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
151	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	General initial conditions	✓	30.383	✓	1.527	Yes
152	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Specific initial condition	✓	8.837	✓	0.577	Yes	Yes
153	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC \(u=x(1-x)\)	✓	14.949	✓	0.467	Yes	Yes
154	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (a)	✓	0.416	✓	2.448	Yes
155	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC hat function	✓	18.465	✓	1.688	Yes	Yes
156	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (b)	✓	1.109	✓	2.759	Yes
157	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (c)	✓ but \(n=3\) should be special case	0.886	✓ handled \(n=3\) case correctly.	5.885	Yes
158	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (d)	✓	0.817	✓	2.198	Yes
159	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✗	8.409	✓	0.643	Yes
160	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✗	10.344	✓	0.562	Yes	Yes
161	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (general case)	✗	31.709	✓	1.676	Yes
162	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (special case)	✓	30.659	✓	1.392	Yes	Yes
163	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (general case)	✗	2.272	✓	2.266	Yes
164	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (special case)	✓	33.479	✓	7.534	Yes	Yes
165	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Math 4567 Exam	✓	18.088	✓	1.585	Yes
166	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	With source	✗	1.951	✓	4.825
167	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	special initial condition	✓	2.705	✗ (Timed out)	600.	Yes
168	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction and Euler-Cauchy Sturm-Liouville	✗	1.185	✓	2.078
169	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Using growth form reaction term	✗	21.792	✓	0.531	Yes
170	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using logistic form for reaction term	✗	1.614	✗	3.782
171	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using Aleee form for reaction term	✗	2.769	✗	5.574
172	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Haberman 2.3.8	✗	1.249	✓	1.476	Yes
173	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✗	1.033	✓	1.533	Yes
174	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	0.65	✓	1.031
175	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✗	40.533	✓	0.819
176	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (general case)	✗	1.257	✗	3.387	Yes
177	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (special case)	✗	31.408	✗ (Timed out)	600.	Yes	Yes
178	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.2 (general case)	✓	0.366	✓	1.772	Yes
179	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Left end zero, right end insulated, no source	✓	0.324	✓	1.086	Yes
180	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	One end insulated	✓	30.734	✓	1.54
181	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.7 (general case)	✓	0.402	✓	1.166	Yes
182	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	specific case	✓	6.115	✓	0.694	Yes	Yes
183	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (a)	✓	0.549	✓	3.674
184	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Both ends insulated, no source	✓	13.565	✓	1.255	Yes
185	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (b) (special case)	✓	0.562	✓	3.112	Yes
186	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (c) (special case)	✓	0.846	✓	6.312	Yes
187	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (d)	✓	0.391	✓	2.8
188	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	both ends insulated	✗	19.042	✓	3.996
189	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✗	30.572	✓	0.796
190	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.25	✗	0.989	✓	7.672
191	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	external source	✗	0.527	✓	18.664
192	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction (general case)	✗	28.869	✓	0.726	Yes	Yes
193	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.23	✓	1.329	✓	26.237
194	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.21	✓	0.374	✓	2.572
195	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Euler-Cauchy Sturm-Liouville	✗	1.665	✗ (Timed out)	600.	Yes
196	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✗	31.058	✓	1.952	Yes
197	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Mixed BC	✗	1.006	✓	3.45
198	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 8.2.1 (f)	✗	0.52	✓	3.463
199	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (general case)	✓	2.295	✓	2.016	Yes
200	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (special case)	✓	5.676	✓	1.126	Yes	Yes
201	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (general case)	✓	32.608	✓	0.957	Yes
202	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (special case)	✓	0.009	✓	2.862	Yes
203	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC, special case	✓	10.052	✓	2.355	Yes	Yes
204	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	convection heat loss	✗	12.542	✗ (Timed out)	600.	Yes
205	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	0.397	✓	5.377
206	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	6.798	✓	1.171
207	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	31.601	✓	9.329	Yes
208	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (general case)	✗	0.49	✓	5.963	Yes
209	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✗	14.632	✓	6.313	Yes	Yes
210	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✗	18.619	✓	4.6	Yes	Yes
211	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	Haberman 8.3.6 (special case)	✗	30.966	✓	16.03	Yes	Yes
212	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	BC depends on time (special case)	✓	7.153	✓	5.204
213	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). IC given	✗	0.492	✓	5.799
214	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). No IC given	✗	0.007	✓	2.203	Yes
215	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). Damped heat PDE. No IC given	✗	0.007	✓	2.229	Yes
216	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends non-homogeneous BC (general case)	✓	31.519	✓	3.014	Yes
217	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	non-homogeneous BC (special case)	✓	19.58	✓	1.43	Yes
218	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.1 (special case)	✓	19.845	✓	1.498	Yes
219	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (general case)	✗	31.438	✓	3.576	Yes
220	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (special case)	✗	30.815	✓	2.017	Yes	Yes
221	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space and time only (general case)	✗	1.307	✓	6.288	Yes
222	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (general case)	✗	8.714	✓	17.586	Yes
223	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (special case)	✓	35.85	✓	13.062	Yes	Yes
224	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends nonhomogeneous	✗	30.804	✓	11.831	Yes
225	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (a) (general case)	✗	0.991	✓	3.046	Yes
226	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (d) (general solution)	✗	31.075	✓	1.733	Yes
227	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source that depends on space only (general solution)	✗	1.194	✗ (Timed out)	600.	Yes
228	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source that depends on space only (special case)	✗	30.834	✓	5.136	Yes	Yes
229	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)	✗	1.318	✓	2.434	Yes
230	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source present (special case)	✗	30.968	✓	7.	Yes	Yes
231	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Pinchover and Rubinstein 6.17	✗	1.967	✓	1.941
232	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	nonhomogeneous BC	✗	30.863	✓	4.252
233	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.2. (a)	✗	0.607	✓	3.23	Yes
234	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.3	✗	31.014	✓	7.222
235	Diffusion in 1D Semi-infinite domain	left end constant (general case)	✓	390.937	✓	0.439	Yes
236	Diffusion in 1D Semi-infinite domain	left end constant (special case)	✓ It fail if assumption \(x>0\) is given. A bug	13.432	✓	0.925	Yes	Yes
237	Diffusion in 1D Semi-infinite domain	Logan p. 76. Left end general function of time (general case)	✓	0.56	✓	1.139	Yes
238	Diffusion in 1D Semi-infinite domain	Left end function of time (special case)	✓	133.902	✓	1.431	Yes	Yes
239	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	390.726	✓	0.51
240	Diffusion in 1D Semi-infinite domain	I.C. not zero	✗ due to IC/BC not zero	0.008	✓	0.599
241	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	394.948	✓	0.55
242	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	38.713	✓	1.905
243	Diffusion in 1D Semi-infinite domain	nonhomogeneous B.C.	✓	3.438	✓	0.99
244	Diffusion in 1D Semi-infinite domain	Unit triangle I.C.	✓	29.149	✓	0.792
245	Diffusion in 1D Semi-infinite domain	I.C. not at \(t=0\)	✗	0.008	✓	0.596
246	Diffusion in 1D Semi-infinite domain	Diffusion with advection	✗	0.828	✓	1.413
247	Diffusion in 1D Semi-infinite domain	Practice exam problem	✓	26.449	✓	0.8
248	Diffusion in 1D Infinite domain	Inverse exponential I.C.	✓	1.542	✓	0.857	Yes
249	Diffusion in 1D Infinite domain	Advection term	✓	0.51	✓	0.824
250	Diffusion in 1D Infinite domain	UnitBox I.C.	✓	1.974	✓	1.225
251	Diffusion in 1D Infinite domain	No source	✓	0.211	✓	1.515
252	Diffusion in 1D Infinite domain	constant as source	✓	7.7	✓	1.086
253	Diffusion in 1D Infinite domain	No intial conditions	✗	0.002	✓	0.109	Yes
254	Diffusion in 1D Infinite domain	piecewise I.C.	✗ due to i.c. not at zero	0.011	✓	0.815	Yes
255	Diffusion in 1D Infinite domain	Practice exam problem	✓	0.008	✓	0.304
256	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	No source	✗	0.003	✓	1.75
257	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Internal source term	✗	0.004	✓	1.602
258	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Articolo 6.6.3	✗	0.003	✓	3.749
259	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (General solution)	✗	2.378	✓ Cant get series solution	15.478	Yes
260	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (Specific solution)	✗	1.88	✓ Do not understand Maple solution	6.748	Yes	Yes
261	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✗	2.27	✓	1.276
262	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	0.914	✓	0.312
263	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.3.5 (General solution)	✗	2.176	✓	1.565	Yes
264	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✗	1.734	✓	1.463	Yes	Yes
265	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✗	1.74	✗	1.765	Yes	Yes
266	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.1	✗	0.004	✗	1.387
267	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.2	✗	0.004	✓	6.82
268	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.2.5 with \(\theta \) dependency (General case)	✗	0.004	✗	0.515	Yes
269	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (Specific example)	✗	0.004	✗	1.213	Yes	Yes
270	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (specific example)	✗	0.004	✗	1.144	Yes	Yes
271	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (General solution)	✗	0.004	✗	2.447	Yes
272	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✗	0.004	✗	2.788	Yes	Yes
273	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✗	0.004	✗	2.141	Yes	Yes
274	Diffusion in 3D Spherical coordinates	No angle dependencies	✗	2.195	✓ Has unresolved Laplace integrals	1.151
275	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (a)	✗	0.004	✓	11.908	Yes	Yes
276	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (b)	✗	0.003	✗	6.707	Yes	Yes
277	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (c)	✗	0.003	✗	6.136
278	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (a)	✗	0.004	✗	1.939
279	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (b)	✗	0.003	✗	2.63
280	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (c)	✗	0.004	✗	1.612