1.3.3 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
151	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	General initial conditions	✓	62.537	✓	1.31	Yes
152	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Specific initial condition	✓	26.091	✓	0.619	Yes	Yes
153	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC \(u=x(1-x)\)	✓	27.383	✓	0.611	Yes	Yes
154	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (a)	✓	0.42	✓	3.055	Yes
155	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	IC hat function	✓	35.813	✓	1.385	Yes	Yes
156	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (b)	✓	1.236	✓	3.697	Yes
157	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (c)	✓ but \(n=3\) should be special case	65.187	✓ handled \(n=3\) case correctly.	5.432	Yes
158	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.3 (d)	✓	545.655	✓	26.013	Yes
159	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	18.891	✓	1.119	Yes
160	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	domain from -1 to +1	✓	62.92	✓	0.753	Yes	Yes
161	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (general case)	✓	42.268	✓	1.606	Yes
162	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space only (special case)	✓	32.815	✓	2.285	Yes	Yes
163	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (general case)	✓	71.123	✓	3.32	Yes
164	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	with source that depends on space and time (special case)	✓	35.558	✓	10.046	Yes	Yes
165	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Math 4567 Exam	✓	18.115	✓	1.938	Yes
166	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	With source	✓	67.959	✓	6.367
167	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	special initial condition	✓	3.189	✗ (Timed out)	600.	Yes
168	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction and Euler-Cauchy Sturm-Liouville	✓	1.816	✓	8.987
169	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Using growth form reaction term	✓	93.031	✓	0.83	Yes
170	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using logistic form for reaction term	✗	4.709	✗	27.471
171	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction, using Aleee form for reaction term	✗	7.192	✗	35.232
172	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Haberman 2.3.8	✓	34.92	✓	6.826	Yes
173	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	34.502	✓	1.482	Yes
174	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✓	0.774	✓	1.497
175	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction	✗	43.242	✓	1.147
176	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (general case)	✓	3.081	✓	2.102	Yes
177	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion convection (special case)	✓	63.055	✓	1.978	Yes	Yes
178	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.2 (general case)	✓	30.562	✓	2.112	Yes
179	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Left end zero, right end insulated, no source	✓	29.39	✓	2.041	Yes
180	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	One end insulated	✓	32.45	✓	2.24
181	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.3.7 (general case)	✓	32.108	✓	1.804	Yes
182	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	specific case	✓	12.122	✓	3.334	Yes	Yes
183	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (a)	✗ (Timed out)	600.	✓	26.393
184	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Both ends insulated, no source	✓	33.1	✓	2.065	Yes
185	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (b) (special case)	✓	1.606	✓	4.518	Yes
186	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (c) (special case)	✓	32.364	✓	6.848	Yes
187	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 2.4.1 (d)	✓	1.387	✓	4.021
188	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	both ends insulated	✓	91.769	✓	4.442
189	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	32.721	✓	3.185
190	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.25	✓	42.65	✓	8.53
191	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	external source	✓	18.726	✓	19.326
192	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction (general case)	✓	91.872	✓	0.938	Yes	Yes
193	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.23	✓	64.245	✓	31.669
194	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Pinchover and Rubinstein 6.21	✓	63.132	✓	3.556
195	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Diffusion Reaction. Euler-Cauchy Sturm-Liouville	✓	2.684	✓	26.115	Yes
196	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	convection heat loss	✓	32.892	✓	1.936	Yes
197	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Mixed BC	✓	33.75	✓	4.725
198	Diffusion in 1D Finite domain (bar), Both ends homogeneous BC	Haberman 8.2.1 (f)	✓	63.62	✓	4.279
199	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (general case)	✓	33.393	✓	4.627	Yes
200	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	left end insulated (special case)	✓	7.485	✓	1.614	Yes	Yes
201	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (general case)	✓	66.082	✓	2.418	Yes
202	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC (special case)	✓	0.01	✓	3.152	Yes
203	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	right end nonhomogeneous BC, special case	✓	10.202	✓	2.215	Yes	Yes
204	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	convection heat loss	✓	39.799	✗ (Timed out)	600.	Yes
205	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✗	2.176	✓	9.692
206	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	92.738	✓	1.904
207	Diffusion in 1D Finite domain (bar), left end homogeneous, right end not	nonhomogeneous BC	✓	34.188	✓	15.72	Yes
208	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (general case)	✓	66.569	✓	11.76	Yes
209	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	88.627	✓	15.321	Yes	Yes
210	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	left end BC depends on time (special case)	✓	86.742	✓	13.846	Yes	Yes
211	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	Haberman 8.3.6 (special case)	✓	95.024	✓	18.548	Yes	Yes
212	Diffusion in 1D Finite domain (bar), right end homogeneous, left end not	BC depends on time (special case)	✓	12.162	✓	5.8
213	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). IC given	✓	15.368	✓	10.554
214	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). No IC given	✓	0.014	✓	1.573	Yes
215	Diffusion in 1D Finite domain (bar), Periodic BC	Periodic BC (general case). Damped heat PDE. No IC given	✗	0.071	✓	1.459	Yes
216	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends non-homogeneous BC (general case)	✓	63.746	✓	4.237	Yes
217	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	non-homogeneous BC (special case)	✓	16.942	✓	2.648	Yes
218	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.1 (special case)	✓	63.188	✓	2.548	Yes
219	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (general case)	✓	52.613	✓	6.341	Yes
220	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space only (special case)	✓	94.816	✓	3.432	Yes	Yes
221	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	With source that depends on space and time only (general case)	✓	69.048	✓	9.333	Yes
222	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (general case)	✓	84.436	✓	17.345	Yes
223	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time (special case)	✓	67.824	✓	28.194	Yes	Yes
224	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	both ends nonhomogeneous	✓	63.852	✓	14.49	Yes
225	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (a) (general case)	✓	38.949	✓	6.474	Yes
226	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.1 (d) (general solution)	✓	54.039	✓	5.482	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)	✓	115.005	✗ (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)	✓	104.862	✓	29.86	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)	✓	79.308	✓	16.293	Yes
230	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Both ends depend on time with source present (special case)	✓	117.548	✓	63.454	Yes	Yes
231	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Pinchover and Rubinstein 6.17	✗	21.971	✓	11.401
232	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	nonhomogeneous BC	✓	111.737	✓	16.879
233	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Haberman 8.2.2. (a)	✓	122.354	✓	21.911	Yes
234	Diffusion in 1D Finite domain (bar), Both ends nonhomogeneous BC	Articolo 8.4.3	✓	5.486	✓	16.523
235	Diffusion in 1D Semi-infinite domain	left end constant (general case)	✓	60.035	✓	2.036	Yes
236	Diffusion in 1D Semi-infinite domain	left end constant (special case)	✓ It fail if assumption \(x>0\) is given. A bug	31.691	✓	2.772	Yes	Yes
237	Diffusion in 1D Semi-infinite domain	Logan p. 76. Left end general function of time (general case)	✓	0.577	✓	6.391	Yes
238	Diffusion in 1D Semi-infinite domain	Left end function of time (special case)	✓	60.047	✓	10.552	Yes	Yes
239	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.029	✓	2.204
240	Diffusion in 1D Semi-infinite domain	I.C. not zero	✓ due to IC/BC not zero	13.357	✓	2.941
241	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.074	✓	2.326
242	Diffusion in 1D Semi-infinite domain	nonhomogeneous BC	✓	60.071	✓	6.131
243	Diffusion in 1D Semi-infinite domain	nonhomogeneous B.C.	✓	60.033	✓	2.684
244	Diffusion in 1D Semi-infinite domain	Unit triangle I.C.	✓	61.013	✓	2.839
245	Diffusion in 1D Semi-infinite domain	I.C. not at \(t=0\)	✓	61.517	✓	4.622
246	Diffusion in 1D Semi-infinite domain	Diffusion with advection	✓	1.75	✓	4.813
247	Diffusion in 1D Semi-infinite domain	Practice exam problem	✓	60.108	✓	2.73
248	Diffusion in 1D Infinite domain	Inverse exponential I.C.	✓	0.869	✓	3.444	Yes
249	Diffusion in 1D Infinite domain	Advection term	✓	0.067	✓	2.516
250	Diffusion in 1D Infinite domain	UnitBox I.C.	✓	60.016	✓	4.32
251	Diffusion in 1D Infinite domain	No source	✓	0.4	✓	5.85
252	Diffusion in 1D Infinite domain	constant as source	✓	0.425	✓	2.873
253	Diffusion in 1D Infinite domain	No intial conditions	✓	0.004	✓	0.676	Yes
254	Diffusion in 1D Infinite domain	piecewise I.C.	✓ due to i.c. not at zero	52.493	✓	3.928	Yes
255	Diffusion in 1D Infinite domain	Practice exam problem	✓	0.065	✓	1.503
256	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	No source	✓	4.249	✓	14.339
257	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Internal source term	✓	10.246	✓	13.191
258	Diffusion in 2D Cartesian coordinates (Rectangle, Square)	Articolo 6.6.3	✓	6.641	✓	20.238
259	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (General solution)	✓	0.97	✓ Cant get series solution	51.553	Yes
260	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency, insulated (Specific solution)	✓	2.743	✓ Do not understand Maple solution	17.502	Yes	Yes
261	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	0.333	✓	1.819
262	Diffusion in 2D Polar coordinates (disk, sector, annulus)	no \(\theta \) dependency	✓	1.131	✓	1.511
263	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.3.5 (General solution)	✓	19.48	✓	5.353	Yes
264	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	3.516	✓	1.148	Yes	Yes
265	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Specific example of the above	✓	2.641	✗	1.184	Yes	Yes
266	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Inside ring	✓ \(n=1\) causs division by zero	3.449	✗ (Timed out)	600.
267	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.1	✓	3.973	✗	0.908
268	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Articolo 6.9.2	✓	4.38	✓	38.487
269	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Haberman 8.2.5 with \(\theta \) dependency (General case)	✓	310.649	✗	0.441	Yes
270	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (Specific example)	✓	8.757	✗	1.062	Yes	Yes
271	Diffusion in 2D Polar coordinates (disk, sector, annulus)	With \(\theta \) dependency (specific example)	✓	10.1	✗	28.405	Yes	Yes
272	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (General solution)	✓	4.65	✗	5.436	Yes
273	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✓	16.146	✗	9.737	Yes	Yes
274	Diffusion in 2D Polar coordinates (disk, sector, annulus)	Insulated with \(\theta \) dependency (Specific example)	✗ (Timed out)	600.	✗	9.36	Yes	Yes
275	Diffusion in 3D Spherical coordinates	No angle dependencies	✓	61.104	✓ Has unresolved Laplace integrals	3.518
276	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (a)	✓	1.883	✓	81.613	Yes	Yes
277	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (b)	✓	6.469	✗	30.911	Yes	Yes
278	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.4 (c)	✓	3.162	✗	27.302
279	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (a)	✗	0.012	✗	4.145
280	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (b)	✗	0.008	✗	4.083
281	Diffusion in 3D Cylinderical coordinates	Haberman 7.9.3 (c)	✗	0.012	✗	4.082