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 \(u_t+ u_x = 0\)

0.046

0.014

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)

0.007

0.061

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

0.023

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)

0.005

0.047

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)

0.005

0.023

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

0.073

0.052

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

0.023

0.157

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

0.006

0.107

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

0.032

0.213

Yes

Yes

10

General first order

Transport equation \(u_t+\frac {1}{x^2+4} u_x= 0\) IC \(u(x,0)=e^{x^3+12 x}\)

0.015

0.085

Yes

Yes

11

General first order

\(3 u_x + 5 u_y = x\)

0.007

0.012

Yes

12

General first order

\(x u_y + y u_x = -4 x y u\) and \(u(x,0)=e^{-x^2}\)

0.024

0.155

Yes

13

General first order

\(u_t + u_x = 0\) and \(u(x,0)=\sin x\) and \(u(0,t)=0\)

0.121

0.374

Yes

14

General first order

\(u_t+ c u_x = 0\) and \(u(x,0)=e^{-x^2}\)

0.005

0.026

Yes

15

General first order

(Haberman 12.2.2) \(\omega _t -3 \omega _x = 0\) and \(\omega (x,0)=\cos x\)

0.004

0.021

Yes

16

General first order

(Haberman 12.2.4) \(\omega _t +c \omega _x = 0\) and \(\omega (x,0)=f(x)\) and \(\omega (0,t)=h(t)\)

1.705


Solution contains unresolved invlaplace calls

0.432

Yes

17

General first order

(Haberman 12.2.5 (a)) \(\omega _t +c \omega _x = e^{2 x}\) and \(\omega (x,0)=f(x)\)

0.048

0.086

Yes

18

General first order

(Haberman 12.2.5 (d)) \(\omega _t +3 t \omega _x = \omega (x,t)\) and \(\omega (x,0)=f(x)\)

1.118

0.096

Yes

19

General first order

\( 2 u_x + 5 u_y = u^2(x,y) + 1\)

0.16

0.038

Yes

20

General first order

Clairaut equation \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\)

0.054

0.191

Yes

21

General first order

Clairaut equation. \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\) with \(u(x,0)= \frac {1}{2} (1-x^2)\)

0.014

0.813

22

General first order

Clairaut equation. \(u = x u_x+ y u_y + \sin ( u_x + u_y )\)

0.054

0.022

23

General first order

Recover a function from its gradient vector

0.032

0.089

24

General first order

\(x f_y - f_x = \frac {g(x)}{h(y)} f^2\)

0.055

0.04

Yes

25

General first order

\(f_x + (f_y)^2 = f(x,y,z)+z\)

0.132

0.533

26

General first order

\(x u_x+y u_y=u\) (Example 3.5.1 in Lokenath Debnath)

0.017

0.011

Yes

27

General first order

\(x u_x+y u_y=n u\) Example 3.5.2 in Lokenath Debnath

0.018

0.01

Yes

28

General first order

\(x^2 u_x+y^2 u_y=(x+y) u\) Example 3.5.3 in Lokenath Debnath

0.12

0.027

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.

2.291

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)

0.489

0.162

Yes

31

General first order

\(u_x-u_y=1\) with \(u(x,0)=x^2\) Example 3.5.6 in Lokenath Debnath

0.005

0.019

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

2.078

0.545

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

0.235

0.063

34

General first order

\(u_t+u u_x=x\) with \(u(x,0)=f(x)\) Example 3.5.11 in Lokenath Debnath.

3.678

0.256

35

General first order

\(u_x=0\) Problem 3.3(a) Lokenath Debnath

0.004

0.003

36

General first order

\(a u_x+b u_y=0\) Problem 3.3(b) Lokenath Debnath

0.009

0.01

37

General first order

\(u_x+y u_y=0\) Problem 3.3(c) Lokenath Debnath

0.029

0.011

38

General first order

\((1+x^2) u_x+ u_y=0\) Problem 3.3(d) Lokenath Debnath

0.009

0.01

39

General first order

\(2 x y u_x+(x^2+y^2)u_y=0\) Problem 3.3(e) Lokenath Debnath

0.126

0.033

40

General first order

\((y+u) u_x+y u_y=x-y\) Problem 3.3(f) Lokenath Debnath

105.34

0.527

41

General first order

\(y^2 u_x- x y u_y=x(u-2 y)\) Problem 3.3(g) Lokenath Debnath

0.137

0.038

42

General first order

\(y u_y - x u_x = 1\) Problem 3.3(h) Lokenath Debnath

0.02

0.009

43

General first order

\(u_x+2 x y^2 u_y=0\) Problem 3.4 Lokenath Debnath

0.126

0.014

44

General first order

\(3 u_x+2 u_y=0\) with \(u(x,0)=\sin x\). Problem 3.5(a) Lokenath Debnath

0.005

0.018

45

General first order

\(y u_x+x u_y=0\) with \(u(0,y)=e^{-y^2}\). Problem 3.5(b) Lokenath Debnath

0.023

0.033

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

0.017

0.008

47

General first order

\(u_x+x u_y=0\) with \(u(0,y)=\sin y\). Problem 3.5(d) Lokenath Debnath

0.006

0.021

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

3.803

0.523

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

0.012

0.097

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

0.021

0.01

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

0.065

0.065

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

0.027

0.07

53

General first order

\(x u_x+y u_y+z u_z=0\) Problem 3.8(a) .Lokenath Debnath

0.032

0.014

54

General first order

\(x^2 u_x+y^2 u_y+z(x+y)u_z=0\) Problem 3.8(b) Lokenath Debnath

0.12

0.019

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

0.049

0.977

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

0.114

0.068

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

45.677

0.255

58

General first order

\(u_x+x u_y=y\) with \(u(0,y)=y^2\) Problem 3.9(a) Lokenath Debnath

0.007

0.021

59

General first order

\(u_x+x u_y=y\) with \(u(1,y)=2 y\) Problem 3.9(b) Lokenath Debnath

0.007

0.009

60

General first order

\((u_x+u_y)^2-u^2=0\). Problem 3.10 Lokenath Debnath

0.011

0.017

61

General first order

\((y+u)u_x+y u_y=x-y\) with \(u(x,1)=1+x\). Problem 3.11 Lokenath Debnath

211.318

1.067

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

0.021

0.161

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

0.034

0.073

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

0.035

0.05

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

0.019

0.016

66

General first order

\(u_x+u u_y=1\) with \(u(0,y)=a y\). Problem 3.16(a) Lokenath Debnath

0.084

0.026

67

General first order

\((y+u)u_x+(x+u)u_y=x+y\). Problem 3.17(a) Lokenath Debnath

(Timed out)

600.

1.372

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

0.065

0.036

69

General first order

\((x+y) u_x + (x-y)u_y =0\). Problem 3.17(c) Lokenath Debnath

0.057

0.052

70

General first order

\(y u_x - x u_y = e^u\) with \(u(0,y)=y^2-1\)

0.163

0.094

Yes

71

General first order

\(y u_x - x u_y = e^u\)

0.074

0.001

Yes

72

General first order

\(u_t + x u_x = 0\) with \(u(x,0)=x^2\). Math 5587

0.007

0.056

Yes

73

General first order

\(u_t + t u_x = 0\) with \(u(x,0)=e^x\)

0.02

0.202

Yes

74

General first order

\(2 u_x + 3 u_y = 1\)

0.008

0.009

Yes

75

General first order

\(x u_t - t u_x = 0\)

0.027

0.02

Yes

76

General first order

\(u_t + u_x = 0\) with \(u(x,1)=\frac {x}{1+x^2}\)

0.005

0.013

Yes

77

General first order

\(u_x u_y = 1\)

0.003

0.02

Yes

78

General first order

\(u_x u_y = u\) with \(u(x,0)=0,u(0,y)=0\)

1.493

0.226

Yes

79

Solved by factoring into two transport equations

\(u_{xx} + u_{xt} - 6 u_{tt} = 0\)

0.012

0.096

Yes

80

Solved by factoring into two transport equations

\(u_{xx} - u_{xt} - 12 u_{tt} = 0\)

0.013

0.276

Yes

81

Solved by factoring into two transport equations

\(u_{xx} - 3 u_{xt} - 4 u_{tt} = 0\)

0.013

2.616

Yes

82

Solved by factoring into two transport equations

\(u_{tt} - 2 u_{xt} - 3 u_{xx} = 0\) with \(u(0,x)=x^2, u_t(x,0)=e^x\)

0.018

2.259

83

Beam PDE

Beam PDE \(u_{tt} + u_{xxxx} = 0\)

1.711

0.205

84

Burger’s PDE

Inviscid Burgers \(u_x + u u_y = 0\)


Implicit solution

0.028

0.025

Yes

85

Burger’s PDE

Inviscid Burgers with I.C. \(u_x+ u u_y = 0\) and \(u(x,0)=\frac {1}{x+1}\)

0.011

0.039

Yes

86

Burger’s PDE

\(u_t+ u u_x = \mu u_{xx}\)

0.039

0.075

87

Burger’s PDE

\(u_t + u u_x + \mu u_{xx}\) with IC

10.918

0.556

88

Burger’s PDE

\(u_t + u u_x + \mu u_{xx}\) IC as UnitBox

39.853

0.606

89

Black Scholes PDE

classic Black Scholes model from finance, European call version

3.061

0.837

90

Black Scholes PDE

Boundary value problem for the Black Scholes equation

4.421

2.12

91

Korteweg-deVries PDE

\(u_{xxx} + u_t -6 u u_x = 0\)

0.031

0.186

92

Tricomi PDE

\(u_{xx} + y u_{yy} = 0\) with \(u(x,0)=0,u_y(x,0)=x^2\)

9.736

3.436

93

Tricomi PDE

\(u_{xx} + x u_{yy} = 0\)

0.011

2.783

94

Keldysh equation

\(x u_{xx} + u_{yy} = 0\)

0.007

4.056

95

Euler-Poisson-Darboux equation

\(u_{xx} + u_{yy} + \frac {\beta }{x} u_x = 0\)

0.008

0.139

96

Euler-Poisson-Darboux equation

\(u_{xx} - u_{yy} + \frac {\beta }{x} u_x = 0\)

0.008

0.487

97

Euler-Poisson-Darboux equation

\(u_{tt} - u_{xx} - \frac {2}{x} u_x = 0\) with \(u(x,0)=0,u_t(x,0)=g(x)\)

2.613

4.295

98

Chaplygin’s equation

\(u_{\theta \theta }+\frac {v^2}{1-\frac {v^2}{c^2}} u_{vv} + v u_v=0\)

0.027

1.266

99

Cauchy Riemann PDE’s

Cauchy Riemann PDE with Prescribe the values of \(u\) and \(v\) on the \(x\) axis

0.011

0.168

100

Cauchy Riemann PDE’s

Cauchy Riemann PDE With extra term on right side

0.003

0.07

101

Hamilton-Jacobi PDE

Hamilton-Jacobi type PDE

0.01

0.191

102

Airy PDE

\(u_t + u_{xxx} = 0\)

0.059

0.115

Yes

103

Nonlinear PDE’s

Bateman-Burgers \(u_t+u u_x = \nu u_{xx}\)

0.03

0.113

104

Nonlinear PDE’s

Benjamin Bona Mahony \(u_t+u_x + u u+x - u_{xxt} = 0\)

0.036

0.115

105

Nonlinear PDE’s

Benjamin Ono \(u_t+H u_{xx} +u u_x = 0\)

0.031

0.106

106

Nonlinear PDE’s

Born Infeld \((1-u_t^2) u_{xx} + 2 u_x u_t u_{xt} - (1+ u_x^2) u_{tt}=0\)

0.014

0.175

107

Nonlinear PDE’s

Boussinesq \(u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0\)

0.058

0.117

108

Nonlinear PDE’s

Boussinesq type \(u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0\)

0.048

0.132

109

Nonlinear PDE’s

Buckmaster \( u_t = (u^4)_{xx} + (u^3)_x\)

0.113


Answer in terms of RootOf.

0.704

110

Nonlinear PDE’s

Camassa Holm \(u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx}\)

0.199


Answer in terms of RootOf.

1.45

111

Nonlinear PDE’s

Chaffee Infante \(u_t = u_{xx} + \lambda (u^3 - u) = 0\)

0.105

0.226

112

Nonlinear PDE’s

Clarke. \(\left ( \theta _t - \gamma e^\theta \right )_{tt} = \left ( \theta _t - e^\theta \right )_{xx}\)

0.013

0.05

113

Nonlinear PDE’s

Degasperis Procesi \(u_t - u_{xxt} + 4 u u_x = 3 u_x u_xx + u u_{xxx}\)

0.195


But still has unresolved ODE’s in solution

0.621

114

Nonlinear PDE’s

Dym equation \(u_t =u^3 u_{xxx}\)

0.096


has RootOf

0.541

115

Nonlinear PDE’s

Estevez Mansfield Clarkson \(u_{tyyy} + \beta u_y u_{yt} + \beta u_{yy} u_t + u_{tt} = 0\)

0.041

0.158

116

Nonlinear PDE’s

Fisher’s \(u_t = u(1-u)+u_{xx}\)

0.063

0.232

117

Nonlinear PDE’s

Hunter Saxton \(\left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2\)

0.051


with RootOf

0.123

118

Nonlinear PDE’s

Kadomtsev Petviashvili \( \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \)

0.073

0.158

119

Nonlinear PDE’s

Klein Gordon \(u_{xx}+u_{yy}+ \lambda u^p=0\)

0.006

0.025

120

Nonlinear PDE’s

Klein Gordon \(u_{xx}+u_{yy}+ u^2=0\)

0.237

0.408

121

Nonlinear PDE’s

Khokhlov Zabolotskaya \(u_{x t} - (u u_x)_x = u_{yy}\)

0.073

0.251

122

Nonlinear PDE’s

Korteweg de Vries (KdV) \(u_t + (u_x)^3+ 6 u u_x = 0\)

0.038

0.115

123

Nonlinear PDE’s

Lin Tsien \(2 u_{tx} + u_x u_{xx} - u_{yy} = 0\)

0.09

0.277

124

Nonlinear PDE’s

Liouville \(u_{xx} + u_{yy} +e^{\lambda u} = 0\)

0.006

0.026

125

Nonlinear PDE’s

Plateau \((1+u_y^2)u_{xx} - 2 u_x u_y y_{xy} + (1+u_x^2) u_{yy} = 0\)

0.039

0.463

126

Nonlinear PDE’s

Rayleigh \(u_{tt} - u_{xx} = \epsilon (u_t - u_t^3)\)

0.099


Has RootOf

0.163

127

Nonlinear PDE’s

Sawada Kotera \(u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \)

0.093

0.183

128

Nonlinear PDE’s

Sine Gordon \(\phi _{tt} - \phi _{xx} + \sin \phi = 0\)

0.01

0.025

129

Nonlinear PDE’s

Sinh Gordon \( u_{xt} = \sinh u\)

0.01

0.028

130

Nonlinear PDE’s

Sinh Poisson \(u_{xx}+u_{yy} + \sinh u=0\)

0.009

0.026

131

Nonlinear PDE’s

Thomas equation \( u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0\)

0.074

0.414

132

Nonlinear PDE’s

phi equation \(\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0\)

0.053

0.138

133

more miscellaneous

\(S S_{xy} + S_x S_y = 1\)

0.04

0.032

134

more miscellaneous

\(u_{rr} + u_{\theta \theta } = 0\)

30.952

0.631

135

more miscellaneous

\( u_{xx} + y u_{yy} = 0\)

8.842

2.995

136

more miscellaneous

\(u_t + u_{xxx} = 0\)

0.191

5.69

137

more miscellaneous

\(u_{xy} = \sin (x) \sin (y) \)

5.14

0.52

138

more miscellaneous

\(w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3}\)

2.907

0.745

139

more miscellaneous

Linear PDE, initial conditions at \(t=t_0\)

3.923

0.718

140

more miscellaneous

second order in time, Linear PDE, initial conditions at \(t=t_0\)

2.226

2.228

141

more miscellaneous

Einstein-Weiner \(u_t = -\beta u_x + D u_{xx}\)

0.046

0.305

142

more miscellaneous

Using integral transforms.

40.116

2.523