	Yes
229	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source present (special case)	✓	171.102	✓	6.242	Yes	Yes
230	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Pinchover and Rubinstein 6.17	✗	15.807	✓	2.019
231	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	nonhomogeneous BC	✓	169.105	✓	4.05
232	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.2. (a)	✓	94.926	✓	3.335	Yes
233	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.3	✓	4.935	✓	2.803
234	Diffusion in 1D Semi-infinite domain	left end constant (general case)	✓	60.028	✓	0.711	Yes
235	Diffusion in 1D Semi-infinite domain	left end constant (special case)	✓ It fail if assumption \(x>0\) is given. A bug	37.776	✓	0.7	Yes	Yes
236	Diffusion in 1D Semi-infinite domain	Logan p. 76. Left end general function of time (general case)	✓	0.454	✓	0.969	Yes
237	Diffusion in 1D Semi-infinite domain	Left end function of time (special case)	✓	30.759	✓	1.326	Yes	Yes
238	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.023	✓	0.432
239	Diffusion in 1D Semi-infinite domain	I.C. not zero	✓ due to IC/BC not zero	14.15	✓	0.692
240	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.051	✓	0.402
241	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.045	✓	1.57
242	Diffusion in 1D Semi-infinite domain	nonhomogeneous B.C.	✓	60.026	✓	0.877
243	Diffusion in 1D Semi-infinite domain	Unit triangle I.C.	✓	31.526	✓	0.918
244	Diffusion in 1D Semi-infinite domain	I.C. not at \(t=0\)	✓	5.371	✓	0.87
245	Diffusion in 1D Semi-infinite domain	Diffusion with advection	✓	2.249	✓	1.263
246	Diffusion in 1D Semi-infinite domain	Practice exam problem	✓	60.043	✓	0.65
247	Diffusion in 1D Infinite domain	Inverse exponential I.C.	✓	22.122	✓	0.617	Yes
248	Diffusion in 1D Infinite domain	Advection term	✓	0.42	✓	0.583
249	Diffusion in 1D Infinite domain	UnitBox I.C.	✓	0.118	✓	0.745
250	Diffusion in 1D Infinite domain	No source	✓	1.365	✓	1.123
251	Diffusion in 1D Infinite domain	constant as source	✓	30.401	✓	0.722
252	Diffusion in 1D Infinite domain	No intial conditions	✓	0.004	✓	0.21	Yes
253	Diffusion in 1D Infinite domain	piecewise I.C.	✓ due to i.c. not at zero	84.545	✓	0.953	Yes
254	Diffusion in 1D Infinite domain	Practice exam problem	✓	20.868	✓	0.553
255	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	No source	✓	3.275	✓	1.801
256	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Internal source term	✓	7.154	✓	2.337
257	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Articolo 6.6.3	✓	5.044	✓	3.939
258	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (General solution)	✓	0.819	✓ Cant get series solution	6.217	Yes
259	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (Specific solution)	✓	2.137	✓ Do not understand Maple solution	2.874	Yes	Yes
260	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	0.232	✓	1.7
261	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	1.152	✓	0.739
262	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.3.5 (General solution)	✓	105.809	✓	1.11	Yes
263	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	3.25	✓	1.269	Yes	Yes
264	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	2.718	✓	1.146	Yes	Yes
265	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.1	✓	3.368	✗	0.758
266	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.2	✓	3.49	✓	10.656
267	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.2.5 with \(\theta \) dependency (General case)	✓	239.203	✗	0.305	Yes
268	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (Specific example)	✓	6.781	✗	0.815	Yes	Yes
269	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (specific example)	✓	7.919	✗	0.866	Yes	Yes
270	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (General solution)	✓	2.886	✗	1.843	Yes
271	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✓	11.87	✗	1.838	Yes	Yes
272	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✗ (Timed out)	600.	✗	1.901	Yes	Yes
273	Diffusion in 3D Spherical coordinates	No angle dependencies	✓	61.011	✓ Has unresolved Laplace integrals	0.525
274	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (a)	✓	1.743	✓	9.997	Yes	Yes
275	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (b)	✓	4.68	✗	4.181	Yes	Yes
276	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (c)	✓	2.333	✗	6.657
277	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (a)	✗	0.015	✗	1.047
278	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (b)	✗	0.01	✗	1.136
279	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (c)	✗	0.012	✗	1.143