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
334	Wave PDE in 1D Finite length string	General solution for both ends fixed. Domain is \(0\dots L\)	✗ Surprising it could not solve this standard wave PDE on string	5.788	✓	5.997	Yes
335	Wave PDE in 1D Finite length string	both ends fixed, inital position zero (special case)	✓	31.082	✓	1.97	Yes	Yes
336	Wave PDE in 1D Finite length string	both ends fixed, inital velocity zero (special case)	✓	31.042	✓	1.723	Yes	Yes
337	Wave PDE in 1D Finite length string	both ends fixed but domain is \(-\pi \dots \pi \). zero intial position, non zero initial velocity	✗	51.161	✓	15.36	Yes	Yes
338	Wave PDE in 1D Finite length string	both ends fixed but domain is \(-1 \dots 1\). intial position is an impulse, zero initial velocity	✗	5.785	✓	1.727	Yes
339	Wave PDE in 1D Finite length string	Logan book, page 28. Both ends fixed	✗	0.422	✓	0.375
340	Wave PDE in 1D Finite length string	non-zero initial velocity. Both ends fixed	✗	27.558	✓	1.625
341	Wave PDE in 1D Finite length string	Logan book page 149)	✗	1.244	✓	2.446
342	Wave PDE in 1D Finite length string	Haberman 8.5.2 (a)	✗	1.242	✓	10.801
343	Wave PDE in 1D Finite length string	Haberman 8.5.2 (b)	✗	1.541	✓	7.278	Yes
344	Wave PDE in 1D Finite length string	Both I.C. not zero	✓	40.463	✓	2.398
345	Wave PDE in 1D Finite length string	With constant source	✗	0.965	✓	6.548
346	Wave PDE in 1D Finite length string	Logan page 213	✗	0.915	✓	4.085
347	Wave PDE in 1D Finite length string	Telegraphy PDE	✗	31.304	✓ But \(n = 1\) should not be included.	3.083
348	Wave PDE in 1D Finite length string	Dispersion term present (general case)	✗ Due to adding dispersion term	1.252	✓	4.715	Yes
349	Wave PDE in 1D Finite length string	Dispersion term present	✗ Due to adding dispersion term	31.593	✓	20.208	Yes
350	Wave PDE in 1D Finite length string	Dispersion term present (specific case)	✗ Due to adding dispersion term	31.468	✓	3.76	Yes	Yes
351	Wave PDE in 1D Finite length string	non-zero initial position	✓ But sum should not include \(n=2\)	31.137	✓ Handled \(n=2\) case correctly	9.71
352	Wave PDE in 1D Finite length string	With source	✗	30.99	✓	10.826
353	Wave PDE in 1D Finite length string	Right end free (general case)	✗	0.969	✓	7.717	Yes
354	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (general case)	✗	0.967	✓	5.08	Yes
355	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (special case)	✗	33.775	✓	2.87	Yes	Yes
356	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (general case)	✗	1.28	✓	6.279	Yes
357	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, underdamped)	✗	49.27	✓	5.674	Yes	Yes
358	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, critical damped)	✗	48.952	✓	5.414	Yes	Yes
359	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, over damped)	✗	49.823	✓	4.132	Yes	Yes
360	Wave PDE in 1D Finite length string	I.C. at different times, right end free, with source	✗	1.153	✓	54.131	Yes
361	Wave PDE in 1D Finite length string	Right end oscillates	✗	30.978	✓	11.337
362	Wave PDE in 1D Finite length string	Perioidic B.C.	✗	7.439	✗	2.564
363	Wave PDE in 1D Finite length string	Mixed B.C.	✗	21.322	✗	3.43
364	Wave PDE in 1D Finite length string	Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions	✗	0.963	✓	4.239	Yes	Yes
365	Wave PDE in 1D Semi-infinite domain	Left end fixed, (general case)	✓	2.14	✗	3.623	Yes
366	Wave PDE in 1D Semi-infinite domain	Left end fixed with specific initial position	✓	22.917	✓	3.145	Yes
367	Wave PDE in 1D Semi-infinite domain	Logan page 115, left end fixed with source	✓	0.183	✓	2.443
368	Wave PDE in 1D Semi-infinite domain	Left moving boundary condition	✓	7.585	✓	2.458
369	Wave PDE in 1D Semi-infinite domain	moving Left end	✓	17.273	✓	1.045
370	Wave PDE in 1D Semi-infinite domain	I.C. at \(t=1\)	✓	0.01	✓	0.33
371	Wave PDE in 1D Semi-infinite domain	B.C. at \(x=1\)	✗	10.107	✓	0.543
372	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (general solution)	✗	2.494	✓	0.341
373	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (Special solution)	✓	3.478	✓	0.713	Yes	Yes
374	Wave PDE in 1D Semi-infinite domain	Left end fixed. zero initial velocity (Special solution)	✓	2.6	✓	0.536	Yes	Yes
375	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial position (general solution)	✗	2.75	✓	0.369
376	Wave PDE in 1D Semi-infinite domain	Left end free. Non zero initial position and velocity (general solution)	✗	3.196	✓	0.321
377	Wave PDE in 1D Semi-infinite domain	Left end free with source	✓	0.182	✓	0.246
378	Wave PDE in 1D Infinite domain	General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)	✓	0.062	✓	0.056
379	Wave PDE in 1D Infinite domain	General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)	✓	0.005	✓	0.195
380	Wave PDE in 1D Infinite domain	\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)	✗	0.008	✓	1.505
381	Wave PDE in 1D Infinite domain	No source. \(u_{tt} = u_{xx}\), with \(u(x,0) =e^{-x^2},u_t(x,0)=1\)	✓	0.048	✓	0.084
382	Wave PDE in 1D Infinite domain	With source term. \(u_{tt} = u_{xx} + m\)	✓	0.01	✓	0.105
383	Wave PDE in 1D Infinite domain	non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)	✓	0.03	✓	0.166	Yes	Yes
384	Wave PDE in 1D Infinite domain	Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)	✓	0.003	✓	0.105
385	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.074	✓	0.086
386	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.031	✓	0.056
387	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	24.937	✓	0.132
388	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.075	✓	0.197	Yes	Yes
389	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.078	✓	0.316	Yes	Yes
390	Wave PDE in 1D Infinite domain	General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)	✓	0.134	✓	0.087
391	Wave PDE in 1D Infinite domain	General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position	✓	0.032	✓	0.118	Yes	Yes
392	Wave PDE in 1D Infinite domain	\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)	✓	0.095	✓	0.116	Yes	Yes
393	Wave PDE in 1D Infinite domain	\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)	✗	35.84	✓	0.06	Yes
394	Wave PDE in 1D Infinite domain	system of 2 inhomogeneous linear hyperbolic system with constant coefficients	✓	0.279	✗	0.359
395	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, General solution	✗	0.028	✓	184.836	Yes
396	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, zero velocity. Specific example	✗	0.023	✓	2.036	Yes	Yes
397	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example	✓	5.053	✓	12.351	Yes	Yes
398	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example, delta in center	✓	1.229	✓	10.169	Yes	Yes
399	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed	✓	0.997	✓	5.191
400	Wave PDE in 2D Cartesian coordinates	All edges fixed (Haberman 8.5.5 (a)	✗	0.003	✗	2.079
401	Wave PDE in 2D Cartesian coordinates	2 edgs fixed, 2 free, zero initial velocity	✗	0.003	✓	9.615
402	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, general solution	✓	0.1	✗ (Timed out)	600.	Yes
403	Wave PDE in 2D Cartesian coordinates	With damping	✗	0.003	✓	18.699
404	Wave PDE in 2D Cartesian coordinates	On the whole plane	✓	0.104	✗	0.086
405	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency, fixed boundary, general case	✓	0.723	✓ Has unresolved Invlaplace calls	0.785	Yes
406	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	31.224	✓ Has unresolved Invlaplace calls	42.668
407	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	31.077	✓ Has unresolved Invlaplace calls. How to get series solution?	6.716	Yes	Yes
408	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	1.772	✓	10.237
409	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	1.954	✓	3.56
410	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, general solution	✗	0.004	✗	1.901	Yes
411	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, general solution	✗	0.004	✗	4.776	Yes
412	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, specific example	✗	0.004	✗	1.127	Yes	Yes
413	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position, specific example	✗	0.005	✗	1.689	Yes	Yes
414	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position with internal source (Haberman 8.5.5. (b)	✗	0.027	✗	0.398	Yes
415	Wave PDE in 3D Spherical coordinates	No I.C. no B.C.	✗	0.031	✓	3.537
416	Wave PDE in 3D Cylindrical coordinates	No I.C. no B.C.	✗	0.003	✓	0.268