1.3.1 Miscellaneous PDE’s

Table 1.6: Miscellaneous PDE’s breakdown of results. Time in seconds

#	PDE	description	Mathematica		Maple		hand solved?	Animated?
			result	time	result	time
1	General first order	Transport equation \(u_t+ u_x = 0\)	✓	0.031	✓	0.097	Yes
2	General first order	Transport equation \(u_t-3 u_x = 0\) IC \(u(0,x)=e^{-x^2}\). Peter Olver textbook, 2.2.2 (a)	✓	0.005	✓	0.128	Yes	Yes
3	General first order	Transport equation \(u_t+2 u_x = 0\) IC \(u(-1,x)=\frac {x}{1+x^2}\). Peter Olver textbook, 2.2.2 (b)	✓	0.006	✓	0.125	Yes	Yes
4	General first order	Transport equation \(u_t+u_x+\frac {1}{2}u = 0\) IC \(u(0,x)=\arctan (x)\). Peter Olver textbook, 2.2.2 (c)	✓	0.004	✓	0.149	Yes	Yes
5	General first order	Transport equation \(u_t-4u_x+u = 0\) IC \(u(0,x)=\frac {1}{1+x^2}\). Peter Olver textbook, 2.2.2 (d)	✓	0.004	✓	0.161	Yes	Yes
6	General first order	Transport equation \(u_t+2 u_x= \sin x\) IC \(u(0,x)=\sin x\). Peter Olver textbook, 2.2.5	✓	0.046	✓	0.21	Yes	Yes
7	General first order	Transport equation \(u_t+\frac {1}{1+x^2} u_x= 0\) IC \(u(x,0)=\frac {1}{1+(3+x)^2}\). Peter Olver textbook, page 27	✓	0.012	✓	0.228	Yes	Yes
8	General first order	Transport equation \(u_t-x u_x= 0\) IC \(u(x,0)=\frac {1}{1+x^2}\). Peter Olver textbook, problem 2.2.17	✓	0.005	✓	0.215	Yes	Yes
9	General first order	Transport equation \(u_t+(1-2 t) u_x= 0\) IC \(u(x,0)=\frac {1}{1+x^2}\). Peter Olver textbook, problem 2.2.29	✓	0.048	✓	0.29	Yes	Yes
10	General first order	Transport equation \(u_t+\frac {1}{x^2+4} u_x= 0\) IC \(u(x,0)=e^{x^3+12 x}\)	✓	0.008	✓	0.275	Yes	Yes
11	General first order	\(3 u_x + 5 u_y = x\)	✓	0.005	✓	0.113	Yes
12	General first order	\(x u_y + y u_x = -4 x y u\) and \(u(x,0)=e^{-x^2}\)	✓	0.019	✓	0.241	Yes
13	General first order	\(u_t + u_x = 0\) and \(u(x,0)=\sin x\) and \(u(0,t)=0\)	✓	0.122	✓	0.384	Yes
14	General first order	\(u_t+ c u_x = 0\) and \(u(x,0)=e^{-x^2}\)	✓	0.005	✓	0.128	Yes
15	General first order	(Haberman 12.2.2) \(\omega _t -3 \omega _x = 0\) and \(\omega (x,0)=\cos x\)	✓	0.005	✓	0.141	Yes
16	General first order	(Haberman 12.2.4) \(\omega _t +c \omega _x = 0\) and \(\omega (x,0)=f(x)\) and \(\omega (0,t)=h(t)\)	✗	1.75	✓ Solution contains unresolved invlaplace calls	0.418	Yes
17	General first order	(Haberman 12.2.5 (a)) \(\omega _t +c \omega _x = e^{2 x}\) and \(\omega (x,0)=f(x)\)	✓	0.057	✓	0.169	Yes
18	General first order	(Haberman 12.2.5 (d)) \(\omega _t +3 t \omega _x = \omega (x,t)\) and \(\omega (x,0)=f(x)\)	✓	0.392	✓	0.281	Yes
19	General first order	\( 2 u_x + 5 u_y = u^2(x,y) + 1\)	✓	0.087	✓	0.214	Yes
20	General first order	Clairaut equation \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\)	✓	0.041	✓	0.224	Yes
21	General first order	Clairaut equation. \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\) with \(u(x,0)= \frac {1}{2} (1-x^2)\)	✓	0.009	✓	0.829
22	General first order	Clairaut equation. \(u = x u_x+ y u_y + \sin ( u_x + u_y )\)	✓	0.034	✓	0.067
23	General first order	Recover a function from its gradient vector	✓	0.045	✓	0.267
24	General first order	\(x f_y - f_x = \frac {g(x)}{h(y)} f^2\)	✓	0.034	✓	0.174	Yes
25	General first order	\(f_x + (f_y)^2 = f(x,y,z)+z\)	✓	0.067	✓	0.627
26	General first order	\(x u_x+y u_y=u\) (Example 3.5.1 in Lokenath Debnath)	✓	0.017	✓	0.104	Yes
27	General first order	\(x u_x+y u_y=n u\) Example 3.5.2 in Lokenath Debnath	✓	0.013	✓	0.105	Yes
28	General first order	\(x^2 u_x+y^2 u_y=(x+y) u\) Example 3.5.3 in Lokenath Debnath	✓	0.069	✓	0.153	Yes
29	General first order	\((y-z) u_x + (z-x) u_y + (x-y) u_z = 0\) (Example 3.5.4 in Lokenath Debnath)	✓	9.864	✓	4.685	Yes
30	General first order	\(u(x+y) u_x+u(x-y) u_y=x^2+y^2\) (Example 3.5.5 in Lokenath Debnath)	✓	0.257	✓	0.858	Yes
31	General first order	\(u_x-u_y=1\) with \(u(x,0)=x^2\) Example 3.5.6 in Lokenath Debnath	✓	0.006	✓	0.11
32	General first order	\(y u_x+x u_y=u\) with \(u(x,0)=x^3\) and \(u(0,y)=y^3\) Example 3.5.8 in Lokenath Debnath	✗	1.238	✗	0.589
33	General first order	\(x u_x+y u_y=x e^{-u}\) with \(u=0\) on \(y=x^2\) Example 3.5.10 in Lokenath Debnath	✓	0.105	✓	0.16
34	General first order	\(u_t+u u_x=x\) with \(u(x,0)=f(x)\) Example 3.5.11 in Lokenath Debnath.	✗	2.793	✓	0.517
35	General first order	\(u_x=0\) Problem 3.3(a) Lokenath Debnath	✓	0.004	✓	0.07
36	General first order	\(a u_x+b u_y=0\) Problem 3.3(b) Lokenath Debnath	✓	0.007	✓	0.155
37	General first order	\(u_x+y u_y=0\) Problem 3.3(c) Lokenath Debnath	✓	0.021	✓	0.238
38	General first order	\((1+x^2) u_x+ u_y=0\) Problem 3.3(d) Lokenath Debnath	✓	0.009	✓	0.198
39	General first order	\(2 x y u_x+(x^2+y^2)u_y=0\) Problem 3.3(e) Lokenath Debnath	✓	0.073	✓	0.373
40	General first order	\((y+u) u_x+y u_y=x-y\) Problem 3.3(f) Lokenath Debnath	✓	7.176	✓	0.912
41	General first order	\(y^2 u_x- x y u_y=x(u-2 y)\) Problem 3.3(g) Lokenath Debnath	✓	0.07	✓	0.421
42	General first order	\(y u_y - x u_x = 1\) Problem 3.3(h) Lokenath Debnath	✓	0.014	✓	0.202
43	General first order	\(u_x+2 x y^2 u_y=0\) Problem 3.4 Lokenath Debnath	✓	0.048	✓	0.22
44	General first order	\(3 u_x+2 u_y=0\) with \(u(x,0)=\sin x\). Problem 3.5(a) Lokenath Debnath	✓	0.004	✓	0.195
45	General first order	\(y u_x+x u_y=0\) with \(u(0,y)=e^{-y^2}\). Problem 3.5(b) Lokenath Debnath	✓	0.017	✓	0.492
46	General first order	\(x u_x+y u_y=2 x y\) with \(u=2\) on \(y=x^2\). Problem 3.5(c) Lokenath Debnath	✓	0.011	✓	0.179
47	General first order	\(u_x+x u_y=0\) with \(u(0,y)=\sin y\). Problem 3.5(d) Lokenath Debnath	✓	0.006	✓	0.237
48	General first order	\(y u_x+x u_y=x y\) with \(u(0,y)=e^{-y^2},u(x,0)=e^{-x^2}\). Problem 3.5(e) Lokenath Debnath	✗	2.921	✗	0.812
49	General first order	\(u_x+x u_y=(y-\frac {1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath	✓	0.008	✓	0.314
50	General first order	\(x u_x+y u_y=u+1\) with \(u=x^2\) on \(y=x^2\) Problem 3.5(g) Lokenath Debnath	✓	0.014	✓	0.179
51	General first order	\(u u_x - u u_y= u^2 + (x+y)^2\) with \(u(x,0)=1\) Problem 3.5(h) Lokenath Debnath	✓	0.031	✓	0.417
52	General first order	\(x u_x+(x+y)u_y=u+1\) with \(u(x,0)=x^2\) Problem 3.5(i) Lokenath Debnath	✓	0.021	✓	0.402
53	General first order	\(x u_x+y u_y+z u_z=0\) Problem 3.8(a) .Lokenath Debnath	✓	0.023	✓	0.17
54	General first order	\(x^2 u_x+y^2 u_y+z(x+y)u_z=0\) Problem 3.8(b) Lokenath Debnath	✓	0.051	✓	0.307
55	General first order	\(x(y-z)u_x+y(z-x)u_y+z(x-y)u_z=0\) Problem 3.8(c) Lokenath Debnath	✓	0.023	✓	1.173
56	General first order	\(y z u_x - x z u_y+ x y (x^2+y^2) u_z=0\) Problem 3.8(d) Lokenath Debnath	✓	0.058	✓	0.379
57	General first order	\(x(y^2-z^2) u_x + y(z^2-y^2) u_y+ z (x^2-y^2) u_z=0\) Problem 3.8(e) Lokenath Debnath	✗	35.91	✗	0.457
58	General first order	\(u_x+x u_y=y\) with \(u(0,y)=y^2\) Problem 3.9(a) Lokenath Debnath	✓	0.006	✓	0.219
59	General first order	\(u_x+x u_y=y\) with \(u(1,y)=2 y\) Problem 3.9(b) Lokenath Debnath	✓	0.007	✓	0.189
60	General first order	\((u_x+u_y)^2-u^2=0\). Problem 3.10 Lokenath Debnath	✓	0.009	✓	0.189
61	General first order	\((y+u)u_x+y u_y=x-y\) with \(u(x,1)=1+x\). Problem 3.11 Lokenath Debnath	✓	14.71	✗	1.484
62	General first order	\(2 x u_x+(x+1) u_y=y\) with \(u(1,y)=2 y\). Problem 3.14(d) Lokenath Debnath	✓	0.022	✓	0.297
63	General first order	\(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath	✓	0.022	✓	0.28
64	General first order	\(y^2 u_x+(x y) u_y=x\) with \(u(x,1)=x^2\). Problem 3.14(f) Lokenath Debnath	✓	0.023	✓	0.387
65	General first order	\(x u_x+y u_y=x y\) with \(u=\frac {x^2}{2}\) at \(y=x\). Problem 3.14(g) Lokenath Debnath	✓	0.016	✓	0.178
66	General first order	\(u_x+u u_y=1\) with \(u(0,y)=a y\). Problem 3.16(a) Lokenath Debnath	✓	0.051	✓	0.275
67	General first order	\((y+u)u_x+(x+u)u_y=x+y\). Problem 3.17(a) Lokenath Debnath	✗	230.268	✓	1.62
68	General first order	\(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath	✓	0.037	✓	0.28
69	General first order	\((x+y) u_x + (x-y)u_y =0\). Problem 3.17(c) Lokenath Debnath	✓	0.039	✓	0.947
70	General first order	\(y u_x - x u_y = e^u\) with \(u(0,y)=y^2-1\)	✓	0.099	✓	0.634	Yes
71	General first order	\(y u_x - x u_y = e^u\)	✓	0.048	✓	0.503	Yes
72	General first order	\(u_t + x u_x = 0\) with \(u(x,0)=x^2\). Math 5587	✓	0.011	✓	0.377	Yes
73	General first order	\(u_t + t u_x = 0\) with \(u(x,0)=e^x\)	✓	0.018	✓	0.457	Yes
74	General first order	\(2 u_x + 3 u_y = 1\)	✓	0.009	✓	0.162	Yes
75	General first order	\(x u_t - t u_x = 0\)	✓	0.019	✓	0.334	Yes
76	General first order	\(u_t + u_x = 0\) with \(u(x,1)=\frac {x}{1+x^2}\)	✓	0.007	✓	0.206	Yes
77	General first order	\(u_x u_y = 1\)	✓	0.004	✓	0.152	Yes
78	General first order	\(u_x u_y = u\) with \(u(x,0)=0,u(0,y)=0\)	✓	1.279	✓	0.436	Yes
79	Solved by factoring into two transport equations	\(u_{xx} + u_{xt} - 6 u_{tt} = 0\)	✓	0.027	✓	0.325	Yes
80	Solved by factoring into two transport equations	\(u_{xx} - u_{xt} - 12 u_{tt} = 0\)	✓	0.013	✓	0.491	Yes
81	Solved by factoring into two transport equations	\(u_{xx} - 3 u_{xt} - 4 u_{tt} = 0\)	✓	0.014	✓	2.669	Yes
82	Solved by factoring into two transport equations	\(u_{tt} - 2 u_{xt} - 3 u_{xx} = 0\) with \(u(0,x)=x^2, u_t(x,0)=e^x\)	✓	0.014	✓	2.492
83	Schrodinger PDE	Logan textbook, page 30	✓	0.406	✓	0.879
84	Schrodinger PDE	In a square, zero potential	✓	0.306	✓	3.28
85	Schrodinger PDE	From Mathematica help pages	✓	0.42	✓	1.514
86	Schrodinger PDE	From Mathematica help pages	✓	0.008	✗ Trivial solution. Maple does not support \(\infty \) in boundary conditions	1.974
87	Schrodinger PDE	David Griffiths, page 47	✓	30.775	✓	1.25
88	Schrodinger PDE	David Griffiths, page 47	✓	0.325	✓	0.796
89	Schrodinger PDE	In a square	✓	0.307	✓	3.898
90	Beam PDE	Beam PDE \(u_{tt} + u_{xxxx} = 0\)	✓	2.485	✓	0.386
91	Burger’s PDE	Inviscid Burgers \(u_x + u u_y = 0\)	✓ Implicit solution	0.03	✓	0.215	Yes
92	Burger’s PDE	Inviscid Burgers with I.C. \(u_x+ u u_y = 0\) and \(u(x,0)=\frac {1}{x+1}\)	✓	0.01	✓	0.26	Yes
93	Burger’s PDE	\(u_t+ u u_x = \mu u_{xx}\)	✓	0.029	✓	0.267
94	Burger’s PDE	\(u_t + u u_x + \mu u_{xx}\) with IC	✓	11.242	✗	0.621
95	Burger’s PDE	\(u_t + u u_x + \mu u_{xx}\) IC as UnitBox	✓	23.532	✗	0.46
96	Black Scholes PDE	classic Black Scholes model from finance, European call version	✓	5.177	✓	0.786
97	Black Scholes PDE	Boundary value problem for the Black Scholes equation	✓	1.849	✓	1.307
98	Korteweg-deVries PDE	\(u_{xxx} + u_t -6 u u_x = 0\)	✓	0.029	✓	0.165
99	Tricomi PDE	\(u_{xx} + y u_{yy} = 0\) with \(u(x,0)=0,u_y(x,0)=x^2\)	✓	8.326	✓	2.058
100	Tricomi PDE	\(u_{xx} + x u_{yy} = 0\)	✗	0.011	✓	1.035
101	Keldysh equation	\(x u_{xx} + u_{yy} = 0\)	✗	0.007	✓	1.319
102	Euler-Poisson-Darboux equation	\(u_{xx} + u_{yy} + \frac {\beta }{x} u_x = 0\)	✗	0.01	✓	0.31
103	Euler-Poisson-Darboux equation	\(u_{xx} - u_{yy} + \frac {\beta }{x} u_x = 0\)	✗	0.009	✓	0.515
104	Euler-Poisson-Darboux equation	\(u_{tt} - u_{xx} - \frac {2}{x} u_x = 0\) with \(u(x,0)=0,u_t(x,0)=g(x)\)	✗	2.679	✓	2.225
105	Chaplygin’s equation	\(u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0\)	✗	0.037	✓	0.82
106	Cauchy Riemann PDE’s	Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis	✓	0.005	✓	0.185
107	Cauchy Riemann PDE’s	Cauchy Riemann PDE With extra term on right side	✗	0.003	✓	0.225
108	Hamilton-Jacobi PDE	Hamilton-Jacobi type PDE	✗	0.041	✓	0.342
109	Airy PDE	\(u_t + u_{xxx} = 0\)	✓	0.012	✓	0.235	Yes
110	Nonlinear PDE’s	Bateman-Burgers \(u_t+u u_x = \nu u_{xx}\)	✓	0.026	✓	0.165
111	Nonlinear PDE’s	Benjamin Bona Mahony \(u_t+u_x + u u+x - u_{xxt} = 0\)	✓	0.032	✓	0.18
112	Nonlinear PDE’s	Benjamin Ono \(u_t+H u_{xx} +u u_x = 0\)	✓	0.023	✓	0.162
113	Nonlinear PDE’s	Born Infeld \((1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0\)	✓	0.013	✓	0.222
114	Nonlinear PDE’s	Boussinesq \(u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0\)	✓	0.042	✓	0.178
115	Nonlinear PDE’s	Boussinesq type \(u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0\)	✓	0.037	✓	0.211
116	Nonlinear PDE’s	Buckmaster \( u_t = (u^4)_{xx} + (u^3)_x\)	✗	0.082	✓ Answer in terms of RootOf.	0.446
117	Nonlinear PDE’s	Camassa Holm \(u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx}\)	✗	0.134	✓ Answer in terms of RootOf.	1.537
118	Nonlinear PDE’s	Chaffee Infante \(u_t = u_{xx} + \lambda (u^3 - u) = 0\)	✗	0.091	✓	0.25
119	Nonlinear PDE’s	Clarke. \(\left ( \theta _t - \gamma e^\theta \right )_{tt} = \left ( \theta _t - e^\theta \right )_{xx}\)	✗	0.012	✗	0.055
120	Nonlinear PDE’s	Degasperis Procesi \(u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx}\)	✗	0.132	✓ But still has unresolved ODE’s in solution	0.593
121	Nonlinear PDE’s	Dym equation \(u_t =u^3 u_{xxx}\)	✗	0.074	✓ has RootOf	0.66
122	Nonlinear PDE’s	Estevez Mansfield Clarkson \(u_{tyyy} + \beta u_y u_{yt} + \beta u_{yy} u_t + u_{tt} = 0\)	✓	0.035	✓	0.2
123	Nonlinear PDE’s	Fisher’s \(u_t = u(1-u)+u_{xx}\)	✓	0.043	✓	0.22
124	Nonlinear PDE’s	Hunter Saxton \(\left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2\)	✗	0.051	✓ with RootOf	0.331
125	Nonlinear PDE’s	Kadomtsev Petviashvili \( \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \)	✓	0.068	✓	0.251
126	Nonlinear PDE’s	Klein Gordon \(u_{xx}+u_{yy}+ \lambda u^p=0\)	✗	0.008	✗	0.073
127	Nonlinear PDE’s	Klein Gordon \(u_{xx}+u_{yy}+ u^2=0\)	✗	0.18	✓	0.828
128	Nonlinear PDE’s	Khokhlov Zabolotskaya \(u_{x t} - (u u_x)_x = u_{yy}\)	✗	0.063	✓	0.772
129	Nonlinear PDE’s	Korteweg de Vries (KdV) \(u_t + (u_x)^3+ 6 u u_x = 0\)	✓	0.028	✓	0.36
130	Nonlinear PDE’s	Lin Tsien \(2 u_{tx} + u_x u_{xx} - u_{yy} = 0\)	✗	0.075	✓	0.524
131	Nonlinear PDE’s	Liouville \(u_{xx} + u_{yy} +e^{\lambda u} = 0\)	✗	0.007	✗	0.103
132	Nonlinear PDE’s	Plateau \((1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0\)	✗	0.042	✓	0.434
133	Nonlinear PDE’s	Rayleigh \(u_{tt} - u_{xx} = \epsilon (u_t - u_t^3)\)	✗	0.074	✓ Has RootOf	0.52
134	Nonlinear PDE’s	Sawada Kotera \(u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \)	✓	0.078	✓	0.397
135	Nonlinear PDE’s	Sine Gordon \(\phi _{tt} - \phi _{xx} + \sin \phi = 0\)	✗	0.009	✗	0.088
136	Nonlinear PDE’s	Sinh Gordon \( u_{xt} = \sinh u\)	✗	0.007	✗	0.154
137	Nonlinear PDE’s	Sinh Poisson \(u_{xx}+u_{yy} + \sinh u=0\)	✗	0.007	✗	0.083
138	Nonlinear PDE’s	Thomas equation \( u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0\)	✗	0.061	✓	0.951
139	Nonlinear PDE’s	phi equation \(\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0\)	✓	0.041	✓	0.303
140	more miscellaneous	\(S S_{xy} + S_x S_y = 1\)	✗	0.042	✓	0.256
141	more miscellaneous	\(u_{rr} + u_{\theta \theta } = 0\)	✓	22.869	✓	0.87
142	more miscellaneous	\( u_{xx} + y u_{yy} = 0\)	✓	8.295	✓	2.664
143	more miscellaneous	\(u_t + u_{xxx} = 0\)	✓	0.176	✓	2.595
144	more miscellaneous	\(u_{xy} = \sin (x) \sin (y) \)	✓	2.243	✓	0.603
145	more miscellaneous	\(w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3}\)	✓	1.274	✓	0.858
146	more miscellaneous	Linear PDE, initial conditions at \(t=t_0\)	✓	1.592	✓	0.706
147	more miscellaneous	second order in time, Linear PDE, initial conditions at \(t=t_0\)	✓	1.011	✓	1.607
148	more miscellaneous	Einstein-Weiner \(u_t = -\beta u_x + D u_{xx}\)	✓	0.031	✓	0.566
149	more miscellaneous	Using integral transforms.	✓	38.845	✓	2.009