1.3.4 Hyperbolic PDE’s (Wave)

Table 1.12: 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 \(0\dots L\)	✓	56.56	✓	2.944	Yes
334	Wave PDE in 1D Finite length string	both ends fixed, inital position zero (special case)	✓	1.922	✓	1.158	Yes	Yes
335	Wave PDE in 1D Finite length string	both ends fixed, inital velocity zero (special case)	✓	1.518	✓	1.221	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	✓	130.55	✓	4.77	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	✓	0.426	✓	1.542	Yes
338	Wave PDE in 1D Finite length string	Logan book, page 28. Both ends fixed	✓	0.617	✓	0.476
339	Wave PDE in 1D Finite length string	non-zero initial velocity. Both ends fixed	✓	114.736	✓	0.92
340	Wave PDE in 1D Finite length string	Logan book page 149)	✓	13.944	✓	1.034
341	Wave PDE in 1D Finite length string	Haberman 8.5.2 (a)	✓	15.499	✓	5.109
342	Wave PDE in 1D Finite length string	Haberman 8.5.2 (b)	✓	79.383	✓	4.204	Yes
343	Wave PDE in 1D Finite length string	Both I.C. not zero	✓	142.574	✓	1.571
344	Wave PDE in 1D Finite length string	With constant source	✓	65.392	✓	3.524
345	Wave PDE in 1D Finite length string	Logan page 213	✓	8.322	✓	2.791
346	Wave PDE in 1D Finite length string	Telegraphy PDE	✓	128.95	✓	1.691
347	Wave PDE in 1D Finite length string	Dispersion term present (general case)	✓ Due to adding dispersion term	94.668	✓	2.861	Yes
348	Wave PDE in 1D Finite length string	Dispersion term present	✓ Due to adding dispersion term	175.105	✓	4.205	Yes
349	Wave PDE in 1D Finite length string	Dispersion term present (specific case)	✓ Due to adding dispersion term	105.105	✓	1.926	Yes	Yes
350	Wave PDE in 1D Finite length string	non-zero initial position	✓ But sum should not include \(n=2\)	110.291	✓ Handled \(n=2\) case correctly	4.448
351	Wave PDE in 1D Finite length string	With source	✓	17.787	✓	4.07
352	Wave PDE in 1D Finite length string	Right end free (general case)	✓	25.165	✓	4.099	Yes
353	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (general case)	✓	11.492	✓	3.48	Yes
354	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (special case)	✓	70.972	✓	1.686	Yes	Yes
355	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (general case)	✓	43.742	✓	5.961	Yes
356	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, underdamped)	✓	96.738	✓	2.967	Yes	Yes
357	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, critical damped)	✓	96.689	✓	3.183	Yes	Yes
358	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, over damped)	✓	98.121	✓	3.162	Yes	Yes
359	Wave PDE in 1D Finite length string	I.C. at different times, right end free, with source	✗	34.86	✓	29.077	Yes
360	Wave PDE in 1D Finite length string	Right end oscillates	✓	72.812	✓	3.504
361	Wave PDE in 1D Finite length string	Perioidic B.C.	✗	1.94	✗	0.419
362	Wave PDE in 1D Finite length string	Mixed B.C.	✗	2.49	✗	0.372
363	Wave PDE in 1D Finite length string	Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions	✓	63.273	✓	3.765	Yes	Yes
364	Wave PDE in 1D Semi-infinite domain	Left end fixed, (general case)	✓	8.161	✓	0.394	Yes
365	Wave PDE in 1D Semi-infinite domain	Left end fixed with specific initial position	✓	2.799	✓	1.048	Yes
366	Wave PDE in 1D Semi-infinite domain	Logan page 115, left end fixed with source	✓	4.677	✓	0.399
367	Wave PDE in 1D Semi-infinite domain	Left moving boundary condition	✓	1.304	✓	0.411
368	Wave PDE in 1D Semi-infinite domain	moving Left end	✓	15.944	✓	0.817
369	Wave PDE in 1D Semi-infinite domain	I.C. at \(t=1\)	✓	7.006	✓	0.386
370	Wave PDE in 1D Semi-infinite domain	B.C. at \(x=1\)	✗	77.586	✓	0.422
371	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (general solution)	✓	8.515	✓	0.31
372	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (Special solution)	✓	3.773	✓	0.431	Yes	Yes
373	Wave PDE in 1D Semi-infinite domain	Left end fixed. zero initial velocity (Special solution)	✓	3.599	✓	0.385	Yes	Yes
374	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial position (general solution)	✓	9.447	✓	0.391
375	Wave PDE in 1D Semi-infinite domain	Left end free. Non zero initial position and velocity (general solution)	✓	30.77	✓	0.409
376	Wave PDE in 1D Semi-infinite domain	Left end free with source	✓	1.403	✓	0.286
377	Wave PDE in 1D Infinite domain	General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)	✓	0.022	✓	0.229
378	Wave PDE in 1D Infinite domain	General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)	✓	0.006	✓	0.286
379	Wave PDE in 1D Infinite domain	\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)	✓	0.11	✓	0.775
380	Wave PDE in 1D Infinite domain	No source. \(u_{tt} = u_{xx}\), with \(u(x,0) =e^{-x^2},u_t(x,0)=1\)	✓	0.004	✓	0.161
381	Wave PDE in 1D Infinite domain	With source term. \(u_{tt} = u_{xx} + m\)	✓	0.011	✓	0.218
382	Wave PDE in 1D Infinite domain	non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)	✓	0.029	✓	0.175	Yes	Yes
383	Wave PDE in 1D Infinite domain	Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)	✓	0.004	✓	0.206
384	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.511	✓	0.225
385	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.003	✓	0.169
386	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	7.194	✓	0.236
387	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.003	✓	0.141	Yes	Yes
388	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.004	✓	0.323	Yes	Yes
389	Wave PDE in 1D Infinite domain	General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)	✓	0.099	✓	0.239
390	Wave PDE in 1D Infinite domain	General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position	✓	0.003	✓	0.185	Yes	Yes
391	Wave PDE in 1D Infinite domain	\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)	✓	1.786	✓	0.253	Yes	Yes
392	Wave PDE in 1D Infinite domain	\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)	✓	81.489	✓	0.18	Yes
393	Wave PDE in 1D Infinite domain	system of 2 inhomogeneous linear hyperbolic system with constant coefficients	✓	0.211	✗	0.396
394	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, General solution	✓	1.341	✓	58.092	Yes
395	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, zero velocity. Specific example	✓	0.294	✓	2.216	Yes	Yes
396	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example	✓	32.485	✓	2.34	Yes	Yes
397	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example, delta in center	✓	8.389	✓	4.691	Yes	Yes
398	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed	✓	4.849	✓	2.317
399	Wave PDE in 2D Cartesian coordinates	All edges fixed (Haberman 8.5.5 (a)	✓	21.957	✗	0.247
400	Wave PDE in 2D Cartesian coordinates	2 edgs fixed, 2 free, zero initial velocity	✓	5.353	✓	2.994
401	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, general solution	✓	1.003	✓	174.896	Yes
402	Wave PDE in 2D Cartesian coordinates	With damping	✓	6.8	✓	6.113
403	Wave PDE in 2D Cartesian coordinates	On the whole plane	✗	0.321	✗	0.208
404	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency, fixed boundary, general case	✓	32.335	✓ Has unresolved Invlaplace calls	0.592	Yes
405	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	30.194	✓ Has unresolved Invlaplace calls	84.162
406	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	20.785	✓ Has unresolved Invlaplace calls. How to get series solution?	1.841	Yes	Yes
407	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✓	3.323	✓	1.446
408	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	2.032	✓	0.544
409	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, general solution	✓	6.547	✗	1.552	Yes
410	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, general solution	✓	3.125	✗	1.579	Yes
411	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, specific example	✓	15.046	✗	0.828	Yes	Yes
412	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position, specific example	✓	5.509	✗	0.872	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)	✗	0.034	✗	0.244	Yes
414	Wave PDE in 3D Spherical coordinates	No I.C. no B.C.	✓	0.031	✓	1.414
415	Wave PDE in 3D Cylindrical coordinates	No I.C. no B.C.	✓	0.004	✓	0.558