1.3.2 Parabolic PDE’s (Diffusion)

Table 1.7: 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	✓	60.751	✓	2.103	Yes
151	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Specific initial condition	✓	9.824	✓	0.916	Yes	Yes
152	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC \(u=x(1-x)\)	✓	14.917	✓	1.108	Yes	Yes
153	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (a)	✓	0.351	✓	3.666	Yes
154	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC hat function	✓	19.338	✓	2.584	Yes	Yes
155	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (b)	✓	0.863	✓	3.852	Yes
156	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (c)	✓ but \(n=3\) should be special case	30.809	✓ handled \(n=3\) case correctly.	9.375	Yes
157	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (d)	✓	499.335	✓	38.43	Yes
158	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	17.557	✓	1.255	Yes
159	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	15.594	✓	1.259	Yes	Yes
160	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (general case)	✓	68.416	✓	2.5	Yes
161	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (special case)	✓	57.962	✓	2.473	Yes	Yes
162	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (general case)	✓	67.009	✓	5.115	Yes
163	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (special case)	✓	64.652	✓	10.528	Yes	Yes
164	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Math 4567 Exam	✓	17.974	✓	2.245	Yes
165	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	With source	✓	64.651	✓	8.138
166	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	special initial condition	✓	2.704	✗ (Timed out)	600.	Yes
167	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction and Euler-Cauchy Sturm-Liouville	✓	12.255	✓	3.718
168	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Using growth form reaction term	✓	81.603	✓	1.02	Yes
169	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using logistic form for reaction term	✗	1.648	✗	4.908
170	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using Aleee form for reaction term	✗	2.839	✗	6.778
171	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Haberman 2.3.8	✓	32.027	✓	2.621	Yes
172	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	31.889	✓	1.887	Yes
173	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	0.537	✓	1.493
174	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✗	39.849	✓	1.266
175	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (general case)	✓	30.778	✓	1.681	Yes
176	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (special case)	✗	31.483	✓	1.295	Yes	Yes
177	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.2 (general case)	✓	30.45	✓	2.846	Yes
178	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Left end zero, right end insulated, no source	✓	30.554	✓	4.609	Yes
179	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	One end insulated	✓	30.898	✓	2.701
180	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.7 (general case)	✓	30.566	✓	2.343	Yes
181	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	specific case	✓	6.322	✓	1.323	Yes	Yes
182	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (a)	✗ (Timed out)	600.	✓	34.522
183	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Both ends insulated, no source	✓	14.057	✓	3.831	Yes
184	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (b) (special case)	✓	0.593	✓	5.284	Yes
185	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (c) (special case)	✓	31.19	✓	11.313	Yes
186	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (d)	✓	0.39	✓	4.62
187	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	both ends insulated	✓	77.322	✓	8.408
188	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	31.507	✓	1.127
189	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.25	✓	68.393	✓	10.324
190	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	external source	✓	47.067	✓	24.241
191	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction (general case)	✓	88.469	✓	1.007	Yes	Yes
192	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.23	✓	1.239	✓	25.969
193	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.21	✓	0.381	✓	5.803
194	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Euler-Cauchy Sturm-Liouville	✓	31.566	✓	32.102	Yes
195	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	30.888	✓	2.48	Yes
196	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Mixed BC	✓	31.407	✓	5.308
197	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 8.2.1 (f)	✓	92.918	✓	6.183
198	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (general case)	✓	32.209	✓	3.295	Yes
199	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (special case)	✓	5.766	✓	1.711	Yes	Yes
200	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (general case)	✓	64.386	✓	3.083	Yes
201	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (special case)	✓	0.009	✓	3.347	Yes
202	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC, special case	✓	11.58	✓	3.084	Yes	Yes
203	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	convection heat loss	✓	20.874	✗ (Timed out)	600.	Yes
204	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	0.422	✓	10.994
205	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	66.959	✓	1.681
206	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	66.762	✓	14.425	Yes
207	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (general case)	✓	80.609	✓	11.675	Yes
208	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	80.316	✓	11.947	Yes	Yes
209	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	85.399	✓	9.148	Yes	Yes
210	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	Haberman 8.3.6 (special case)	✓	123.182	✓	18.416	Yes	Yes
211	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	BC depends on time (special case)	✓	10.802	✓	5.644
212	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). IC given	✓	17.304	✓	10.775
213	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). No IC given	✓	0.013	✓	1.554	Yes
214	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). Damped heat PDE. No IC given	✗	0.026	✓	1.55	Yes
215	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends non-homogeneous BC (general case)	✓	61.831	✓	4.59	Yes
216	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	non-homogeneous BC (special case)	✓	21.41	✓	2.569	Yes
217	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.1 (special case)	✓	19.417	✓	2.634	Yes
218	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (general case)	✓	80.332	✓	5.211	Yes
219	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (special case)	✓	123.312	✓	4.407	Yes	Yes
220	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space and time only (general case)	✓	66.991	✓	9.974	Yes
221	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (general case)	✓	99.038	✓	21.379	Yes
222	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (special case)	✓	65.856	✓	22.126	Yes	Yes
223	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends nonhomogeneous	✓	93.199	✓	16.422	Yes
224	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (a) (general case)	✓	37.261	✓	5.292	Yes
225	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (d) (general solution)	✓	78.393	✓	6.201	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)	✓	140.923	✗ (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)	✓	132.866	✓	33.465	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)	✓	77.304	✓	16.904	Yes
229	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source present (special case)	✓	141.893	✓	52.566	Yes	Yes
230	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Pinchover and Rubinstein 6.17	✗	1.972	✓	8.3
231	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	nonhomogeneous BC	✓	139.915	✓	18.675
232	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.2. (a)	✓	119.345	✓	28.079	Yes
233	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.3	✓	34.029	✓	15.807
234	Diffusion in 1D Semi-infinite domain	left end constant (general case)	✓	60.035	✓	1.84	Yes
235	Diffusion in 1D Semi-infinite domain	left end constant (special case)	✓ It fail if assumption \(x>0\) is given. A bug	34.512	✓	2.836	Yes	Yes
236	Diffusion in 1D Semi-infinite domain	Logan p. 76. Left end general function of time (general case)	✓	0.553	✓	6.489	Yes
237	Diffusion in 1D Semi-infinite domain	Left end function of time (special case)	✓	60.044	✓	7.712	Yes	Yes
238	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.029	✓	2.753
239	Diffusion in 1D Semi-infinite domain	I.C. not zero	✓ due to IC/BC not zero	13.338	✓	2.967
240	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.073	✓	2.69
241	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.07	✓	7.536
242	Diffusion in 1D Semi-infinite domain	nonhomogeneous B.C.	✓	60.034	✓	2.384
243	Diffusion in 1D Semi-infinite domain	Unit triangle I.C.	✓	61.013	✓	3.709
244	Diffusion in 1D Semi-infinite domain	I.C. not at \(t=0\)	✓	61.506	✓	2.31
245	Diffusion in 1D Semi-infinite domain	Diffusion with advection	✓	1.674	✓	4.866
246	Diffusion in 1D Semi-infinite domain	Practice exam problem	✓	60.118	✓	3.146
247	Diffusion in 1D Infinite domain	Inverse exponential I.C.	✓	0.753	✓	3.941	Yes
248	Diffusion in 1D Infinite domain	Advection term	✓	0.513	✓	2.835
249	Diffusion in 1D Infinite domain	UnitBox I.C.	✓	60.017	✓	6.335
250	Diffusion in 1D Infinite domain	No source	✓	0.384	✓	6.612
251	Diffusion in 1D Infinite domain	constant as source	✓	0.416	✓	3.597
252	Diffusion in 1D Infinite domain	No intial conditions	✓	0.002	✓	0.707	Yes
253	Diffusion in 1D Infinite domain	piecewise I.C.	✓ due to i.c. not at zero	51.763	✓	7.01	Yes
254	Diffusion in 1D Infinite domain	Practice exam problem	✓	0.066	✓	1.631
255	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	No source	✓	8.192	✓	11.596
256	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Internal source term	✓	16.973	✓	11.128
257	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Articolo 6.6.3	✓	12.205	✓	17.917
258	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (General solution)	✓	0.926	✓ Cant get series solution	53.213	Yes
259	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (Specific solution)	✓	2.87	✓ Do not understand Maple solution	17.698	Yes	Yes
260	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	0.329	✓	5.741
261	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	1.125	✓	1.125
262	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.3.5 (General solution)	✓	18.268	✓	5.17	Yes
263	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	3.388	✓	4.	Yes	Yes
264	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	2.366	✗	6.497	Yes	Yes
265	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.1	✓	4.106	✗	3.571
266	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.2	✓	4.525	✓	42.538
267	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.2.5 with \(\theta \) dependency (General case)	✓	326.327	✗	1.684	Yes
268	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (Specific example)	✓	8.153	✗	4.289	Yes	Yes
269	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (specific example)	✓	9.657	✗	4.13	Yes	Yes
270	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (General solution)	✓	6.323	✗	10.47	Yes
271	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✓	19.174	✗	7.694	Yes	Yes
272	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✗ (Timed out)	600.	✗	7.392	Yes	Yes
273	Diffusion in 3D Spherical coordinates	No angle dependencies	✓	61.069	✓ Has unresolved Laplace integrals	3.78
274	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (a)	✗	0.033	✓	75.741	Yes	Yes
275	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (b)	✗	0.028	✗	26.434	Yes	Yes
276	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (c)	✗	0.029	✗	26.128
277	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (a)	✗	0.006	✗	4.344
278	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (b)	✗	0.006	✗	4.188
279	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (c)	✗	0.006	✗	4.214