|
Table 1.6: Miscellaneous PDE’s breakdown of results. Time in seconds
|
||||||||
|
# |
PDE |
description |
Mathematica
| Maple
|
hand solved? |
Animated? |
||
|
|
|
|
result |
time |
result |
time |
|
|
|
|
||||||||
|
1 |
Transport equation \(u_t+ u_x = 0\) |
✓ |
0.024 |
✓ |
0.089 |
Yes |
|
|
|
2 |
Transport equation \(u_t-3 u_x = 0\) IC \(u(0,x)=e^{-x^2}\). Peter Olver textbook, 2.2.2 (a) |
✓ |
0.004 |
✓ |
0.04 |
Yes |
Yes |
|
|
3 |
Transport equation \(u_t+2 u_x = 0\) IC \(u(-1,x)=\frac {x}{1+x^2}\). Peter Olver textbook, 2.2.2 (b) |
✓ |
0.003 |
✓ |
0.017 |
Yes |
Yes |
|
|
4 |
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 |
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 |
Transport equation \(u_t+2 u_x= \sin x\) IC \(u(0,x)=\sin x\). Peter Olver textbook, 2.2.5 |
✓ |
0.049 |
✓ |
0.049 |
Yes |
Yes |
|
|
7 |
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 |
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 |
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 |
✓ |
0.013 |
✓ |
0.055 |
Yes |
Yes |
|
|
10 |
Transport equation \(u_t+\frac {1}{x^2+4} u_x= 0\) IC \(u(x,0)=e^{x^3+12 x}\) |
✓ |
0.011 |
✓ |
0.065 |
Yes |
Yes |
|
|
11 |
\(3 u_x + 5 u_y = x\) |
✓ |
0.004 |
✓ |
0.007 |
Yes |
|
|
|
12 |
\(x u_y + y u_x = -4 x y u\) and \(u(x,0)=e^{-x^2}\) |
✓ |
0.01 |
✓ |
0.049 |
Yes |
|
|
|
13 |
\(u_t + u_x = 0\) and \(u(x,0)=\sin x\) and \(u(0,t)=0\) |
✓ |
0.052 |
✓ |
0.371 |
Yes |
|
|
|
14 |
\(u_t+ c u_x = 0\) and \(u(x,0)=e^{-x^2}\) |
✓ |
0.003 |
✓ |
0.019 |
Yes |
|
|
|
15 |
(Haberman 12.2.2) \(\omega _t -3 \omega _x = 0\) and \(\omega (x,0)=\cos x\) |
✓ |
0.003 |
✓ |
0.016 |
Yes |
|
|
|
16 |
(Haberman 12.2.4) \(\omega _t +c \omega _x = 0\) and \(\omega (x,0)=f(x)\) and \(\omega (0,t)=h(t)\) |
✗ |
10.387 |
✓ |
0.243 |
Yes |
|
|
|
17 |
(Haberman 12.2.5 (a)) \(\omega _t +c \omega _x = e^{2 x}\) and \(\omega (x,0)=f(x)\) |
✓ |
0.027 |
✓ |
0.037 |
Yes |
|
|
|
18 |
(Haberman 12.2.5 (d)) \(\omega _t +3 t \omega _x = \omega (x,t)\) and \(\omega (x,0)=f(x)\) |
✓ |
0.023 |
✓ |
0.05 |
Yes |
|
|
|
19 |
\( 2 u_x + 5 u_y = u^2(x,y) + 1\) |
✓ |
0.083 |
✓ |
0.03 |
Yes |
|
|
|
20 |
Clairaut equation \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\) |
✓ |
0.012 |
✓ |
0.196 |
Yes |
|
|
|
21 |
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 |
Clairaut equation. \(u = x u_x+ y u_y + \sin ( u_x + u_y )\) |
✓ |
0.016 |
✓ |
0.013 |
|
|
|
|
23 |
Recover a function from its gradient vector |
✓ |
0.021 |
✓ |
0.059 |
|
|
|
|
24 |
\(x f_y - f_x = \frac {g(x)}{h(y)} f^2\) |
✓ |
0.038 |
✓ |
0.036 |
Yes |
|
|
|
25 |
\(f_x + (f_y)^2 = f(x,y,z)+z\) |
✓ |
0.047 |
✓ |
0.464 |
|
|
|
|
26 |
\(x u_x+y u_y=u\) (Example 3.5.1 in Lokenath Debnath) |
✓ |
0.007 |
✓ |
0.01 |
Yes |
|
|
|
27 |
\(x u_x+y u_y=n u\) Example 3.5.2 in Lokenath Debnath |
✓ |
0.007 |
✓ |
0.009 |
Yes |
|
|
|
28 |
\(x^2 u_x+y^2 u_y=(x+y) u\) Example 3.5.3 in Lokenath Debnath |
✓ |
0.082 |
✓ |
0.023 |
Yes |
|
|
|
29 |
\((y-z) u_x + (z-x) u_y + (x-y) u_z = 0\) (Example 3.5.4 in Lokenath Debnath) |
✗ (Timed out) |
600. |
✓ |
1.899 |
Yes |
|
|
|
30 |
\(u(x+y) u_x+u(x-y) u_y=x^2+y^2\) (Example 3.5.5 in Lokenath Debnath) |
✓ |
0.361 |
✓ |
0.154 |
Yes |
|
|
|
31 |
\(u_x-u_y=1\) with \(u(x,0)=x^2\) Example 3.5.6 in Lokenath Debnath |
✓ |
0.003 |
✓ |
0.014 |
|
|
|
|
32 |
\((y-x)u_x+(u-x)u_y=x-y\) with \(u=0\) on \(x y=1\) Example 3.5.7 in Lokenath Debnath |
✗ |
120.019 |
✗ (Timed out) |
600. |
|
|
|
|
33 |
\(y u_x+x u_y=u\) with \(u(x,0)=x^3\) and \(u(0,y)=y^3\) Example 3.5.8 in Lokenath Debnath |
✗ |
1.297 |
✗ |
0.429 |
|
|
|
|
34 |
\(x u_x+y u_y=x e^{-u}\) with \(u=0\) on \(y=x^2\) Example 3.5.10 in Lokenath Debnath |
✓ |
0.1 |
✓ |
0.057 |
|
|
|
|
35 |
\(u_t+u u_x=x\) with \(u(x,0)=f(x)\) Example 3.5.11 in Lokenath Debnath. |
✗ |
0.645 |
✓ |
0.184 |
|
|
|
|
36 |
\(u_x=0\) Problem 3.3(a) Lokenath Debnath |
✓ |
0.003 |
✓ |
0.002 |
|
|
|
|
37 |
\(a u_x+b u_y=0\) Problem 3.3(b) Lokenath Debnath |
✓ |
0.004 |
✓ |
0.007 |
|
|
|
|
38 |
\(u_x+y u_y=0\) Problem 3.3(c) Lokenath Debnath |
✓ |
0.013 |
✓ |
0.009 |
|
|
|
|
39 |
\((1+x^2) u_x+ u_y=0\) Problem 3.3(d) Lokenath Debnath |
✓ |
0.006 |
✓ |
0.009 |
|
|
|
|
40 |
\(2 x y u_x+(x^2+y^2)u_y=0\) Problem 3.3(e) Lokenath Debnath |
✓ |
0.102 |
✓ |
0.023 |
|
|
|
|
41 |
\((y+u) u_x+y u_y=x-y\) Problem 3.3(f) Lokenath Debnath |
✗ |
4.092 |
✓ |
0.319 |
|
|
|
|
42 |
\(y^2 u_x- x y u_y=x(u-2 y)\) Problem 3.3(g) Lokenath Debnath |
✓ |
0.029 |
✓ |
0.038 |
|
|
|
|
43 |
\(y u_y - x u_x = 1\) Problem 3.3(h) Lokenath Debnath |
✓ |
0.007 |
✓ |
0.009 |
|
|
|
|
44 |
\(u_x+2 x y^2 u_y=0\) Problem 3.4 Lokenath Debnath |
✓ |
0.074 |
✓ |
0.01 |
|
|
|
|
45 |
\(3 u_x+2 u_y=0\) with \(u(x,0)=\sin x\). Problem 3.5(a) Lokenath Debnath |
✓ |
0.003 |
✓ |
0.015 |
|
|
|
|
46 |
\(y u_x+x u_y=0\) with \(u(0,y)=e^{-y^2}\). Problem 3.5(b) Lokenath Debnath |
✓ |
0.01 |
✓ |
0.024 |
|
|
|
|
47 |
\(x u_x+y u_y=2 x y\) with \(u=2\) on \(y=x^2\). Problem 3.5(c) Lokenath Debnath |
✓ |
0.006 |
✓ |
0.008 |
|
|
|
|
48 |
\(u_x+x u_y=0\) with \(u(0,y)=\sin y\). Problem 3.5(d) Lokenath Debnath |
✓ |
0.003 |
✓ |
0.149 |
|
|
|
|
49 |
\(y u_x+x u_y=x y\) with \(u(0,y)=e^{-y^2},u(x,0)=e^{-x^2}\). Problem 3.5(e) Lokenath Debnath |
✗ |
1.138 |
✗ |
0.371 |
|
|
|
|
50 |
\(u_x+x u_y=(y-\frac {1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath |
✓ |
0.005 |
✓ |
0.075 |
|
|
|
|
51 |
\(x u_x+y u_y=u+1\) with \(u=x^2\) on \(y=x^2\) Problem 3.5(g) Lokenath Debnath |
✓ |
0.007 |
✓ |
0.01 |
|
|
|
|
52 |
\(u u_x - u u_y= u^2 + (x+y)^2\) with \(u(x,0)=1\) Problem 3.5(h) Lokenath Debnath |
✓ |
0.04 |
✓ |
0.057 |
|
|
|
|
53 |
\(x u_x+(x+y)u_y=u+1\) with \(u(x,0)=x^2\) Problem 3.5(i) Lokenath Debnath |
✓ |
0.011 |
✓ |
0.039 |
|
|
|
|
54 |
\(x u_x+y u_y+z u_z=0\) Problem 3.8(a) .Lokenath Debnath |
✓ |
0.01 |
✓ |
0.011 |
|
|
|
|
55 |
\(x^2 u_x+y^2 u_y+z(x+y)u_z=0\) Problem 3.8(b) Lokenath Debnath |
✓ |
0.089 |
✓ |
0.016 |
|
|
|
|
56 |
\(x(y-z)u_x+y(z-x)u_y+z(x-y)u_z=0\) Problem 3.8(c) Lokenath Debnath |
✓ |
0.027 |
✓ |
0.68 |
|
|
|
|
57 |
\(y z u_x - x z u_y+ x y (x^2+y^2) u_z=0\) Problem 3.8(d) Lokenath Debnath |
✓ |
0.068 |
✓ |
0.034 |
|
|
|
|
58 |
\(x(y^2-z^2) u_x + y(z^2-y^2) u_y+ z (x^2-y^2) u_z=0\) Problem 3.8(e) Lokenath Debnath |
✗ |
44.854 |
✗ |
0.368 |
|
|
|
|
59 |
\(u_x+x u_y=y\) with \(u(0,y)=y^2\) Problem 3.9(a) Lokenath Debnath |
✓ |
0.004 |
✓ |
0.016 |
|
|
|
|
60 |
\(u_x+x u_y=y\) with \(u(1,y)=2 y\) Problem 3.9(b) Lokenath Debnath |
✓ |
0.005 |
✓ |
0.007 |
|
|
|
|
61 |
\((u_x+u_y)^2-u^2=0\). Problem 3.10 Lokenath Debnath |
✓ |
0.007 |
✓ |
0.012 |
|
|
|
|
62 |
\((y+u)u_x+y u_y=x-y\) with \(u(x,1)=1+x\). Problem 3.11 Lokenath Debnath |
✗ |
7.967 |
✗ |
0.708 |
|
|
|
|
63 |
\(2 x u_x+(x+1) u_y=y\) with \(u(1,y)=2 y\). Problem 3.14(d) Lokenath Debnath |
✓ |
0.009 |
✓ |
0.124 |
|
|
|
|
64 |
\(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath |
✓ |
0.01 |
✓ |
0.033 |
|
|
|
|
65 |
\(y^2 u_x+(x y) u_y=x\) with \(u(x,1)=x^2\). Problem 3.14(f) Lokenath Debnath |
✓ |
0.016 |
✓ |
0.037 |
|
|
|
|
66 |
\(x u_x+y u_y=x y\) with \(u=\frac {x^2}{2}\) at \(y=x\). Problem 3.14(g) Lokenath Debnath |
✓ |
0.007 |
✓ |
0.009 |
|
|
|
|
67 |
\(u_x+u u_y=1\) with \(u(0,y)=a y\). Problem 3.16(a) Lokenath Debnath |
✓ |
0.018 |
✓ |
0.02 |
|
|
|
|
68 |
\((y+u)u_x+(x+u)u_y=x+y\). Problem 3.17(a) Lokenath Debnath |
✗ |
30.116 |
✓ |
20.194 |
|
|
|
|
69 |
\(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath |
✓ |
0.027 |
✓ |
0.037 |
|
|
|
|
70 |
\((x+y) u_x + (x-y)u_y =0\). Problem 3.17(c) Lokenath Debnath |
✓ |
0.028 |
✓ |
0.045 |
|
|
|
|
71 |
\(y u_x - x u_y = e^u\) with \(u(0,y)=y^2-1\) |
✓ |
0.12 |
✓ |
0.085 |
Yes |
|
|
|
72 |
\(y u_x - x u_y = e^u\) |
✓ |
0.054 |
✓ |
0.002 |
Yes |
|
|
|
73 |
\(u_t + x u_x = 0\) with \(u(x,0)=x^2\). Math 5587 |
✓ |
0.005 |
✓ |
0.02 |
Yes |
|
|
|
74 |
\(u_t + t u_x = 0\) with \(u(x,0)=e^x\) |
✓ |
0.009 |
✓ |
0.03 |
Yes |
|
|
|
75 |
\(2 u_x + 3 u_y = 1\) |
✓ |
0.004 |
✓ |
0.007 |
Yes |
|
|
|
76 |
\(x u_t - t u_x = 0\) |
✓ |
0.009 |
✓ |
0.013 |
Yes |
|
|
|
77 |
\(u_t + u_x = 0\) with \(u(x,1)=\frac {x}{1+x^2}\) |
✓ |
0.003 |
✓ |
0.008 |
Yes |
|
|
|
78 |
\(u_x u_y = 1\) |
✓ |
0.002 |
✓ |
0.013 |
Yes |
|
|
|
79 |
\(u_x u_y = u\) with \(u(x,0)=0,u(0,y)=0\) |
✗ |
1.074 |
✓ |
0.556 |
Yes |
|
|
|
80 |
\(u_{xx} + u_{xt} - 6 u_{tt} = 0\) |
✓ |
0.514 |
✓ |
0.082 |
Yes |
|
|
|
81 |
\(u_{xx} - u_{xt} - 12 u_{tt} = 0\) |
✗ |
1.554 |
✓ |
0.246 |
Yes |
|
|
|
82 |
\(u_{xx} - 3 u_{xt} - 4 u_{tt} = 0\) |
✗ |
0.467 |
✓ |
2.034 |
Yes |
|
|
|
83 |
\(u_{tt} - 2 u_{xt} - 3 u_{xx} = 0\) with \(u(0,x)=x^2, u_t(x,0)=e^x\) |
✗ |
0.392 |
✓ |
1.312 |
|
|
|
|
84 |
Logan textbook, page 30 |
✓ |
0.377 |
✓ |
0.487 |
|
|
|
|
85 |
In a square, zero potential |
✓ |
0.259 |
✓ |
2.934 |
|
|
|
|
86 |
From Mathematica help pages |
✓ |
0.66 |
✓ |
0.68 |
|
|
|
|
87 |
From Mathematica help pages |
✓ |
0.005 |
✗ |
3.897 |
|
|
|
|
88 |
David Griffiths, page 47 |
✓ |
35.841 |
✓ |
1.52 |
|
|
|
|
89 |
David Griffiths, page 47 |
✓ |
0.011 |
✓ |
0.782 |
|
|
|
|
90 |
In a square |
✓ |
0.011 |
✓ |
2.282 |
|
|
|
|
91 |
Beam PDE \(u_{tt} + u_{xxxx} = 0\) |
✓ |
0.206 |
✓ |
0.229 |
|
|
|
|
92 |
Inviscid Burgers \(u_x + u u_y = 0\) |
✓ |
0.027 |
✓ |
0.019 |
Yes |
|
|
|
93 |
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 |
\(u_t+ u u_x = \mu u_{xx}\) |
✓ |
0.026 |
✓ |
0.064 |
|
|
|
|
95 |
\(u_t + u u_x + \mu u_{xx}\) with IC |
✓ |
21.53 |
✗ |
0.655 |
|
|
|
|
96 |
\(u_t + u u_x + \mu u_{xx}\) IC as UnitBox |
✓ |
24.232 |
✗ |
0.788 |
|
|
|
|
97 |
classic Black Scholes model from finance, European call version |
✓ |
3.046 |
✓ |
0.931 |
|
|
|
|
98 |
Boundary value problem for the Black Scholes equation |
✓ |
2.766 |
✓ |
2.334 |
|
|
|
|
99 |
\(u_{xxx} + u_t -6 u u_x = 0\) |
✓ |
0.029 |
✓ |
0.177 |
|
|
|
|
100 |
\(u_{xx} + y u_{yy} = 0\) with \(u(x,0)=0,u_y(x,0)=x^2\) |
✓ |
4.196 |
✓ |
3.212 |
|
|
|
|
101 |
\(u_{xx} + x u_{yy} = 0\) |
✗ |
0.005 |
✓ |
1.635 |
|
|
|
|
102 |
\(x u_{xx} + u_{yy} = 0\) |
✗ |
0.004 |
✓ |
1.7 |
|
|
|
|
103 |
\(u_{xx} + u_{yy} + \frac {\beta }{x} u_x = 0\) |
✗ |
0.007 |
✓ |
0.082 |
|
|
|
|
104 |
\(u_{xx} - u_{yy} + \frac {\beta }{x} u_x = 0\) |
✗ |
0.006 |
✓ |
0.26 |
|
|
|
|
105 |
\(u_{tt} - u_{xx} - \frac {2}{x} u_x = 0\) with \(u(x,0)=0,u_t(x,0)=g(x)\) |
✗ |
1.963 |
✓ |
3.855 |
|
|
|
|
106 |
\(u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0\) |
✗ |
0.021 |
✓ |
0.889 |
|
|
|
|
107 |
Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis |
✓ |
0.005 |
✓ |
0.13 |
|
|
|
|
108 |
Cauchy Riemann PDE With extra term on right side |
✗ |
0.001 |
✓ |
0.052 |
|
|
|
|
109 |
Hamilton-Jacobi type PDE |
✗ |
0.007 |
✓ |
0.157 |
|
|
|
|
110 |
\(u_t + u_{xxx} = 0\) |
✗ |
0.002 |
✓ |
0.064 |
Yes |
|
|
|
111 |
Bateman-Burgers \(u_t+u u_x = \nu u_{xx}\) |
✓ |
0.02 |
✓ |
0.06 |
|
|
|
|
112 |
Benjamin Bona Mahony \(u_t+u_x + u u+x - u_{xxt} = 0\) |
✓ |
0.031 |
✓ |
0.075 |
|
|
|
|
113 |
Benjamin Ono \(u_t+H u_{xx} +u u_x = 0\) |
✓ |
0.02 |
✓ |
0.062 |
|
|
|
|
114 |
Born Infeld \((1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0\) |
✓ |
0.008 |
✓ |
0.135 |
|
|
|
|
115 |
Boussinesq \(u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0\) |
✓ |
0.042 |
✓ |
0.086 |
|
|
|
|
116 |
Boussinesq type \(u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0\) |
✓ |
0.041 |
✓ |
0.106 |
|
|
|
|
117 |
Buckmaster \( u_t = (u^4)_{xx} + (u^3)_x\) |
✗ |
0.07 |
✓ |
0.352 |
|
|
|
|
118 |
Camassa Holm \(u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx}\) |
✗ |
0.174 |
✓ |
1.52 |
|
|
|
|
119 |
Chaffee Infante \(u_t = u_{xx} + \lambda (u^3 - u) = 0\) |
✗ |
0.093 |
✓ |
0.176 |
|
|
|
|
120 |
Clarke. \(\left ( \theta _t - \gamma e^\theta \right )_{tt} = \left ( \theta _t - e^\theta \right )_{xx}\) |
✗ |
0.01 |
✗ |
0.025 |
|
|
|
|
121 |
Degasperis Procesi \(u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx}\) |
✗ |
0.169 |
✓ |
0.52 |
|
|
|
|
122 |
Dym equation \(u_t =u^3 u_{xxx}\) |
✗ |
0.083 |
✓ |
0.375 |
|
|
|
|
123 |
Estevez Mansfield Clarkson \(u_{tyyy} + \beta u_y u_{yt} + \beta u_{yy} u_t + u_{tt} = 0\) |
✓ |
0.033 |
✓ |
0.121 |
|
|
|
|
124 |
Fisher’s \(u_t = u(1-u)+u_{xx}\) |
✓ |
0.05 |
✓ |
0.165 |
|
|
|
|
125 |
Hunter Saxton \(\left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2\) |
✗ |
0.044 |
✓ |
0.107 |
|
|
|
|
126 |
Kadomtsev Petviashvili \( \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \) |
✓ |
0.071 |
✓ |
0.125 |
|
|
|
|
127 |
Klein Gordon \(u_{xx}+u_{yy}+ \lambda u^p=0\) |
✗ |
0.004 |
✗ |
0.017 |
|
|
|
|
128 |
Klein Gordon \(u_{xx}+u_{yy}+ u^2=0\) |
✗ |
0.228 |
✓ |
0.346 |
|
|
|
|
129 |
Khokhlov Zabolotskaya \(u_{x t} - (u u_x)_x = u_{yy}\) |
✗ |
0.057 |
✓ |
0.237 |
|
|
|
|
130 |
Korteweg de Vries (KdV) \(u_t + (u_x)^3+ 6 u u_x = 0\) |
✓ |
0.029 |
✓ |
0.092 |
|
|
|
|
131 |
Lin Tsien \(2 u_{tx} + u_x u_{xx} - u_{yy} = 0\) |
✗ |
0.072 |
✓ |
0.243 |
|
|
|
|
132 |
Liouville \(u_{xx} + u_{yy} +e^{\lambda u} = 0\) |
✗ |
0.004 |
✗ |
0.017 |
|
|
|
|
133 |
Plateau \((1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0\) |
✗ |
0.032 |
✓ |
0.156 |
|
|
|
|
134 |
Rayleigh \(u_{tt} - u_{xx} = \epsilon (u_t - u_t^3)\) |
✗ |
0.078 |
✓ |
0.127 |
|
|
|
|
135 |
Sawada Kotera \(u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \) |
✓ |
0.085 |
✓ |
0.187 |
|
|
|
|
136 |
Sine Gordon \(\phi _{tt} - \phi _{xx} + \sin \phi = 0\) |
✗ |
0.007 |
✗ |
0.013 |
|
|
|
|
137 |
Sinh Gordon \( u_{xt} = \sinh u\) |
✗ |
0.007 |
✗ |
0.014 |
|
|
|
|
138 |
Sinh Poisson \(u_{xx}+u_{yy} + \sinh u=0\) |
✗ |
0.006 |
✗ |
0.011 |
|
|
|
|
139 |
Thomas equation \( u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0\) |
✗ |
0.06 |
✓ |
0.339 |
|
|
|
|
140 |
phi equation \(\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0\) |
✓ |
0.043 |
✓ |
0.472 |
|
|
|
|
141 |
\(S S_{xy} + S_x S_y = 1\) |
✗ |
0.031 |
✓ |
0.023 |
|
|
|
|
142 |
\(u_{rr} + u_{\theta \theta } = 0\) |
✗ |
0.012 |
✓ |
0.513 |
|
|
|
|
143 |
\( u_{xx} + y u_{yy} = 0\) |
✓ |
3.551 |
✓ |
2.448 |
|
|
|
|
144 |
\(u_t + u_{xxx} = 0\) |
✗ |
0.413 |
✓ |
5.663 |
|
|
|
|
145 |
\(u_{xy} = \sin (x) \sin (y) \) |
✗ |
0.271 |
✓ |
0.426 |
|
|
|
|
146 |
\(w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3}\) |
✗ |
0.003 |
✓ |
0.621 |
|
|
|
|
147 |
Linear PDE, initial conditions at \(t=t_0\) |
✗ |
0.003 |
✓ |
0.616 |
|
|
|
|
148 |
second order in time, Linear PDE, initial conditions at \(t=t_0\) |
✗ |
0.003 |
✓ |
2.179 |
|
|
|
|
149 |
Einstein-Weiner \(u_t = -\beta u_x + D u_{xx}\) |
✗ |
0.004 |
✓ |
0.194 |
|
|
|
|
150 |
Using integral transforms. \({x}^{2}{\frac {\partial ^{2}}{\partial {x}^{2}}}u \left ( x,y \right ) +x{\frac {\partial }{\partial x}}u \left ( x,y \right ) +{\frac {\partial ^{2}}{\partial {y}^{2}}}u \left ( x,y \right ) =0\) |
✗ |
0.982 |
✓ |
2.129 |
|
|
|
|
|
||||||||
|
|
||||||||
|
|
||||||||
|
|
||||||||
|
|
||||||||
