1.4 Result lookup table on one web page

Miscellaneous PDE’s

Table 1.12: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

0.085

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

0.046

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

0.02

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

0.05

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

0.02

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

0.046

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

0.077

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

0.05

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

0.135

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

0.072

Yes

Yes










11

General first order

\(3 u_x + 5 u_y = x\)

0.007

0.011

Yes










12

General first order

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

0.024

0.058

Yes










13

General first order

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

0.121

0.303

Yes










14

General first order

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

0.005

0.03

Yes










15

General first order

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

0.004

0.022

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

0.32

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

0.042

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

0.062

Yes










19

General first order

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

0.16

0.039

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

0.126

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

0.724










22

General first order

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

0.054

0.134










23

General first order

Recover a function from its gradient vector

0.032

0.085










24

General first order

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

0.055

0.039

Yes










25

General first order

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

0.132

0.562










26

General first order

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

0.017

0.01

Yes










27

General first order

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

0.018

0.012

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.

2.163

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

0.174

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

0.018










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

0.679










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

0.06










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

0.24










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

0.009










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

0.009










39

General first order

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

0.126

0.024










40

General first order

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

105.34

0.37










41

General first order

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

0.137

0.037










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

0.011










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

0.016










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

0.028










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

0.009










47

General first order

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

0.006

0.017










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

0.616










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

0.079










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

0.012










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

0.07










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

0.054










53

General first order

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

0.032

0.013










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

0.934










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

0.044










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

0.256










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

0.01










60

General first order

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

0.011

0.021










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

0.913










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

0.154










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

0.042










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

0.045










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

0.012










66

General first order

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

0.084

0.045










67

General first order

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

(Timed out)

600.

1.518










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

0.044










69

General first order

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

0.057

0.047










70

General first order

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

0.163

0.098

Yes










71

General first order

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

0.074

0.004

Yes










72

General first order

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

0.007

0.023

Yes










73

General first order

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

0.02

0.03

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

0.011

Yes










76

General first order

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

0.005

0.01

Yes










77

General first order

\(u_x u_y = 1\)

0.003

0.018

Yes










78

General first order

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

1.493

0.272

Yes










79

Solved by factoring into two transport equations

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

0.012

0.102

Yes










80

Solved by factoring into two transport equations

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

0.013

0.298

Yes










81

Solved by factoring into two transport equations

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

0.013

2.132

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

1.895










83

Schrodinger PDE

pict

Logan textbook, page 30

0.527

0.472










84

Schrodinger PDE

pict

In a square, zero potential

0.707

4.106










85

Schrodinger PDE

pict

From Mathematica help pages

0.756

0.929










86

Schrodinger PDE

pict

From Mathematica help pages

0.006


Trivial solution. Maple does not support \(\infty \) in boundary conditions

9.75










87

Schrodinger PDE

pict

David Griffiths, page 47

30.831

2.01










88

Schrodinger PDE

pict

David Griffiths, page 47

0.678

1.058










89

Schrodinger PDE

pict

In a square

0.648

6.622










90

Beam PDE

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

1.711

0.234










91

Burger’s PDE

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


Implicit solution

0.028

0.026

Yes










92

Burger’s PDE

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

0.011

0.062

Yes










93

Burger’s PDE

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

0.039

0.112










94

Burger’s PDE

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

10.918

0.92










95

Burger’s PDE

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

39.853

0.658










96

Black Scholes PDE

classic Black Scholes model from finance, European call version

3.061

1.014










97

Black Scholes PDE

Boundary value problem for the Black Scholes equation

4.421

2.492










98

Korteweg-deVries PDE

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

0.031

0.193










99

Tricomi PDE

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

9.736

3.933










100

Tricomi PDE

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

0.011

3.122










101

Keldysh equation

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

0.007

3.77










102

Euler-Poisson-Darboux equation

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

0.008

0.117










103

Euler-Poisson-Darboux equation

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

0.008

0.49










104

Euler-Poisson-Darboux equation

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

2.613

5.677










105

Chaplygin’s equation

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

0.027

1.459










106

Cauchy Riemann PDE’s

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

0.011

0.177










107

Cauchy Riemann PDE’s

Cauchy Riemann PDE With extra term on right side

0.003

0.077










108

Hamilton-Jacobi PDE

Hamilton-Jacobi type PDE

0.01

0.237










109

Airy PDE

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

0.059

0.104

Yes










110

Nonlinear PDE’s

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

0.03

0.081










111

Nonlinear PDE’s

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

0.036

0.092










112

Nonlinear PDE’s

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

0.031

0.078










113

Nonlinear PDE’s

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

0.014

0.163










114

Nonlinear PDE’s

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

0.058

0.11










115

Nonlinear PDE’s

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

0.048

0.402










116

Nonlinear PDE’s

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

0.113


Answer in terms of RootOf.

0.465










117

Nonlinear PDE’s

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

0.199


Answer in terms of RootOf.

1.628










118

Nonlinear PDE’s

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

0.105

0.226










119

Nonlinear PDE’s

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

0.013

0.06










120

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

0.671










121

Nonlinear PDE’s

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

0.096


has RootOf

1.051










122

Nonlinear PDE’s

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

0.041

0.15










123

Nonlinear PDE’s

Fisher’s \(u_t = u(1-u)+u_{xx}\)

0.063

0.162










124

Nonlinear PDE’s

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

0.051


with RootOf

0.155










125

Nonlinear PDE’s

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

0.073

0.132










126

Nonlinear PDE’s

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

0.006

0.025










127

Nonlinear PDE’s

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

0.237

0.387










128

Nonlinear PDE’s

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

0.073

0.275










129

Nonlinear PDE’s

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

0.038

0.094










130

Nonlinear PDE’s

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

0.09

0.265










131

Nonlinear PDE’s

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

0.006

0.037










132

Nonlinear PDE’s

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

0.039

0.173










133

Nonlinear PDE’s

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

0.099


Has RootOf

0.167










134

Nonlinear PDE’s

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

0.093

0.176










135

Nonlinear PDE’s

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

0.01

0.026










136

Nonlinear PDE’s

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

0.01

0.031










137

Nonlinear PDE’s

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

0.009

0.016










138

Nonlinear PDE’s

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

0.074

0.45










139

Nonlinear PDE’s

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

0.053

0.165










140

more miscellaneous

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

0.04

0.052










141

more miscellaneous

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

30.952

0.746










142

more miscellaneous

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

8.842

3.441










143

more miscellaneous

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

0.191

15.76










144

more miscellaneous

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

5.14

0.619










145

more miscellaneous

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

2.907

0.847










146

more miscellaneous

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

3.923

0.825










147

more miscellaneous

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

2.226

2.962










148

more miscellaneous

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

0.046

0.362










149

more miscellaneous

Using integral transforms.

40.116

2.906



















Parabolic PDE’s (Diffusion)

Table 1.13:Parabolic PDE’s (Diffusion) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










150

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

General initial conditions

62.537

2.083

Yes










151

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Specific initial condition

26.091

1.237

Yes

Yes










152

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

IC \(u=x(1-x)\)

27.383

0.931

Yes

Yes










153

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (a)

0.42

3.671

Yes










154

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

IC hat function

35.813

2.606

Yes

Yes










155

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (b)

1.236

4.116

Yes










156

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (c)


but \(n=3\) should be special case

65.187


handled \(n=3\) case correctly.

15.515

Yes










157

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (d)

545.655

40.644

Yes










158

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

18.891

1.61

Yes










159

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

62.92

1.042

Yes

Yes










160

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (general case)

42.268

2.872

Yes










161

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (special case)

32.815

2.741

Yes

Yes










162

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space and time (general case)

71.123

4.79

Yes










163

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space and time (special case)

35.558

14.278

Yes

Yes










164

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Math 4567 Exam

18.115

3.305

Yes










165

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

With source

67.959

8.844










166

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

special initial condition

3.189

(Timed out)

600.

Yes










167

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction and Euler-Cauchy Sturm-Liouville

1.816

7.57










168

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Using growth form reaction term

93.031

1.13

Yes










169

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using logistic form for reaction term

4.709

6.938










170

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using Aleee form for reaction term

7.192

11.236










171

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Haberman 2.3.8

34.92

3.22

Yes










172

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

34.502

1.977

Yes










173

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

0.774

2.005










174

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

43.242

1.68










175

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (general case)

3.081

2.905

Yes










176

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (special case)

63.055

2.459

Yes

Yes










177

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.2 (general case)

30.562

4.834

Yes










178

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Left end zero, right end insulated, no source

29.39

2.623

Yes










179

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

One end insulated

32.45

2.755










180

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.7 (general case)

32.108

2.419

Yes










181

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

specific case

12.122

1.513

Yes

Yes










182

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (a)

(Timed out)

600.

39.665










183

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Both ends insulated, no source

33.1

3.24

Yes










184

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (b) (special case)

1.606

5.717

Yes










185

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (c) (special case)

32.364

19.146

Yes










186

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (d)

1.387

5.056










187

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

both ends insulated

91.769

12.132










188

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

32.721

1.529










189

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.25

42.65

13.906










190

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

external source

18.726

38.169










191

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction (general case)

91.872

1.319

Yes

Yes










192

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.23

64.245

24.371










193

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.21

63.132

5.933










194

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Euler-Cauchy Sturm-Liouville

2.684

38.371

Yes










195

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

32.892

2.715

Yes










196

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Mixed BC

33.75

5.871










197

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 8.2.1 (f)

63.62

7.951










198

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

left end insulated (general case)

33.393

3.734

Yes










199

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

left end insulated (special case)

7.485

2.226

Yes

Yes










200

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

right end nonhomogeneous BC (general case)

66.082

3.428

Yes










201

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

right end nonhomogeneous BC (special case)

0.01

3.907

Yes










202

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

right end nonhomogeneous BC, special case

10.202

3.405

Yes

Yes










203

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

convection heat loss

39.799

(Timed out)

600.

Yes










204

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

nonhomogeneous BC

2.176

10.863










205

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

nonhomogeneous BC

92.738

2.184










206

Diffusion in 1D

Finite domain (bar), left end homogeneous, right end not

pict

nonhomogeneous BC

34.188

20.303

Yes










207

Diffusion in 1D

Finite domain (bar), right end homogeneous, left end not

pict

left end BC depends on time (general case)

66.569

15.038

Yes










208

Diffusion in 1D

Finite domain (bar), right end homogeneous, left end not

pict

left end BC depends on time (special case)

88.627

15.167

Yes

Yes










209

Diffusion in 1D

Finite domain (bar), right end homogeneous, left end not

pict

left end BC depends on time (special case)

86.742

9.942

Yes

Yes










210

Diffusion in 1D

Finite domain (bar), right end homogeneous, left end not

pict

Haberman 8.3.6 (special case)

95.024

25.764

Yes

Yes










211

Diffusion in 1D

Finite domain (bar), right end homogeneous, left end not

pict

BC depends on time (special case)

12.162

5.486










212

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). IC given

15.368

9.892










213

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). No IC given

0.014

1.958

Yes










214

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). Damped heat PDE. No IC given

0.071

2.001

Yes










215

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends non-homogeneous BC (general case)

63.746

6.998

Yes










216

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

non-homogeneous BC (special case)

16.942

3.002

Yes










217

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.1 (special case)

63.188

2.931

Yes










218

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (general case)

52.613

5.634

Yes










219

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (special case)

94.816

3.601

Yes

Yes










220

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space and time only (general case)

69.048

13.045

Yes










221

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (general case)

84.436

29.074

Yes










222

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (special case)

67.824

34.132

Yes

Yes










223

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends nonhomogeneous

63.852

33.68

Yes










224

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (a) (general case)

38.949

8.092

Yes










225

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (d) (general solution)

54.039

6.713

Yes










226

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time with source that depends on space only (general solution)

115.005

(Timed out)

600.

Yes










227

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time with source that depends on space only (special case)

104.862

37.183

Yes

Yes










228

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time with source that depends on time and space (general solution)

79.308

18.269

Yes










229

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time with source present (special case)

117.548

74.248

Yes

Yes










230

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Pinchover and Rubinstein 6.17

21.971

12.96










231

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

nonhomogeneous BC

111.737

31.023










232

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.2. (a)

122.354

25.014

Yes










233

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.3

5.486

18.149










234

Diffusion in 1D

Semi-infinite domain

pict

left end constant (general case)

60.035

2.283

Yes










235

Diffusion in 1D

Semi-infinite domain

pict

left end constant (special case)


It fail if assumption \(x>0\) is given. A bug

31.691

3.029

Yes

Yes










236

Diffusion in 1D

Semi-infinite domain

pict

Logan p. 76. Left end general function of time (general case)

0.577

7.363

Yes










237

Diffusion in 1D

Semi-infinite domain

pict

Left end function of time (special case)

60.047

12.418

Yes

Yes










238

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.029

2.499










239

Diffusion in 1D

Semi-infinite domain

pict

I.C. not zero


due to IC/BC not zero

13.357

3.183










240

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.074

2.483










241

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.071

6.834










242

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous B.C.

60.033

3.126










243

Diffusion in 1D

Semi-infinite domain

pict

Unit triangle I.C.

61.013

4.208










244

Diffusion in 1D

Semi-infinite domain

pict

I.C. not at \(t=0\)

61.517

6.016










245

Diffusion in 1D

Semi-infinite domain

pict

Diffusion with advection

1.75

5.401










246

Diffusion in 1D

Semi-infinite domain

pict

Practice exam problem

60.108

3.071










247

Diffusion in 1D

Infinite domain

pict

Inverse exponential I.C.

0.869

3.994

Yes










248

Diffusion in 1D

Infinite domain

pict

Advection term

0.067

2.934










249

Diffusion in 1D

Infinite domain

pict

UnitBox I.C.

60.016

6.347










250

Diffusion in 1D

Infinite domain

pict

No source

0.4

6.829










251

Diffusion in 1D

Infinite domain

pict

constant as source

0.425

3.457










252

Diffusion in 1D

Infinite domain

pict

No intial conditions

0.004

0.754

Yes










253

Diffusion in 1D

Infinite domain

pict

piecewise I.C.


due to i.c. not at zero

52.493

5.05

Yes










254

Diffusion in 1D

Infinite domain

pict

Practice exam problem

0.065

1.716










255

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

No source

4.249

15.159










256

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Internal source term

10.246

13.63










257

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Articolo 6.6.3

6.641

22.307










258

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency, insulated (General solution)

0.97


Cant get series solution

60.266

Yes










259

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency, insulated (Specific solution)

2.743


Do not understand Maple solution

21.679

Yes

Yes










260

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

0.343

6.467










261

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

1.131

1.321










262

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Haberman 8.3.5 (General solution)

19.48

5.786

Yes










263

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

3.518

4.102

Yes

Yes










264

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

2.722

6.169

Yes

Yes










265

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.1

4.127

3.6










266

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.2

4.582

41.933










267

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Haberman 8.2.5 with \(\theta \) dependency (General case)

372.358

1.783

Yes










268

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

With \(\theta \) dependency (Specific example)

8.97

5.392

Yes

Yes










269

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

With \(\theta \) dependency (specific example)

10.372

8.127

Yes

Yes










270

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Insulated with \(\theta \) dependency (General solution)

4.818

8.399

Yes










271

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Insulated with \(\theta \) dependency (Specific example)

16.146

7.529

Yes

Yes










272

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Insulated with \(\theta \) dependency (Specific example)

(Timed out)

600.

7.495

Yes

Yes










273

Diffusion in 3D

Spherical coordinates

pict

No angle dependencies

61.104


Has unresolved Laplace integrals

3.881










274

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (a)

1.883

73.092

Yes

Yes










275

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (b)

6.469

44.451

Yes

Yes










276

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (c)

3.162

43.613










277

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (a)

0.012

4.304










278

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (b)

0.008

4.246










279

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (c)

0.012

4.19



















Elliptic PDE’s (Laplace, Poisson, Helmholtz)

Table 1.14:Elliptic PDE’s (Laplace, Poisson, Helmholtz) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










280

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge not

1.78

78.191

Yes










281

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, right edge not

1.83

8.617

Yes










282

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge has impulse

4.12

27.954

Yes










283

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (a)

0.988

64.358










284

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (b)

0.944

65.826










285

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (c)

4.685

54.033










286

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (d)

4.133

82.87










287

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (e)

19.674

76.588

Yes










288

Laplace in 2D

Cartesian coordinates

pict

Unit triangle B.C.

0.985

21.389










289

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

3.586

10.857

Yes










290

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.813

7.947

Yes










291

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

3.514

7.078

Yes










292

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

41.351

66.157

Yes










293

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.994

85.598










294

Laplace in 2D

Cartesian coordinates

Laplace PDE in 2D Cartessian with boundary condition as Dirac function

0.044

3.888










295

Laplace in 2D

Cartesian coordinates

pict

One side homogeneous

1.692

114.204










296

Laplace in 2D

Cartesian coordinates

pict

In right half plane

21.834

4.683










297

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.434

49.432










298

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem Upper half

1.371

5.602










299

Laplace in 2D

Cartesian coordinates

pict

Right half-plane

19.623

8.136










300

Laplace in 2D

Cartesian coordinates

pict

First quadrant

14.632

35.432










301

Laplace in 2D

Cartesian coordinates

pict

Neumann problem upper half-plane

4.77


used convert(sol,Int).

9.987










302

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

2.303

12.565










303

Laplace in 2D

Cartesian coordinates

pict

Strip in upper half

60.258

13.794










304

Laplace in 2D

Cartesian coordinates

pict

in Rectangle, right edge at infinity

2.353

48.434










305

Laplace in 2D

Polar coordinates

pict

Laplace PDE inside quarter disk, Neumann BC at edge

3.459

6.01










306

Laplace in 2D

Polar coordinates

pict

\(r=4\) and \(u=x^4\) at boundary of disk

150.086

20.028

Yes










307

Laplace in 2D

Polar coordinates

pict

\(r=1\) and \(u_r=x\) at boundary of disk

2.589

2.19

Yes










308

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. General solution

0.624

12.795










309

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. Specific boundary conditions

3.936

11.652

Yes










310

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.5 (c)

3.989

5.669

Yes










311

Laplace in 2D

Polar coordinates

pict

semi-circle

2.08

2.38










312

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.8 (b)

8.91

250.

Yes










313

Laplace in 2D

Polar coordinates

pict

Circular annulus

0.519

6.887










314

Laplace in 2D

Polar coordinates

pict

Outside a disk

6.915

9.466










315

Laplace in 2D

Polar coordinates

pict

Outside a disk

4.21

16.356

Yes










316

Laplace in 3D

Spherical coordinates

In a sphere

0.036

3.632










317

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (a)

0.018

1.206










318

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (b)

0.015

0.948










319

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (c)

0.015

0.984










320

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (d)

0.014

1.03










321

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (e)

0.015

1.014










322

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (a)

0.015

1.875










323

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (b)

0.015

54.412










324

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (c)

0.02

55.061










325

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (d)

0.014

1.882










326

Poisson in 2D

Cartesian coordinates

pict

All boundaries at zero

0.015

79.61

Yes










327

Poisson in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

1.368

0.457










328

Poisson in 2D

Cartesian coordinates

pict

Poisson PDE in whole 2D plane

31.393

0.07

Yes










329

Helmholtz in 2D

Cartesian coordinates

pict

In rectangle

64.18

35.915










330

Helmholtz in 2D

Cartesian coordinates

pict

On whole plane


why? It solved earlier with BC?

0.004

0.282










331

Helmholtz in 2D

Cartesian coordinates

pict

Reduced Helmholtz Inside square

62.146

98.688










332

Helmholtz in 3D

Spherical coordinates

Chain reaction PDE

0.123


trivial solution

2.657

Yes



















Hyperbolic PDE’s (Wave)

Table 1.15: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

pict

General solution for both ends fixed. Domain is \(0\dots L\)

33.33

27.737

Yes










334

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital position zero (special case)

3.324

15.475

Yes

Yes










335

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital velocity zero (special case)

2.518

10.231

Yes

Yes










336

Wave PDE in 1D

Finite length string

pict

both ends fixed but domain is \(-\pi \dots \pi \). zero intial position, non zero initial velocity

6.978

85.345

Yes

Yes










337

Wave PDE in 1D

Finite length string

pict

both ends fixed but domain is \(-1 \dots 1\). intial position is an impulse, zero initial velocity

1.749

23.163

Yes










338

Wave PDE in 1D

Finite length string

pict

Logan book, page 28. Both ends fixed

0.974

2.69










339

Wave PDE in 1D

Finite length string

pict

non-zero initial velocity. Both ends fixed

86.507

12.629










340

Wave PDE in 1D

Finite length string

pict

Logan book page 149)

20.85

13.871










341

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (a)

23.852

57.95










342

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (b)

38.58

37.831

Yes










343

Wave PDE in 1D

Finite length string

pict

Both I.C. not zero

22.264

18.398










344

Wave PDE in 1D

Finite length string

pict

With constant source

34.232

28.917










345

Wave PDE in 1D

Finite length string

pict

Logan page 213

4.763

20.611










346

Wave PDE in 1D

Finite length string

pict

Telegraphy PDE

86.878

20.632










347

Wave PDE in 1D

Finite length string

pict

Dispersion term present (general case)


Due to adding dispersion term

25.235

26.439

Yes










348

Wave PDE in 1D

Finite length string

pict

Dispersion term present


Due to adding dispersion term

92.86

110.806

Yes










349

Wave PDE in 1D

Finite length string

pict

Dispersion term present (specific case)


Due to adding dispersion term

94.897

25.956

Yes

Yes










350

Wave PDE in 1D

Finite length string

pict

non-zero initial position


But sum should not include \(n=2\)

7.991


Handled \(n=2\) case correctly

86.579










351

Wave PDE in 1D

Finite length string

pict

With source

10.847

83.973










352

Wave PDE in 1D

Finite length string

pict

Right end free (general case)

25.353

36.437

Yes










353

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (general case)

7.149

26.58

Yes










354

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (special case)

89.446

20.785

Yes

Yes










355

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (general case)

52.43

32.879

Yes










356

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, underdamped)

114.95

34.069

Yes

Yes










357

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, critical damped)

114.627

45.159

Yes

Yes










358

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity, damping present (special case, over damped)

116.345

32.581

Yes

Yes










359

Wave PDE in 1D

Finite length string

pict

I.C. at different times, right end free, with source

40.028

87.878

Yes










360

Wave PDE in 1D

Finite length string

pict

Right end oscillates

66.861

67.83










361

Wave PDE in 1D

Finite length string

pict

Perioidic B.C.

2.412

1.579










362

Wave PDE in 1D

Finite length string

pict

Mixed B.C.

15.603

1.116










363

Wave PDE in 1D

Finite length string

pict

Left end fixed, right end non-homogeneous Neumann BC. Zero initial conditions

63.343

27.013

Yes

Yes










364

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed, (general case)

3.996

1.541

Yes










365

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed with specific initial position

15.596

19.95

Yes










366

Wave PDE in 1D

Semi-infinite domain

pict

Logan page 115, left end fixed with source

7.298

2.138










367

Wave PDE in 1D

Semi-infinite domain

pict

Left moving boundary condition

1.94

1.46










368

Wave PDE in 1D

Semi-infinite domain

pict

moving Left end

9.201

3.219










369

Wave PDE in 1D

Semi-infinite domain

pict

I.C. at \(t=1\)

9.147

1.544










370

Wave PDE in 1D

Semi-infinite domain

pict

B.C. at \(x=1\)

76.551

3.115










371

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (general solution)

7.123

5.26










372

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (Special solution)

4.202

3.402

Yes

Yes










373

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed. zero initial velocity (Special solution)

4.373

3.247

Yes

Yes










374

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial position (general solution)

8.691

1.785










375

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. Non zero initial position and velocity (general solution)

33.339

1.563










376

Wave PDE in 1D

Semi-infinite domain

pict

Left end free with source

1.122

1.379










377

Wave PDE in 1D

Infinite domain

pict

General case. \(u_{tt} = u_{xx}\) with \(u(x,0)=f(x),u_t(x,0)=g(x)\)

0.019

0.699










378

Wave PDE in 1D

Infinite domain

pict

General case. No IC given. \(u_{tt} + u_{xt} = c^2 u_{xx}\)

0.005

0.538










379

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}= c^2 u_{xx} + f(x,t)\), IC at \(t=1\),\(u(x,1) = g(x),u_t(x,1)=h(x)\)

0.158

14.809










380

Wave PDE in 1D

Infinite domain

pict

No source. \(u_{tt} = u_{xx}\), with \(u(x,0) =e^{-x^2},u_t(x,0)=1\)

0.002

0.589










381

Wave PDE in 1D

Infinite domain

pict

With source term. \(u_{tt} = u_{xx} + m\)

0.011

0.437










382

Wave PDE in 1D

Infinite domain

pict

non-linear (Solitons) \(u_t +6 u(x,t) u_x + u_{xxx} = 0\)

0.028

0.458

Yes

Yes










383

Wave PDE in 1D

Infinite domain

pict

Inhomogeneous PDE \(3 u_{xx}- u_{tt} + u_{xt}=1\)

0.003

0.329










384

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.029

0.388










385

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.002

0.304










386

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

8.846

0.407










387

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.002

0.309

Yes

Yes










388

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.004

4.24

Yes

Yes










389

Wave PDE in 1D

Infinite domain

pict

General case \(u_{tt} = u_{xx}\) with \(u(x,0)=\sin x, u_t(x,0)=-2 x e^{-x^2}\)

0.1

0.401










390

Wave PDE in 1D

Infinite domain

pict

General case. \(u_{tt} =u_{xx}\) dAlembert solution, box function as initial position

0.003

0.341

Yes

Yes










391

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}=4 u_{xx}+ \cos (t)\) dAlembert solution with \(u(x,0)=\sin x,u_t(x,0)=\cos x\)

0.096

0.737

Yes

Yes










392

Wave PDE in 1D

Infinite domain

pict

\(u_{tt}=c^2 u_{xx}\) dAlembert solution with \(u(x,0)=\delta (x-a),u_t(x,0)=0\)

63.956

0.737

Yes










393

Wave PDE in 1D

Infinite domain

system of 2 inhomogeneous linear hyperbolic system with constant coefficients

0.267

1.359










394

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, General solution

2.271

559.481

Yes










395

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, zero velocity. Specific example

0.424

30.771

Yes

Yes










396

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example

33.848

53.935

Yes

Yes










397

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example, delta in center

9.295

53.67

Yes

Yes










398

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed

4.949

29.946










399

Wave PDE in 2D

Cartesian coordinates

pict

All edges fixed (Haberman 8.5.5 (a)

27.715

0.932










400

Wave PDE in 2D

Cartesian coordinates

pict

2 edgs fixed, 2 free, zero initial velocity

5.952

58.452










401

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, general solution

1.411

(Timed out)

600.

Yes










402

Wave PDE in 2D

Cartesian coordinates

pict

With damping

9.074

87.58










403

Wave PDE in 2D

Cartesian coordinates

pict

On the whole plane

0.171

0.574










404

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency, fixed boundary, general case

3.01


Has unresolved Invlaplace calls

2.532

Yes










405

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency. Specific example. Both initial conditions not zero

2.684


Has unresolved Invlaplace calls

93.789










406

Wave PDE in 2D

Polar coordinates

pict

no \(\theta \) dependency. Specific example. Both initial conditions not zero

2.549


Has unresolved Invlaplace calls. How to get series solution?

39.994

Yes

Yes










407

Wave PDE in 2D

Polar coordinates

no \(\theta \) dependency. Using integral transforms. Source present. Specific example

3.628

18.789










408

Wave PDE in 2D

Polar coordinates

no \(\theta \) dependency. Using integral transforms. Source present. Specific example

2.511

2.416










409

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, general solution

9.409

6.898

Yes










410

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial velocity, general solution

4.687

5.48

Yes










411

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial velocity, specific example

15.449

7.102

Yes

Yes










412

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial position, specific example

7.747

6.869

Yes

Yes










413

Wave PDE in 2D

Polar coordinates

pict

\(\theta \) dependency, fixed on edges, zero initial position with internal source (Haberman 8.5.5. (b)

0.042

0.692

Yes










414

Wave PDE in 3D

Spherical coordinates

pict

No I.C. no B.C.

0.039

5.077










415

Wave PDE in 3D

Cylindrical coordinates

pict

No I.C. no B.C.

0.006

0.887



















Handbook of first order partial differential equations

Table 1.16:Handbook of first order partial differential equations breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










416

chapter 1

problem number 1

0.005

0.089










417

chapter 1

problem number 2

0.004

0.007










418

chapter 1

problem number 3

0.017

0.006

Yes










419

chapter 1

problem number 4

0.016

0.004










420

chapter 1

problem number 5

0.032

0.021

Yes










421

chapter 1

problem number 6

0.03

0.018










422

chapter 2

2.1

problem number 1

0.006

0.097

Yes










423

chapter 2

2.1

problem number 2

0.01

0.094

Yes










424

chapter 2

2.1

problem number 3

0.054

0.263

Yes










425

chapter 2

2.1

problem number 4

0.021

0.161

Yes










426

chapter 2

2.1

problem number 5

0.024

0.42

Yes










427

chapter 2

2.1

problem number 6

0.05

0.317

Yes










428

chapter 2

2.1

problem number 7

0.055

4.207

Yes










429

chapter 2

2.1

problem number 8

2.383

7.648










430

chapter 2

2.2

problem number 1

0.009

0.011










431

chapter 2

2.2

problem number 2

0.244

0.467










432

chapter 2

2.2

problem number 3

0.106

0.101










433

chapter 2

2.2

problem number 4

0.385

0.328










434

chapter 2

2.2

problem number 5

0.25

1.324










435

chapter 2

2.2

problem number 6

0.33

0.549










436

chapter 2

2.2

problem number 7

0.353

0.937










437

chapter 2

2.2

problem number 8

0.511

0.727










438

chapter 2

2.2

problem number 9

0.299

0.65










439

chapter 2

2.2

problem number 10

0.171

0.309










440

chapter 2

2.2

problem number 11

0.318

0.417










441

chapter 2

2.2

problem number 12

0.822

1.457










442

chapter 2

2.2

problem number 13

0.113

0.017










443

chapter 2

2.2

problem number 14

0.27

0.961










444

chapter 2

2.2

problem number 15

0.7

0.774










445

chapter 2

2.2

problem number 16

1.564

0.519










446

chapter 2

2.2

problem number 17

0.951

0.966










447

chapter 2

2.2

problem number 18

0.292

0.121










448

chapter 2

2.2

problem number 19

0.294

0.89










449

chapter 2

2.2

problem number 20

0.261

1.122










450

chapter 2

2.2

problem number 21

0.196

1.139










451

chapter 2

2.2

problem number 22

0.348

1.8










452

chapter 2

2.2

problem number 23

0.438

1.015










453

chapter 2

2.2

problem number 24

0.313

0.85










454

chapter 2

2.2

problem number 25

(Timed out)

600.

(Timed out)

600.










455

chapter 2

2.2

problem number 26

165.853

33.849










456

chapter 2

2.2

problem number 27

101.598

1.361










457

chapter 2

2.2

problem number 28

0.528


solution contains RootOf

0.111










458

chapter 2

2.2

problem number 29

101.315

1.211










459

chapter 2

2.2

problem number 30

0.305

0.132










460

chapter 2

2.2

problem number 31, Hesse’s equation

235.84

9.247










461

chapter 2

2.3

problem number 1


But it can’t solve it when assuming \(b>0\) which is strange.

0.491

0.53










462

chapter 2

2.3

problem number 2

0.074

0.161










463

chapter 2

2.3

problem number 3

0.179

0.486










464

chapter 2

2.3

problem number 4

0.277

0.627










465

chapter 2

2.3

problem number 5

0.932


Answer contains RootOf

0.075










466

chapter 2

2.3

problem number 6

0.232


Answer contains RootOf

0.035










467

chapter 2

2.3

problem number 7

0.234

0.083










468

chapter 2

2.3

problem number 8

0.265

0.042










469

chapter 2

2.3

problem number 9

(Timed out)

600.

0.036










470

chapter 2

2.4

problem number 1

0.037

0.025










471

chapter 2

2.4

problem number 2

0.268

0.262










472

chapter 2

2.4

problem number 3

0.196

0.194










473

chapter 2

2.4

problem number 4

0.467

0.309










474

chapter 2

2.4

problem number 5

0.53

(Timed out)

600.










475

chapter 2

2.4

problem number 6

1.952

0.254










476

chapter 2

2.4

problem number 7

1.872

3.05










477

chapter 2

2.5

problem number 1

0.058

0.087










478

chapter 2

2.5

problem number 2

0.134

0.57










479

chapter 2

2.5

problem number 3

0.198

0.098










480

chapter 2

2.5

problem number 4

20.865

0.613










481

chapter 2

2.5

problem number 5

0.933

0.379










482

chapter 2

2.5

problem number 6

1.853

0.662










483

chapter 2

2.5

problem number 7

0.458

0.081










484

chapter 2

2.5

problem number 8

23.262

1.414










485

chapter 2

2.5

problem number 9

1.15

0.698










486

chapter 2

2.5

problem number 10

2.964

1.692










487

chapter 2

2.5

problem number 11

26.182

6.795










488

chapter 2

2.5

problem number 12

25.253

4.863










489

chapter 2

2.5