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 \(0\dots L\)	✓	31.256	✓	5.257	Yes
334	Wave PDE in 1D Finite length string	both ends fixed, inital position zero (special case)	✓	30.778	✓	2.093	Yes	Yes
335	Wave PDE in 1D Finite length string	both ends fixed, inital velocity zero (special case)	✓	28.551	✓	1.545	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	✓	30.489	✓	13.026	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	✓	4.061	✓	3.058	Yes
338	Wave PDE in 1D Finite length string	Logan book, page 28. Both ends fixed	✓	0.932	✓	0.754
339	Wave PDE in 1D Finite length string	non-zero initial velocity. Both ends fixed	✓	87.438	✓	1.256
340	Wave PDE in 1D Finite length string	Logan book page 149)	✓	18.954	✓	2.06
341	Wave PDE in 1D Finite length string	Haberman 8.5.2 (a)	✓	21.679	✓	11.875
342	Wave PDE in 1D Finite length string	Haberman 8.5.2 (b)	✓	44.734	✓	7.368	Yes
343	Wave PDE in 1D Finite length string	Both I.C. not zero	✓	39.364	✓	2.855
344	Wave PDE in 1D Finite length string	With constant source	✓	38.044	✓	6.007
345	Wave PDE in 1D Finite length string	Logan page 213	✓	13.653	✓	4.996
346	Wave PDE in 1D Finite length string	Telegraphy PDE	✓	86.112	✓	2.658
347	Wave PDE in 1D Finite length string	Dispersion term present (general case)	✓ Due to adding dispersion term	34.446	✓	5.641	Yes
348	Wave PDE in 1D Finite length string	Dispersion term present	✓ Due to adding dispersion term	91.097	✓	17.417	Yes
349	Wave PDE in 1D Finite length string	Dispersion term present (specific case)	✓ Due to adding dispersion term	93.203	✓	4.549	Yes	Yes
350	Wave PDE in 1D Finite length string	non-zero initial position	✓ But sum should not include \(n=2\)	31.018	✓ Handled \(n=2\) case correctly	9.898
351	Wave PDE in 1D Finite length string	With source	✓	38.874	✓	15.175
352	Wave PDE in 1D Finite length string	Right end free (general case)	✓	24.766	✓	8.484	Yes
353	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (general case)	✓	5.795	✓	7.003	Yes
354	Wave PDE in 1D Finite length string	Right end free, zero initial velocity (special case)	✓	94.767	✓	4.104	Yes	Yes
355	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (general case)	✓	49.358	✓	8.853	Yes
356	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, underdamped)	✓	112.752	✓	5.928	Yes	Yes
357	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, critical damped)	✓	112.322	✓	7.007	Yes	Yes
358	Wave PDE in 1D Finite length string	Right end free, zero initial velocity, damping present (special case, over damped)	✓	114.153	✓	6.029	Yes	Yes
359	Wave PDE in 1D Finite length string	I.C. at different times, right end free, with source	✗	1.199	✓	60.707	Yes
360	Wave PDE in 1D Finite length string	Right end oscillates	✓	95.279	✓	9.051
361	Wave PDE in 1D Finite length string	Perioidic B.C.	✗	7.374	✗	1.578
362	Wave PDE in 1D Finite length string	Mixed B.C.	✗	22.566	✗	2.016
363	Wave PDE in 1D Finite length string	Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions	✓	72.198	✓	8.039	Yes	Yes
364	Wave PDE in 1D Semi-infinite domain	Left end fixed, (general case)	✓	2.204	✗	1.63	Yes
365	Wave PDE in 1D Semi-infinite domain	Left end fixed with specific initial position	✓	13.27	✓	2.888	Yes
366	Wave PDE in 1D Semi-infinite domain	Logan page 115, left end fixed with source	✓	0.169	✓	1.659
367	Wave PDE in 1D Semi-infinite domain	Left moving boundary condition	✓	7.618	✓	1.341
368	Wave PDE in 1D Semi-infinite domain	moving Left end	✓	13.244	✓	1.02
369	Wave PDE in 1D Semi-infinite domain	I.C. at \(t=1\)	✓	9.057	✓	0.283
370	Wave PDE in 1D Semi-infinite domain	B.C. at \(x=1\)	✓	71.399	✓	0.565
371	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (general solution)	✓	5.943	✓	0.347
372	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial velocity (Special solution)	✓	3.816	✓	0.73	Yes	Yes
373	Wave PDE in 1D Semi-infinite domain	Left end fixed. zero initial velocity (Special solution)	✓	2.893	✓	0.72	Yes	Yes
374	Wave PDE in 1D Semi-infinite domain	Left end free. zero initial position (general solution)	✓	7.456	✓	0.349
375	Wave PDE in 1D Semi-infinite domain	Left end free. Non zero initial position and velocity (general solution)	✓	30.559	✓	0.332
376	Wave PDE in 1D Semi-infinite domain	Left end free with source	✓	0.156	✓	0.253
377	Wave PDE in 1D Infinite domain	General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)	✓	0.017	✓	0.108
378	Wave PDE in 1D Infinite domain	General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)	✓	0.004	✓	0.178
379	Wave PDE in 1D Infinite domain	\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)	✓	0.149	✓	2.117
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	✓	0.095
381	Wave PDE in 1D Infinite domain	With source term. \(u_{tt} = u_{xx} + m\)	✓	0.01	✓	0.088
382	Wave PDE in 1D Infinite domain	non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)	✓	0.028	✓	0.132	Yes	Yes
383	Wave PDE in 1D Infinite domain	Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)	✓	0.003	✓	0.095
384	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.028	✓	0.075
385	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	0.002	✓	0.041
386	Wave PDE in 1D Infinite domain	Practice exam problem Math 5587	✓	71.005	✓	0.113
387	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.002	✓	0.043	Yes	Yes
388	Wave PDE in 1D Infinite domain	zero initial velocity	✓	0.004	✓	0.766	Yes	Yes
389	Wave PDE in 1D Infinite domain	General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)	✓	0.104	✓	0.079
390	Wave PDE in 1D Infinite domain	General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position	✓	0.003	✓	0.073	Yes	Yes
391	Wave PDE in 1D Infinite domain	\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)	✓	0.093	✓	0.115	Yes	Yes
392	Wave PDE in 1D Infinite domain	\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)	✓	0.472	✓	0.125	Yes
393	Wave PDE in 1D Infinite domain	system of 2 inhomogeneous linear hyperbolic system with constant coefficients	✓	0.262	✗	0.371
394	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, General solution	✗	2.344	✓	435.606	Yes
395	Wave PDE in 2D Cartesian coordinates	Rectangular membrane. Fixed on all edges, zero velocity. Specific example	✗	0.424	✓	6.533	Yes	Yes
396	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example	✓	35.568	✓	12.471	Yes	Yes
397	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, Specific example, delta in center	✓	9.778	✓	13.637	Yes	Yes
398	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed	✓	5.245	✓	6.812
399	Wave PDE in 2D Cartesian coordinates	All edges fixed (Haberman 8.5.5 (a)	✓	27.223	✗	1.017
400	Wave PDE in 2D Cartesian coordinates	2 edgs fixed, 2 free, zero initial velocity	✓	6.344	✓	11.118
401	Wave PDE in 2D Cartesian coordinates	All 4 edges fixed, zero initial velocity, general solution	✓	1.434	✗ (Timed out)	600.	Yes
402	Wave PDE in 2D Cartesian coordinates	With damping	✓	9.472	✓	28.236
403	Wave PDE in 2D Cartesian coordinates	On the whole plane	✓	0.133	✗	0.161
404	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency, fixed boundary, general case	✓	31.99	✓ Has unresolved Invlaplace calls	4.367	Yes
405	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	31.872	✓ Has unresolved Invlaplace calls	51.951
406	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Specific example. Both initial conditions not zero	✓	31.175	✓ Has unresolved Invlaplace calls. How to get series solution?	54.262	Yes	Yes
407	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	1.746	✓	16.58
408	Wave PDE in 2D Polar coordinates	no \(\theta \) dependency. Using integral transforms. Source present. Specific example	✗	1.853	✓	5.803
409	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, general solution	✓	9.756	✗	6.959	Yes
410	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, general solution	✓	4.485	✗	6.933	Yes
411	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial velocity, specific example	✓	15.091	✗	6.368	Yes	Yes
412	Wave PDE in 2D Polar coordinates	\(\theta \) dependency, fixed on edges, zero initial position, specific example	✓	7.533	✗	5.072	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)	✗	0.038	✗	4.683	Yes
414	Wave PDE in 3D Spherical coordinates	No I.C. no B.C.	✓	0.038	✓	5.734
415	Wave PDE in 3D Cylindrical coordinates	No I.C. no B.C.	✓	0.006	✓	0.309