1.3.4 Hyperbolic PDE’s (Wave)

Table 1.9: Hyperbolic PDE’s (Wave) breakdown of results. Time in seconds

#	PDE	description	Mathematica		Maple		hand solved?	Animated?
			result	time	result	time
333	Wave PDE in 1D Finite length string	General solution for both ends fixed. Domain is \(0\dots L\)	✓	33.33	✓	27.737	Yes
334	Wave PDE in 1D Finite length string	both ends fixed, inital position zero (special case)	✓	3.324	✓	15.475	Yes	Yes
335	Wave PDE in 1D Finite length string	both ends fixed, inital velocity zero (special case)	✓	2.518	✓	10.231	Yes	Yes
336	Wave PDE in 1D Finite length string	both ends fixed but domain is \(-\pi \dots \pi \). zero intial position, non zero initial velocity	✓	6.978	✓	85.345	Yes	Yes
337	Wave PDE in 1D Finite length string	both ends fixed but domain is \(-1 \dots 1\). intial position is an impulse, zero initial velocity	✓	1.749	✓	23.163	Yes
338	Wave PDE in 1D Finite length string	Logan book, page 28. Both ends fixed	✓	0.974	✓	2.69
339	Wave PDE in 1D Finite length string	non-zero initial velocity. Both ends fixed	✓	86.507	✓	12.629
340	Wave PDE in 1D Finite length string	Logan book page 149)	✓	20.85	✓	13.871
341	Wave PDE in 1D Finite length string	Haberman 8.5.2 (a)	✓	23.852	✓	57.95
342	Wave PDE in 1D Finite length string	Haberman 8.5.2 (b)	✓	38.58	✓	37.831	Yes
343	Wave PDE in 1D Finite length string	Both I.C. not zero	✓	22.264	✓	18.398
344	Wave PDE in 1D Finite length string	With constant source	✓	34.232	✓	28.917
345	Wave PDE in 1D Finite length string	Logan page 213	✓	4.763	✓	20.611
346	Wave PDE in 1D Finite length string	Telegraphy PDE	✓	86.878	✓	20.632
347	Wave PDE in 1D Finite length string	Dispersion term present (general case)	✓ Due to adding dispersion term	25.235	✓	26.439	Yes
348	Wave PDE in 1D Finite length string	Dispersion term present	✓ Due to adding dispersion term	92.86	✓	110.806	Yes
349	Wave PDE in 1D Finite length string	Dispersion term present (specific case)	✓ Due to adding dispersion term	94.897	✓	25.956	Yes	Yes
350	Wave PDE in 1D Finite length string	non-zero initial position	✓ But sum should not include \(n=2\)	7.991	✓ Handled \(n=2\) case correctly	86.579
351	Wave PDE in 1D Finite length string	With source	✓	10.847	✓	83.973
352	Wave PDE in 1D Finite length string	Right end free (general case)	✓	25.353	✓	36.437	Yes
353	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (general case)	✓	7.149	✓	26.58	Yes
354	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (special case)	✓	89.446	✓	20.785	Yes	Yes
355	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (general case)	✓	52.43	✓	32.879	Yes
356	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, underdamped)	✓	114.95	✓	34.069	Yes	Yes
357	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, critical damped)	✓	114.627	✓	45.159	Yes	Yes
358	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, over damped)	✓	116.345	✓	32.581	Yes	Yes
359	Wave PDE in 1D Finite length string	I.C. at different times, right end free, with source	✗	40.028	✓	87.878	Yes
360	Wave PDE in 1D Finite length string	Right end oscillates	✓	66.861	✓	67.83
361	Wave PDE in 1D Finite length string	Perioidic B.C.	✗	2.412	✗	1.579
362	Wave PDE in 1D Finite length string	Mixed B.C.	✗	15.603	✗	1.116
363	Wave PDE in 1D Finite length string	Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions	✓	63.343	✓	27.013	Yes	Yes
364	Wave PDE in 1D Semi-infinite domain	Left end fixed, (general case)	✓	3.996	✗	1.541	Yes
365	Wave PDE in 1D Semi-infinite domain	Left end fixed with specific initial position	✓	15.596	✓	19.95	Yes
366	Wave PDE in 1D Semi-infinite domain	Logan page 115, left end fixed with source	✓	7.298	✓	2.138
367	Wave PDE in 1D Semi-infinite domain	Left moving boundary condition	✓	1.94	✓	1.46
368	Wave PDE in 1D Semi-infinite domain	moving Left end	✓	9.201	✓	3.219
369	Wave PDE in 1D Semi-infinite domain	I.C. at \(t=1\)	✓	9.147	✓	1.544
370	Wave PDE in 1D Semi-infinite domain	B.C. at \(x=1\)	✓	76.551	✓	3.115
371	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (general solution)	✓	7.123	✓	5.26
372	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (Special solution)	✓	4.202	✓	3.402	Yes	Yes
373	Wave PDE in 1D Semi-infinite domain	Left end fixed. zero initial velocity (Special solution)	✓	4.373	✓	3.247	Yes	Yes
374	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial position (general solution)	✓	8.691	✓	1.785
375	Wave PDE in 1D Semi-infinite domain	Left end free. Non zero initial position and velocity (general solution)	✓	33.339	✓	1.563
376	Wave PDE in 1D Semi-infinite domain	Left end free with source	✓	1.122	✓	1.379
377	Wave PDE in 1D Infinite domain	General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)	✓	0.019	✓	0.699
378	Wave PDE in 1D Infinite domain	General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)	✓	0.005	✓	0.538
379	Wave PDE in 1D Infinite domain	\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)	✓	0.158	✓	14.809
380	Wave PDE in 1D Infinite domain	No source. \(u_{tt} = u_{xx}\), with \(u(x,0) =e^{-x^2},u_t(x,0)=1\)	✓	0.002	✓	0.589
381	Wave PDE in 1D Infinite domain	With source term. \(u_{tt} = u_{xx} + m\)	✓	0.011	✓	0.437
382	Wave PDE in 1D Infinite domain	non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)	✓	0.028	✓	0.458	Yes	Yes
383	Wave PDE in 1D Infinite domain	Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)	✓	0.003	✓	0.329
384	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.029	✓	0.388
385	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.002	✓	0.304
386	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	8.846	✓	0.407
387	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.002	✓	0.309	Yes	Yes
388	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.004	✓	4.24	Yes	Yes
389	Wave PDE in 1D Infinite domain	General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)	✓	0.1	✓	0.401
390	Wave PDE in 1D Infinite domain	General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position	✓	0.003	✓	0.341	Yes	Yes
391	Wave PDE in 1D Infinite domain	\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)	✓	0.096	✓	0.737	Yes	Yes
392	Wave PDE in 1D Infinite domain	\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)	✓	63.956	✓	0.737	Yes
393	Wave PDE in 1D Infinite domain	system of 2 inhomogeneous linear hyperbolic system with constant coefficients	✓	0.267	✗	1.359
394	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, General solution	✗	2.271	✓	559.481	Yes
395	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, zero velocity. Specific example	✗	0.424	✓	30.771	Yes	Yes
396	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example	✓	33.848	✓	53.935	Yes	Yes
397	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example, delta in center	✓	9.295	✓	53.67	Yes	Yes
398	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed	✓	4.949	✓	29.946
399	Wave PDE in 2D Cartesian coordinates	All edges fixed (Haberman 8.5.5 (a)	✓	27.715	✗	0.932
400	Wave PDE in 2D Cartesian coordinates	2 edgs fixed, 2 free, zero initial velocity	✓	5.952	✓	58.452
401	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, general solution	✓	1.411	✗ (Timed out)	600.	Yes
402	Wave PDE in 2D Cartesian coordinates	With damping	✓	9.074	✓	87.58
403	Wave PDE in 2D Cartesian coordinates	On the whole plane	✓	0.171	✗	0.574
404	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency, fixed boundary, general case	✓	3.01	✓ Has unresolved Invlaplace calls	2.532	Yes
405	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	2.684	✓ Has unresolved Invlaplace calls	93.789
406	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	2.549	✓ Has unresolved Invlaplace calls. How to get series solution?	39.994	Yes	Yes
407	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✓	3.628	✓	18.789
408	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	2.511	✓	2.416
409	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, general solution	✓	9.409	✗	6.898	Yes
410	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, general solution	✓	4.687	✗	5.48	Yes
411	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, specific example	✓	15.449	✗	7.102	Yes	Yes
412	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position, specific example	✓	7.747	✗	6.869	Yes	Yes
413	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position with internal source (Haberman 8.5.5. (b)	✗	0.042	✗	0.692	Yes
414	Wave PDE in 3D Spherical coordinates	No I.C. no B.C.	✓	0.039	✓	5.077
415	Wave PDE in 3D Cylindrical coordinates	No I.C. no B.C.	✓	0.006	✓	0.887