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 $u_t+u_x=0$	✓	0.024	✓	0.089	Yes
2	General first order	Transport equation $u_t-3u_x=0$ IC $u(0,x)=e^{-x^2}$. Peter Olver textbook, 2.2.2 (a)	✓	0.004	✓	0.04	Yes	Yes
3	General first order	Transport equation $u_t+2u_x=0$ IC $u(-1,x)=\frac{x}{1+x^2}$. Peter Olver textbook, 2.2.2 (b)	✓	0.003	✓	0.017	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)	✓	0.003	✓	0.055	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)	✓	0.003	✓	0.016	Yes	Yes
6	General first order	Transport equation $u_t+2u_x=\sin x$ IC $u(0,x)=\sin x$. Peter Olver textbook, 2.2.5	✓	0.049	✓	0.049	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	✓	0.014	✓	0.073	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	✓	0.004	✓	0.045	Yes	Yes
9	General first order	Transport equation $u_t+(1-2t)u_x=0$ IC $u(x,0)=\frac{1}{1+x^2}$. Peter Olver textbook, problem 2.2.29	✓	0.013	✓	0.055	Yes	Yes
10	General first order	Transport equation $u_t+\frac{1}{x^2+4}{u}_x=0$ IC $u(x,0)=e^{x^3+12x}$	✓	0.011	✓	0.065	Yes	Yes
11	General first order	$3u_x+5u_y=x$	✓	0.004	✓	0.007	Yes
12	General first order	$x{u}_y+y{u}_x=-4xyu$ and $u(x,0)=e^{-x^2}$	✓	0.01	✓	0.049	Yes
13	General first order	$u_t+u_x=0$ and $u(x,0)=\sin x$ and $u(0,t)=0$	✓	0.052	✓	0.371	Yes
14	General first order	$u_t+c{u}_x=0$ and $u(x,0)=e^{-x^2}$	✓	0.003	✓	0.019	Yes
15	General first order	(Haberman 12.2.2) ${\omega }_t-3{\omega }_x=0$ and $\omega (x,0)=\cos x$	✓	0.003	✓	0.016	Yes
16	General first order	(Haberman 12.2.4) ${\omega }_t+c{\omega }_x=0$ and $\omega (x,0)=f(x)$ and $\omega (0,t)=h(t)$	✗	10.387	✓ Solution contains unresolved invlaplace calls	0.243	Yes
17	General first order	(Haberman 12.2.5 (a)) ${\omega }_t+c{\omega }_x=e^{2x}$ and $\omega (x,0)=f(x)$	✓	0.027	✓	0.037	Yes
18	General first order	(Haberman 12.2.5 (d)) ${\omega }_t+3t{\omega }_x=\omega (x,t)$ and $\omega (x,0)=f(x)$	✓	0.023	✓	0.05	Yes
19	General first order	$2u_x+5u_y=u^2(x,y)+1$	✓	0.083	✓	0.03	Yes
20	General first order	Clairaut equation $x{u}_x+y{u}_y+\frac{1}{2}((u_x{)}^2+(u_y{)}^2)=0$	✓	0.012	✓	0.196	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	✓	0.611
22	General first order	Clairaut equation. $u=x{u}_x+y{u}_y+\sin (u_x+u_y)$	✓	0.016	✓	0.013
23	General first order	Recover a function from its gradient vector	✓	0.021	✓	0.059
24	General first order	$x{f}_y-f_x=\frac{g(x)}{h(y)}{f}^2$	✓	0.038	✓	0.036	Yes
25	General first order	$f_x+(f_y{)}^2=f(x,y,z)+z$	✓	0.047	✓	0.464
26	General first order	$x{u}_x+y{u}_y=u$ (Example 3.5.1 in Lokenath Debnath)	✓	0.007	✓	0.01	Yes
27	General first order	$x{u}_x+y{u}_y=nu$ Example 3.5.2 in Lokenath Debnath	✓	0.007	✓	0.009	Yes
28	General first order	$x^2{u}_x+y^2{u}_y=(x+y)u$ Example 3.5.3 in Lokenath Debnath	✓	0.082	✓	0.023	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)	✗ (Timed out)	600.	✓	1.899	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)	✓	0.361	✓ Maple does not accept this form of Cauchy data	0.154	Yes
31	General first order	$u_x-u_y=1$ with $u(x,0)=x^2$ Example 3.5.6 in Lokenath Debnath	✓	0.003	✓	0.014
32	General first order	$(y-x)u_x+(u-x)u_y=x-y$ with $u=0$ on $xy=1$ Example 3.5.7 in Lokenath Debnath	✗ Kernel error	120.019	✗ (Timed out)	600.
33	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	✗	1.297	✗	0.429
34	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	✓	0.1	✓ Maple does not accept this form of Cauchy data	0.057
35	General first order	$u_t+u{u}_x=x$ with $u(x,0)=f(x)$ Example 3.5.11 in Lokenath Debnath.	✗ Kernel error	0.645	✓	0.184
36	General first order	$u_x=0$ Problem 3.3(a) Lokenath Debnath	✓	0.003	✓	0.002
37	General first order	$a{u}_x+b{u}_y=0$ Problem 3.3(b) Lokenath Debnath	✓	0.004	✓	0.007
38	General first order	$u_x+y{u}_y=0$ Problem 3.3(c) Lokenath Debnath	✓	0.013	✓	0.009
39	General first order	$(1+x^2)u_x+u_y=0$ Problem 3.3(d) Lokenath Debnath	✓	0.006	✓	0.009
40	General first order	$2xy{u}_x+(x^2+y^2)u_y=0$ Problem 3.3(e) Lokenath Debnath	✓	0.102	✓	0.023
41	General first order	$(y+u)u_x+y{u}_y=x-y$ Problem 3.3(f) Lokenath Debnath	✗	4.092	✓	0.319
42	General first order	$y^2{u}_x-xy{u}_y=x(u-2y)$ Problem 3.3(g) Lokenath Debnath	✓	0.029	✓	0.038
43	General first order	$y{u}_y-x{u}_x=1$ Problem 3.3(h) Lokenath Debnath	✓	0.007	✓	0.009
44	General first order	$u_x+2x{y}^2{u}_y=0$ Problem 3.4 Lokenath Debnath	✓	0.074	✓	0.01
45	General first order	$3u_x+2u_y=0$ with $u(x,0)=\sin x$. Problem 3.5(a) Lokenath Debnath	✓	0.003	✓	0.015
46	General first order	$y{u}_x+x{u}_y=0$ with $u(0,y)=e^{-y^2}$. Problem 3.5(b) Lokenath Debnath	✓	0.01	✓	0.024
47	General first order	$x{u}_x+y{u}_y=2xy$ with $u=2$ on $y=x^2$. Problem 3.5(c) Lokenath Debnath	✓	0.006	✓ Maple does not accept this form of Cauchy data as I.C.	0.008
48	General first order	$u_x+x{u}_y=0$ with $u(0,y)=\sin y$. Problem 3.5(d) Lokenath Debnath	✓	0.003	✓	0.149
49	General first order	$y{u}_x+x{u}_y=xy$ with $u(0,y)=e^{-y^2},u(x,0)=e^{-x^2}$. Problem 3.5(e) Lokenath Debnath	✗	1.138	✗	0.371
50	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	✓	0.005	✓	0.075
51	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	✓	0.007	✓ Maple does not accept this form of Cauchy data as I.C.	0.01
52	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	✓	0.04	✓	0.057
53	General first order	$x{u}_x+(x+y)u_y=u+1$ with $u(x,0)=x^2$ Problem 3.5(i) Lokenath Debnath	✓	0.011	✓	0.039
54	General first order	$x{u}_x+y{u}_y+z{u}_z=0$ Problem 3.8(a) .Lokenath Debnath	✓	0.01	✓	0.011
55	General first order	$x^2{u}_x+y^2{u}_y+z(x+y)u_z=0$ Problem 3.8(b) Lokenath Debnath	✓	0.089	✓	0.016
56	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	✓	0.027	✓	0.68
57	General first order	$yz{u}_x-xz{u}_y+xy(x^2+y^2)u_z=0$ Problem 3.8(d) Lokenath Debnath	✓	0.068	✓	0.034
58	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	✗	44.854	✗	0.368
59	General first order	$u_x+x{u}_y=y$ with $u(0,y)=y^2$ Problem 3.9(a) Lokenath Debnath	✓	0.004	✓	0.016
60	General first order	$u_x+x{u}_y=y$ with $u(1,y)=2y$ Problem 3.9(b) Lokenath Debnath	✓	0.005	✓	0.007
61	General first order	$(u_x+u_y{)}^2-u^2=0$. Problem 3.10 Lokenath Debnath	✓	0.007	✓	0.012
62	General first order	$(y+u)u_x+y{u}_y=x-y$ with $u(x,1)=1+x$. Problem 3.11 Lokenath Debnath	✗	7.967	✗	0.708
63	General first order	$2x{u}_x+(x+1)u_y=y$ with $u(1,y)=2y$. Problem 3.14(d) Lokenath Debnath	✓	0.009	✓	0.124
64	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	✓	0.01	✓	0.033
65	General first order	$y^2{u}_x+(xy)u_y=x$ with $u(x,1)=x^2$. Problem 3.14(f) Lokenath Debnath	✓	0.016	✓	0.037
66	General first order	$x{u}_x+y{u}_y=xy$ with $u=\frac{x^2}{2}$ at $y=x$. Problem 3.14(g) Lokenath Debnath	✓ Mathematica does not support this Cauchy data I.C.	0.007	✓ Maple does not support this Cauchy data I.C.	0.009
67	General first order	$u_x+u{u}_y=1$ with $u(0,y)=ay$. Problem 3.16(a) Lokenath Debnath	✓	0.018	✓	0.02
68	General first order	$(y+u)u_x+(x+u)u_y=x+y$. Problem 3.17(a) Lokenath Debnath	✗	30.116	✓	20.194
69	General first order	$xu(u^2+xy)u_x-yu(u^2+xy)u_y=x^4$. Problem 3.17(b) Lokenath Debnath	✓	0.027	✓	0.037
70	General first order	$(x+y)u_x+(x-y)u_y=0$. Problem 3.17(c) Lokenath Debnath	✓	0.028	✓	0.045
71	General first order	$y{u}_x-x{u}_y=e^u$ with $u(0,y)=y^2-1$	✓	0.12	✓	0.085	Yes
72	General first order	$y{u}_x-x{u}_y=e^u$	✓	0.054	✓	0.002	Yes
73	General first order	$u_t+x{u}_x=0$ with $u(x,0)=x^2$. Math 5587	✓	0.005	✓	0.02	Yes
74	General first order	$u_t+t{u}_x=0$ with $u(x,0)=e^x$	✓	0.009	✓	0.03	Yes
75	General first order	$2u_x+3u_y=1$	✓	0.004	✓	0.007	Yes
76	General first order	$x{u}_t-t{u}_x=0$	✓	0.009	✓	0.013	Yes
77	General first order	$u_t+u_x=0$ with $u(x,1)=\frac{x}{1+x^2}$	✓	0.003	✓	0.008	Yes
78	General first order	$u_x{u}_y=1$	✓	0.002	✓	0.013	Yes
79	General first order	$u_x{u}_y=u$ with $u(x,0)=0,u(0,y)=0$	✗	1.074	✓	0.556	Yes
80	Solved by factoring into two transport equations	$u_{xx}+u_{xt}-6u_{tt}=0$	✓	0.514	✓	0.082	Yes
81	Solved by factoring into two transport equations	$u_{xx}-u_{xt}-12u_{tt}=0$	✗	1.554	✓	0.246	Yes
82	Solved by factoring into two transport equations	$u_{xx}-3u_{xt}-4u_{tt}=0$	✗	0.467	✓	2.034	Yes
83	Solved by factoring into two transport equations	$u_{tt}-2u_{xt}-3u_{xx}=0$ with $u(0,x)=x^2,u_t(x,0)=e^x$	✗	0.392	✓	1.312
84	Schrodinger PDE	Logan textbook, page 30	✓	0.377	✓	0.487
85	Schrodinger PDE	In a square, zero potential	✓	0.259	✓	2.934
86	Schrodinger PDE	From Mathematica help pages	✓	0.66	✓	0.68
87	Schrodinger PDE	From Mathematica help pages	✓	0.005	✗ Trivial solution. Maple does not support $\mathrm{\infty }$ in boundary conditions	3.897
88	Schrodinger PDE	David Griffiths, page 47	✓	35.841	✓	1.52
89	Schrodinger PDE	David Griffiths, page 47	✓	0.011	✓	0.782
90	Schrodinger PDE	In a square	✓	0.011	✓	2.282
91	Beam PDE	Beam PDE $u_{tt}+u_{xxxx}=0$	✓	0.206	✓	0.229
92	Burger’s PDE	Inviscid Burgers $u_x+u{u}_y=0$	✓ Implicit solution	0.027	✓	0.019	Yes
93	Burger’s PDE	Inviscid Burgers with I.C. $u_x+u{u}_y=0$ and $u(x,0)=\frac{1}{x+1}$	✓	0.008	✓	0.047	Yes
94	Burger’s PDE	$u_t+u{u}_x=\mu u_{xx}$	✓	0.026	✓	0.064
95	Burger’s PDE	$u_t+u{u}_x+\mu u_{xx}$ with IC	✓	21.53	✗	0.655
96	Burger’s PDE	$u_t+u{u}_x+\mu u_{xx}$ IC as UnitBox	✓	24.232	✗	0.788
97	Black Scholes PDE	classic Black Scholes model from finance, European call version	✓	3.046	✓	0.931
98	Black Scholes PDE	Boundary value problem for the Black Scholes equation	✓	2.766	✓	2.334
99	Korteweg-deVries PDE	$u_{xxx}+u_t-6u{u}_x=0$	✓	0.029	✓	0.177
100	Tricomi PDE	$u_{xx}+y{u}_{yy}=0$ with $u(x,0)=0,u_y(x,0)=x^2$	✓	4.196	✓	3.212
101	Tricomi PDE	$u_{xx}+x{u}_{yy}=0$	✗	0.005	✓	1.635
102	Keldysh equation	$x{u}_{xx}+u_{yy}=0$	✗	0.004	✓	1.7
103	Euler-Poisson-Darboux equation	$u_{xx}+u_{yy}+\frac{\beta }{x}{u}_x=0$	✗	0.007	✓	0.082
104	Euler-Poisson-Darboux equation	$u_{xx}-u_{yy}+\frac{\beta }{x}{u}_x=0$	✗	0.006	✓	0.26
105	Euler-Poisson-Darboux equation	$u_{tt}-u_{xx}-\frac{2}{x}{u}_x=0$ with $u(x,0)=0,u_t(x,0)=g(x)$	✗	1.963	✓	3.855
106	Chaplygin’s equation	$u_{\theta \theta }+\frac{v^2}{1-\frac{v^2}{c^2}}{u}_{vv}+v{u}_v=0$	✗	0.021	✓	0.889
107	Cauchy Riemann PDE’s	Cauchy Riemann PDE with Prescribe the values of $u$ and $v$ on the $x$ axis	✓	0.005	✓	0.13
108	Cauchy Riemann PDE’s	Cauchy Riemann PDE With extra term on right side	✗	0.001	✓	0.052
109	Hamilton-Jacobi PDE	Hamilton-Jacobi type PDE	✗	0.007	✓	0.157
110	Airy PDE	$u_t+u_{xxx}=0$	✗	0.002	✓	0.064	Yes
111	Nonlinear PDE’s	Bateman-Burgers $u_t+u{u}_x=\nu u_{xx}$	✓	0.02	✓	0.06
112	Nonlinear PDE’s	Benjamin Bona Mahony $u_t+u_x+uu+x-u_{xxt}=0$	✓	0.031	✓	0.075
113	Nonlinear PDE’s	Benjamin Ono $u_t+H{u}_{xx}+u{u}_x=0$	✓	0.02	✓	0.062
114	Nonlinear PDE’s	Born Infeld $(1-u_t^2)u_{xx}+2u_x{u}_t{u}_{xt}-(1+u_x^2)u_{tt}=0$	✓	0.008	✓	0.135
115	Nonlinear PDE’s	Boussinesq $u_{tt}-u_{xx}-u_{xxxx}-3(u^2{)}_{xx}=0$	✓	0.042	✓	0.086
116	Nonlinear PDE’s	Boussinesq type $u_{tt}-u_{xx}-2\alpha (u{u}_x{)}_x-\beta u_{xxtt}=0$	✓	0.041	✓	0.106
117	Nonlinear PDE’s	Buckmaster $u_t=(u^4{)}_{xx}+(u^3{)}_x$	✗	0.07	✓ Answer in terms of RootOf.	0.352
118	Nonlinear PDE’s	Camassa Holm $u_t+2k{u}_x-u_{xxt}+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$	✗	0.174	✓ Answer in terms of RootOf.	1.52
119	Nonlinear PDE’s	Chaffee Infante $u_t=u_{xx}+\lambda (u^3-u)=0$	✗	0.093	✓	0.176
120	Nonlinear PDE’s	Clarke. ${({\theta }_t-\gamma e^{\theta })}_{tt}={({\theta }_t-e^{\theta })}_{xx}$	✗	0.01	✗	0.025
121	Nonlinear PDE’s	Degasperis Procesi $u_t-u_{xxt}+4u{u}_x=3u_x{u}_{x}x+u{u}_{xxx}$	✗	0.169	✓ But still has unresolved ODE’s in solution	0.52
122	Nonlinear PDE’s	Dym equation $u_t=u^3{u}_{xxx}$	✗	0.083	✓ has RootOf	0.375
123	Nonlinear PDE’s	Estevez Mansfield Clarkson $u_{tyyy}+\beta u_y{u}_{yt}+\beta u_{yy}{u}_t+u_{tt}=0$	✓	0.033	✓	0.121
124	Nonlinear PDE’s	Fisher’s $u_t=u(1-u)+u_{xx}$	✓	0.05	✓	0.165
125	Nonlinear PDE’s	Hunter Saxton ${(u_t+u{u}_x))}_x=\frac{1}{2}(u_x{)}^2$	✗	0.044	✓ with RootOf	0.107
126	Nonlinear PDE’s	Kadomtsev Petviashvili ${(u_t+u{u}_x+{ϵ}^2{u}_{xxx})}_x+\lambda u_{yy}=0$	✓	0.071	✓	0.125
127	Nonlinear PDE’s	Klein Gordon $u_{xx}+u_{yy}+\lambda u^p=0$	✗	0.004	✗	0.017
128	Nonlinear PDE’s	Klein Gordon $u_{xx}+u_{yy}+u^2=0$	✗	0.228	✓	0.346
129	Nonlinear PDE’s	Khokhlov Zabolotskaya $u_{xt}-(u{u}_x{)}_x=u_{yy}$	✗	0.057	✓	0.237
130	Nonlinear PDE’s	Korteweg de Vries (KdV) $u_t+(u_x{)}^3+6u{u}_x=0$	✓	0.029	✓	0.092
131	Nonlinear PDE’s	Lin Tsien $2u_{tx}+u_x{u}_{xx}-u_{yy}=0$	✗	0.072	✓	0.243
132	Nonlinear PDE’s	Liouville $u_{xx}+u_{yy}+e^{\lambda u}=0$	✗	0.004	✗	0.017
133	Nonlinear PDE’s	Plateau $(1+u_y^2)u_{xx}-2u_x{u}_y{y}_{xy}+(1+u_x^2)u_{yy}=0$	✗	0.032	✓	0.156
134	Nonlinear PDE’s	Rayleigh $u_{tt}-u_{xx}=ϵ(u_t-u_t^3)$	✗	0.078	✓ Has RootOf	0.127
135	Nonlinear PDE’s	Sawada Kotera $u_t+45u^2{u}_x+15u_x{u}_{xx}+15u{u}_{xxx}+u_{xxxxx}=0$	✓	0.085	✓	0.187
136	Nonlinear PDE’s	Sine Gordon ${\varphi }_{tt}-{\varphi }_{xx}+\sin \varphi =0$	✗	0.007	✗	0.013
137	Nonlinear PDE’s	Sinh Gordon $u_{xt}=\sinh u$	✗	0.007	✗	0.014
138	Nonlinear PDE’s	Sinh Poisson $u_{xx}+u_{yy}+\sinh u=0$	✗	0.006	✗	0.011
139	Nonlinear PDE’s	Thomas equation $u_{xy}+\alpha u_x+\beta u_y+\nu u_x{u}_y=0$	✗	0.06	✓	0.339
140	Nonlinear PDE’s	phi equation ${\varphi }_{tt}-{\varphi }_{xx}-\varphi +{\varphi }^3=0$	✓	0.043	✓	0.472
141	more miscellaneous	$S{S}_{xy}+S_x{S}_y=1$	✗	0.031	✓	0.023
142	more miscellaneous	$u_{rr}+u_{\theta \theta }=0$	✗	0.012	✓	0.513
143	more miscellaneous	$u_{xx}+y{u}_{yy}=0$	✓	3.551	✓	2.448
144	more miscellaneous	$u_t+u_{xxx}=0$	✗	0.413	✓	5.663
145	more miscellaneous	$u_{xy}=\sin (x)\sin (y)$	✗	0.271	✓	0.426
146	more miscellaneous	$w_t=w_{x_1{x}_1}+w_{x_2{x}_2}+w_{x_3{x}_3}$	✗	0.003	✓	0.621
147	more miscellaneous	Linear PDE, initial conditions at $t=t_0$	✗	0.003	✓	0.616
148	more miscellaneous	second order in time, Linear PDE, initial conditions at $t=t_0$	✗	0.003	✓	2.179
149	more miscellaneous	Einstein-Weiner $u_t=-\beta u_x+D{u}_{xx}$	✗	0.004	✓	0.194
150	more miscellaneous	Using integral transforms. $x^2\frac{{\partial }^2}{\partial x^2}u(x,y)+x\frac{\partial }{\partial x}u(x,y)+\frac{{\partial }^2}{\partial y^2}u(x,y)=0$	✗	0.982	✓	2.129