1.4 Result lookup table on one web page

Miscellaneous PDE’s

Table 1.13: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 \(u_t+ u_x = 0\)

0.041

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)

0.005

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)

0.004

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

0.058

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

0.018

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

0.005

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

0.07

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

\(3 u_x + 5 u_y = x\)

0.006

0.012

Yes










12

General first order

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

0.023

0.155

Yes










13

General first order

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

0.104

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 \(\omega (0,t)=h(t)\)

1.669


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 \(\omega (x,0)=f(x)\)

0.042

0.086

Yes










18

General first order

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

0.396

0.096

Yes










19

General first order

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

0.138

0.038

Yes










20

General first order

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

0.055

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 \(u(x,0)= \frac {1}{2} (1-x^2)\)

0.01

0.813










22

General first order

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

0.058

0.022










23

General first order

Recover a function from its gradient vector

0.026

0.089










24

General first order

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

0.049

0.04

Yes










25

General first order

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

0.107

0.533










26

General first order

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

0.016

0.011

Yes










27

General first order

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

0.017

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

0.104

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)

0.382

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

0.004

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

2.248

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

0.123

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.

3.382

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

0.007

0.01










37

General first order

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

0.023

0.011










38

General first order

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

0.008

0.01










39

General first order

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

0.101

0.033










40

General first order

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

9.993

0.527










41

General first order

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

0.1

0.038










42

General first order

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

0.016

0.009










43

General first order

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

0.07

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

0.004

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

0.019

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

0.014

0.008










47

General first order

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

0.004

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

3.612

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

0.01

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

0.017

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

0.046

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

0.026

0.07










53

General first order

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

0.027

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

0.045

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

0.035

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

0.091

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

44.354

0.255










58

General first order

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

0.006

0.021










59

General first order

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

0.006

0.009










60

General first order

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

0.01

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

20.216

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

0.018

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

0.022

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

0.027

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

0.018

0.016










66

General first order

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

0.081

0.026










67

General first order

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

139.518

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

0.047

0.036










69

General first order

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

0.048

0.052










70

General first order

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

0.14

0.094

Yes










71

General first order

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

0.067

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 \(u(x,0)=e^x\)

0.018

0.202

Yes










74

General first order

\(2 u_x + 3 u_y = 1\)

0.006

0.009

Yes










75

General first order

\(x u_t - t u_x = 0\)

0.018

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

\(u_x u_y = 1\)

0.002

0.02

Yes










78

General first order

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

1.415

0.226

Yes










79

Solved by factoring into two transport equations

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

0.03

0.096

Yes










80

Solved by factoring into two transport equations

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

0.008

0.276

Yes










81

Solved by factoring into two transport equations

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

0.008

2.616

Yes










82

Solved by factoring into two transport equations

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

0.011

2.259










83

Beam PDE

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

1.254

0.205










84

Burger’s PDE

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


Implicit solution

0.037

0.025

Yes










85

Burger’s PDE

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

0.009

0.039

Yes










86

Burger’s PDE

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

0.026

0.075










87

Burger’s PDE

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

9.278

0.556










88

Burger’s PDE

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

20.599

0.606










89

Black Scholes PDE

classic Black Scholes model from finance, European call version

5.655

0.837










90

Black Scholes PDE

Boundary value problem for the Black Scholes equation

4.522

2.12










91

Korteweg-deVries PDE

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

0.03

0.186










92

Tricomi PDE

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

9.249

3.436










93

Tricomi PDE

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

0.009

2.783










94

Keldysh equation

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

0.007

4.056










95

Euler-Poisson-Darboux equation

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

0.012

0.139










96

Euler-Poisson-Darboux equation

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

0.01

0.487










97

Euler-Poisson-Darboux equation

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

2.637

4.295










98

Chaplygin’s equation

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

0.048

1.266










99

Cauchy Riemann PDE’s

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

0.005

0.168










100

Cauchy Riemann PDE’s

Cauchy Riemann PDE With extra term on right side

0.002

0.07










101

Hamilton-Jacobi PDE

Hamilton-Jacobi type PDE

0.051

0.191










102

Airy PDE

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

0.053

0.115

Yes










103

Nonlinear PDE’s

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

0.021

0.113










104

Nonlinear PDE’s

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

0.033

0.115










105

Nonlinear PDE’s

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

0.022

0.106










106

Nonlinear PDE’s

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

0.012

0.175










107

Nonlinear PDE’s

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

0.043

0.117










108

Nonlinear PDE’s

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

0.044

0.132










109

Nonlinear PDE’s

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

0.108


Answer in terms of RootOf.

0.704










110

Nonlinear PDE’s

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

0.191


Answer in terms of RootOf.

1.45










111

Nonlinear PDE’s

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

0.102

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

0.18


But still has unresolved ODE’s in solution

0.621










114

Nonlinear PDE’s

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

0.089


has RootOf

0.541










115

Nonlinear PDE’s

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

0.034

0.158










116

Nonlinear PDE’s

Fisher’s \(u_t = u(1-u)+u_{xx}\)

0.051

0.232










117

Nonlinear PDE’s

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

0.046


with RootOf

0.123










118

Nonlinear PDE’s

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

0.066

0.158










119

Nonlinear PDE’s

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

0.008

0.025










120

Nonlinear PDE’s

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

0.222

0.408










121

Nonlinear PDE’s

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

0.067

0.251










122

Nonlinear PDE’s

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

0.028

0.115










123

Nonlinear PDE’s

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

0.085

0.277










124

Nonlinear PDE’s

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

0.008

0.026










125

Nonlinear PDE’s

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

0.038

0.463










126

Nonlinear PDE’s

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

0.082


Has RootOf

0.163










127

Nonlinear PDE’s

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

0.085

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

0.01

0.026










131

Nonlinear PDE’s

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

0.067

0.414










132

Nonlinear PDE’s

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

0.045

0.138










133

more miscellaneous

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

0.035

0.032










134

more miscellaneous

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

27.793

0.631










135

more miscellaneous

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

8.389

2.995










136

more miscellaneous

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

0.174

5.69










137

more miscellaneous

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

4.171

0.52










138

more miscellaneous

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

2.84

0.745










139

more miscellaneous

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

3.384

0.718










140

more miscellaneous

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

2.229

2.228










141

more miscellaneous

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

0.05

0.305










142

more miscellaneous

Using integral transforms.

39.73

2.523



















Schrodinger PDE

Table 1.14:Schrodinger PDE breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










143

1D

pict

Logan textbook, page 30

0.439

0.474










144

1D

pict

From Mathematica help pages

0.653

1.235










145

1D

pict

David Griffiths, page 47

30.722

1.26










146

1D

pict

David Griffiths, page 47

0.597

0.933










147

1D

Deep well


Does not handle \(n=2\) case correctly. Division by zero

81.529

6.925










148

1D

pict

From Mathematica help pages

0.008


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

2.546










149

2D

pict

In a square, zero potential

0.614

4.08










150

2D

pict

In a square

0.675

4.622



















Parabolic PDE’s (Diffusion)

Table 1.15:Parabolic PDE’s (Diffusion) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










151

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

General initial conditions

87.065

1.31

Yes










152

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Specific initial condition

29.392

0.619

Yes

Yes










153

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

32.123

0.611

Yes

Yes










154

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (a)

0.402

3.055

Yes










155

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

IC hat function

32.292

1.385

Yes

Yes










156

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (b)

0.963

3.697

Yes










157

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (c)


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

94.078


handled \(n=3\) case correctly.

5.432

Yes










158

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (d)

20.406

26.013

Yes










159

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

24.972

1.119

Yes










160

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

92.53

0.753

Yes

Yes










161

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (general case)

71.052

1.606

Yes










162

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (special case)

59.534

2.285

Yes

Yes










163

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

69.6

3.32

Yes










164

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

65.981

10.046

Yes

Yes










165

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Math 4567 Exam

22.702

1.938

Yes










166

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

With source

67.131

6.367










167

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

special initial condition

2.675

(Timed out)

600.

Yes










168

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction and Euler-Cauchy Sturm-Liouville

1.479

8.987










169

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Using growth form reaction term

152.58

0.83

Yes










170

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using logistic form for reaction term

4.483

27.471










171

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using Aleee form for reaction term

7.125

35.232










172

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Haberman 2.3.8

94.774

6.826

Yes










173

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

93.921

1.482

Yes










174

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

0.622

1.497










175

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

42.986

1.147










176

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (general case)

2.377

2.102

Yes










177

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (special case)

62.434

1.978

Yes

Yes










178

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.2 (general case)

27.698

2.112

Yes










179

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Left end zero, right end insulated, no source

29.368

2.041

Yes










180

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

One end insulated

32.017

2.24










181

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.7 (general case)

31.697

1.804

Yes










182

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

specific case

10.587

3.334

Yes

Yes










183

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (a)

(Timed out)

600.

26.393










184

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Both ends insulated, no source

32.949

2.065

Yes










185

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (b) (special case)

1.496

4.518

Yes










186

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (c) (special case)

32.276

6.848

Yes










187

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (d)

1.346

4.021










188

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

both ends insulated

91.714

4.442










189

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

32.722

3.185










190

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.25

44.137

8.53










191

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

external source

19.439

19.326










192

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction (general case)

91.768

0.938

Yes

Yes










193

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.23

64.376

31.669










194

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.21

63.149

3.556










195

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Euler-Cauchy Sturm-Liouville

2.654

26.115

Yes










196

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

32.75

1.936

Yes










197

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Mixed BC

33.604

4.725










198

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 8.2.1 (f)

63.638

4.279










199

Diffusion in 1D

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

pict

left end insulated (general case)

33.381

4.627

Yes










200

Diffusion in 1D

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

pict

left end insulated (special case)

8.047

1.614

Yes

Yes










201

Diffusion in 1D

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

pict

right end nonhomogeneous BC (general case)

96.438

2.418

Yes










202

Diffusion in 1D

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

pict

right end nonhomogeneous BC (special case)

0.01

3.152

Yes










203

Diffusion in 1D

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

pict

right end nonhomogeneous BC, special case

12.063

2.215

Yes

Yes










204

Diffusion in 1D

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

pict

convection heat loss

40.163

(Timed out)

600.

Yes










205

Diffusion in 1D

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

pict

nonhomogeneous BC

2.233

9.692










206

Diffusion in 1D

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

pict

nonhomogeneous BC

92.576

1.904










207

Diffusion in 1D

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

pict

nonhomogeneous BC

44.428

15.72

Yes










208

Diffusion in 1D

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

pict

left end BC depends on time (general case)

66.568

11.76

Yes










209

Diffusion in 1D

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

pict

left end BC depends on time (special case)

89.752

15.321

Yes

Yes










210

Diffusion in 1D

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

pict

left end BC depends on time (special case)

88.034

13.846

Yes

Yes










211

Diffusion in 1D

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

pict

Haberman 8.3.6 (special case)

125.302

18.548

Yes

Yes










212

Diffusion in 1D

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

pict

BC depends on time (special case)

14.123

5.8










213

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). IC given

15.983

10.554










214

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). No IC given

0.015

1.573

Yes










215

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

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

0.029

1.459

Yes










216

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends non-homogeneous BC (general case)

94.077

4.237

Yes










217

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

non-homogeneous BC (special case)

18.468

2.648

Yes










218

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.1 (special case)

93.164

2.548

Yes










219

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (general case)

83.766

6.341

Yes










220

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (special case)

124.882

3.432

Yes

Yes










221

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

69.358

9.333

Yes










222

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (general case)

85.769

17.345

Yes










223

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (special case)

98.168

28.194

Yes

Yes










224

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends nonhomogeneous

63.93

14.49

Yes










225

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (a) (general case)

39.141

6.474

Yes










226

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (d) (general solution)

82.22

5.482

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 (general solution)

117.169

(Timed out)

600.

Yes










228

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)

135.484

29.86

Yes

Yes










229

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)

80.197

16.293

Yes










230

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

148.347

63.454

Yes

Yes










231

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Pinchover and Rubinstein 6.17

26.441

11.401










232

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

nonhomogeneous BC

142.061

16.879










233

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.2. (a)

125.581

21.911

Yes










234

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.3

5.558

16.523










235

Diffusion in 1D

Semi-infinite domain

pict

left end constant (general case)

42.075

2.036

Yes










236

Diffusion in 1D

Semi-infinite domain

pict

left end constant (special case)


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

34.038

2.772

Yes

Yes










237

Diffusion in 1D

Semi-infinite domain

pict

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

0.578

6.391

Yes










238

Diffusion in 1D

Semi-infinite domain

pict

Left end function of time (special case)

60.047

10.552

Yes

Yes










239

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

41.502

2.204










240

Diffusion in 1D

Semi-infinite domain

pict

I.C. not zero


due to IC/BC not zero

14.608

2.941










241

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

47.487

2.326










242

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.072

6.131










243

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous B.C.

45.447

2.684










244

Diffusion in 1D

Semi-infinite domain

pict

Unit triangle I.C.

35.215

2.839










245

Diffusion in 1D

Semi-infinite domain

pict

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

6.613

4.622










246

Diffusion in 1D

Semi-infinite domain

pict

Diffusion with advection

1.78

4.813










247

Diffusion in 1D

Semi-infinite domain

pict

Practice exam problem

60.111

2.73










248

Diffusion in 1D

Infinite domain

pict

Inverse exponential I.C.

18.765

3.444

Yes










249

Diffusion in 1D

Infinite domain

pict

Advection term

0.072

2.516










250

Diffusion in 1D

Infinite domain

pict

UnitBox I.C.

0.08

4.32










251

Diffusion in 1D

Infinite domain

pict

No source

1.586

5.85










252

Diffusion in 1D

Infinite domain

pict

constant as source

30.487

2.873










253

Diffusion in 1D

Infinite domain

pict

No intial conditions

0.004

0.676

Yes










254

Diffusion in 1D

Infinite domain

pict

piecewise I.C.


due to i.c. not at zero

80.483

3.928

Yes










255

Diffusion in 1D

Infinite domain

pict

Practice exam problem

20.46

1.503










256

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

No source

4.275

14.339










257

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Internal source term

9.849

13.191










258

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Articolo 6.6.3

6.824

20.238










259

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

1.016


Cant get series solution

51.553

Yes










260

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

2.649


Do not understand Maple solution

17.502

Yes

Yes










261

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

0.359

1.819










262

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

1.181

1.511










263

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Haberman 8.3.5 (General solution)

20.342

5.353

Yes










264

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

2.901

1.148

Yes

Yes










265

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

2.482

1.184

Yes

Yes










266

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Inside ring


\(n=1\) causs division by zero

3.69

(Timed out)

600.










267

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.1

3.909

0.908










268

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.2

4.548

38.487










269

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

380.997

0.441

Yes










270

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

8.155

1.062

Yes

Yes










271

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

9.736

28.405

Yes

Yes










272

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

5.089

5.436

Yes










273

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

15.47

9.737

Yes

Yes










274

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

(Timed out)

600.

9.36

Yes

Yes










275

Diffusion in 3D

Spherical coordinates

pict

No angle dependencies

61.114


Has unresolved Laplace integrals

3.518










276

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (a)

2.161

81.613

Yes

Yes










277

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (b)

7.282

30.911

Yes

Yes










278

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (c)

3.412

27.302










279

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (a)

0.012

4.145










280

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (b)

0.009

4.083










281

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (c)

0.012

4.082



















Elliptic PDE’s (Laplace, Poisson, Helmholtz)

Table 1.16:Elliptic PDE’s (Laplace, Poisson, Helmholtz) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










282

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge not

1.631

79.735

Yes










283

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, right edge not

1.743

8.797

Yes










284

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge has impulse

3.989

26.447

Yes










285

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (a)

0.935

72.504










286

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (b)

0.905

83.332










287

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (c)

4.763

58.768










288

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (d)

4.235

83.677










289

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (e)

20.264

73.192

Yes










290

Laplace in 2D

Cartesian coordinates

pict

Unit triangle B.C.

1.039

18.79










291

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

3.505

7.27

Yes










292

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.766

9.875

Yes










293

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

3.464

7.235

Yes










294

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

41.177

41.03

Yes










295

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.962

32.109










296

Laplace in 2D

Cartesian coordinates

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

0.043

3.511










297

Laplace in 2D

Cartesian coordinates

pict

One side homogeneous

1.619

51.323










298

Laplace in 2D

Cartesian coordinates

pict

In right half plane

22.988

4.04










299

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

5.902

23.138










300

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem Upper half

1.282

6.671










301

Laplace in 2D

Cartesian coordinates

pict

Right half-plane

22.63

4.481










302

Laplace in 2D

Cartesian coordinates

pict

First quadrant

14.85

31.473










303

Laplace in 2D

Cartesian coordinates

pict

Neumann problem upper half-plane

4.935


used convert(sol,Int).

8.997










304

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

2.225

12.098










305

Laplace in 2D

Cartesian coordinates

pict

Strip in upper half

60.268

12.959










306

Laplace in 2D

Cartesian coordinates

pict

in Rectangle, right edge at infinity

5.899

19.588










307

Laplace in 2D

Polar coordinates

pict

Laplace PDE inside quarter disk, Neumann BC at edge

3.591

2.842










308

Laplace in 2D

Polar coordinates

pict

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

253.737

22.021

Yes










309

Laplace in 2D

Polar coordinates

pict

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

2.745

1.888

Yes










310

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. General solution

0.655

11.463










311

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. Specific boundary conditions

3.688

13.769

Yes










312

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.5 (c)

4.073

2.611

Yes










313

Laplace in 2D

Polar coordinates

pict

semi-circle

2.093

2.322










314

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.8 (b)

9.474

361.819

Yes










315

Laplace in 2D

Polar coordinates

pict

Circular annulus

0.547

9.499










316

Laplace in 2D

Polar coordinates

pict

Outside a disk

7.343

13.615










317

Laplace in 2D

Polar coordinates

pict

Outside a disk

4.194

15.946

Yes










318

Laplace in 3D

Spherical coordinates

In a sphere

0.036

3.401










319

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (a)

0.018

3.513










320

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (b)

0.015

3.773










321

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (c)

0.015

3.813










322

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (d)

0.015

3.881










323

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (e)

0.014

3.827










324

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (a)

0.016

4.865










325

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (b)

0.015

39.671










326

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (c)

0.02

53.644










327

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (d)

0.015

4.81










328

Poisson in 2D

Cartesian coordinates

pict

All boundaries at zero

0.02

82.75

Yes










329

Poisson in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

1.285

1.599










330

Poisson in 2D

Cartesian coordinates

pict

Poisson PDE in whole 2D plane

30.87

0.066

Yes










331

Helmholtz in 2D

Cartesian coordinates

pict

In rectangle

64.088

25.968










332

Helmholtz in 2D

Cartesian coordinates

pict

On whole plane


why? It solved earlier with BC?

0.004

0.289










333

Helmholtz in 2D

Cartesian coordinates

pict

Reduced Helmholtz Inside square

62.012

26.367










334

Helmholtz in 3D

Spherical coordinates

Chain reaction PDE

0.123


trivial solution

4.683

Yes



















Hyperbolic PDE’s (Wave)

Table 1.17:Hyperbolic PDE’s (Wave) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










335

Wave PDE in 1D

Finite length string

pict

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

39.729

26.077

Yes










336

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital position zero (special case)

2.745

13.872

Yes

Yes










337

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital velocity zero (special case)

1.654

10.655

Yes

Yes










338

Wave PDE in 1D

Finite length string

pict

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

6.485

60.958

Yes

Yes










339

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

0.651

23.098

Yes










340

Wave PDE in 1D

Finite length string

pict

Logan book, page 28. Both ends fixed

1.011

2.843










341

Wave PDE in 1D

Finite length string

pict

non-zero initial velocity. Both ends fixed

102.57

8.794










342

Wave PDE in 1D

Finite length string

pict

Logan book page 149)

22.208

15.192










343

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (a)

25.092

54.316










344

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (b)

56.658

36.438

Yes










345

Wave PDE in 1D

Finite length string

pict

Both I.C. not zero

26.507

18.372










346

Wave PDE in 1D

Finite length string

pict

With constant source

42.579

30.72










347

Wave PDE in 1D

Finite length string

pict

Logan page 213

4.687

17.921










348

Wave PDE in 1D

Finite length string

pict

Telegraphy PDE

148.548

19.516










349

Wave PDE in 1D

Finite length string

pict

Dispersion term present (general case)


Due to adding dispersion term

66.087

25.109

Yes










350

Wave PDE in 1D

Finite length string

pict

Dispersion term present


Due to adding dispersion term

154.94

89.473

Yes










351

Wave PDE in 1D

Finite length string

pict

Dispersion term present (specific case)


Due to adding dispersion term

158.32

22.849

Yes

Yes










352

Wave PDE in 1D

Finite length string

pict

non-zero initial position


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

7.67


Handled \(n=2\) case correctly

75.615










353

Wave PDE in 1D

Finite length string

pict

With source

11.457

79.233










354

Wave PDE in 1D

Finite length string

pict

Right end free (general case)

26.182

34.789

Yes










355

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (general case)

7.292

25.612

Yes










356

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (special case)

90.868

15.144

Yes

Yes










357

Wave PDE in 1D

Finite length string

pict

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

53.247

33.813

Yes










358

Wave PDE in 1D

Finite length string

pict

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

110.821

29.971

Yes

Yes










359

Wave PDE in 1D

Finite length string

pict

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

110.575

36.941

Yes

Yes










360

Wave PDE in 1D

Finite length string

pict

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

112.386

22.824

Yes

Yes










361

Wave PDE in 1D

Finite length string

pict

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

39.847

85.616

Yes










362

Wave PDE in 1D

Finite length string

pict

Right end oscillates

73.611

62.784










363

Wave PDE in 1D

Finite length string

pict

Perioidic B.C.

2.343

4.542










364

Wave PDE in 1D

Finite length string

pict

Mixed B.C.

15.952

5.86










365

Wave PDE in 1D

Finite length string

pict

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

63.293

29.737

Yes

Yes










366

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed, (general case)

4.332

5.185

Yes










367

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed with specific initial position

16.204

13.644

Yes










368

Wave PDE in 1D

Semi-infinite domain

pict

Logan page 115, left end fixed with source

7.221

5.306










369

Wave PDE in 1D

Semi-infinite domain

pict

Left moving boundary condition

1.807

4.509










370

Wave PDE in 1D

Semi-infinite domain

pict

moving Left end

9.432

6.497










371

Wave PDE in 1D

Semi-infinite domain

pict

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

9.345

1.518










372

Wave PDE in 1D

Semi-infinite domain

pict

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

77.621

2.819










373

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (general solution)

10.585

1.703










374

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (Special solution)

4.216

6.313

Yes

Yes










375

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed. zero initial velocity (Special solution)

4.331

8.789

Yes

Yes










376

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial position (general solution)

13.593

1.712










377

Wave PDE in 1D

Semi-infinite domain

pict

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

34.866

1.483










378

Wave PDE in 1D

Semi-infinite domain

pict

Left end free with source

1.029

1.232










379

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

0.69










380

Wave PDE in 1D

Infinite domain

pict

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

0.005

0.55










381

Wave PDE in 1D

Infinite domain

pict

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

0.159

6.92










382

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

0.307










383

Wave PDE in 1D

Infinite domain

pict

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

0.01

0.444










384

Wave PDE in 1D

Infinite domain

pict

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

0.029

0.404

Yes

Yes










385

Wave PDE in 1D

Infinite domain

pict

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

0.004

0.319










386

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.032

0.365










387

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.002

0.29










388

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

8.986

0.491










389

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.002

0.306

Yes

Yes










390

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.004

4.527

Yes

Yes










391

Wave PDE in 1D

Infinite domain

pict

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

0.104

0.378










392

Wave PDE in 1D

Infinite domain

pict

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

0.003

0.347

Yes

Yes










393

Wave PDE in 1D

Infinite domain

pict

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

0.101

0.699

Yes

Yes










394

Wave PDE in 1D

Infinite domain

pict

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

63.765

0.695

Yes










395

Wave PDE in 1D

Infinite domain

system of 2 inhomogeneous linear hyperbolic system with constant coefficients

0.274

1.19










396

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, General solution

3.116

399.349

Yes










397

Wave PDE in 2D

Cartesian coordinates

pict

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

0.471

32.39

Yes

Yes










398

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example

34.18

57.503

Yes

Yes










399

Wave PDE in 2D

Cartesian coordinates

pict

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

9.562

54.177

Yes

Yes










400

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed

4.864

32.94










401

Wave PDE in 2D

Cartesian coordinates

pict

All edges fixed (Haberman 8.5.5 (a)

29.711

3.128










402

Wave PDE in 2D

Cartesian coordinates

pict

2 edgs fixed, 2 free, zero initial velocity

6.093

63.49










403

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, general solution

1.518

(Timed out)

600.

Yes










404

Wave PDE in 2D

Cartesian coordinates

pict

With damping

7.018

88.935










405

Wave PDE in 2D

Cartesian coordinates

pict

On the whole plane

0.187

0.388










406

Wave PDE in 2D

Polar coordinates

pict

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

2.339


Has unresolved Invlaplace calls

8.424

Yes










407

Wave PDE in 2D

Polar coordinates

pict

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

2.425


Has unresolved Invlaplace calls

73.886










408

Wave PDE in 2D

Polar coordinates

pict

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

2.25


Has unresolved Invlaplace calls. How to get series solution?

68.303

Yes

Yes










409

Wave PDE in 2D

Polar coordinates

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

3.815

29.343










410

Wave PDE in 2D

Polar coordinates

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

2.47

7.253










411

Wave PDE in 2D

Polar coordinates

pict

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

10.075

16.112

Yes










412

Wave PDE in 2D

Polar coordinates

pict

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

4.977

12.441

Yes










413

Wave PDE in 2D

Polar coordinates

pict

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

15.007

10.359

Yes

Yes










414

Wave PDE in 2D

Polar coordinates

pict

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

6.402

12.831

Yes

Yes










415

Wave PDE in 2D

Polar coordinates

pict

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

0.043

4.539

Yes










416

Wave PDE in 3D

Spherical coordinates

pict

No I.C. no B.C.

0.042

6.306










417

Wave PDE in 3D

Cylindrical coordinates

pict

No I.C. no B.C.

0.006

0.668



















Handbook of first order partial differential equations

Table 1.18:Handbook of first order partial differential equations breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










418

chapter 1

problem number 1

0.006

0.072










419

chapter 1

problem number 2

0.004

0.005










420

chapter 1

problem number 3

0.017

0.006

Yes










421

chapter 1

problem number 4

0.016

0.003










422

chapter 1

problem number 5

0.034

0.031

Yes










423

chapter 1

problem number 6

0.031

0.014










424

chapter 2

2.1

problem number 1

0.007

0.075

Yes










425

chapter 2

2.1

problem number 2

0.01

0.072

Yes










426

chapter 2

2.1

problem number 3

0.055

0.208

Yes










427

chapter 2

2.1

problem number 4

0.021

0.171

Yes










428

chapter 2

2.1

problem number 5

0.025

1.405

Yes










429

chapter 2

2.1

problem number 6

0.051

1.384

Yes










430

chapter 2

2.1

problem number 7

0.059

3.135

Yes










431

chapter 2

2.1

problem number 8

2.391

18.946










432

chapter 2

2.2

problem number 1

0.009

0.01










433

chapter 2

2.2

problem number 2

0.251

0.582










434

chapter 2

2.2

problem number 3

0.112

0.183










435

chapter 2

2.2

problem number 4

0.392

0.413










436

chapter 2

2.2

problem number 5

0.51

0.994










437

chapter 2

2.2

problem number 6

0.335

0.435










438

chapter 2

2.2

problem number 7

0.363

0.805










439

chapter 2

2.2

problem number 8

0.529

0.707










440

chapter 2

2.2

problem number 9

0.31

0.525










441

chapter 2

2.2

problem number 10

0.173

0.3










442

chapter 2

2.2

problem number 11

0.323

2.757










443

chapter 2

2.2

problem number 12

0.837

1.533










444

chapter 2

2.2

problem number 13

0.113

0.3










445

chapter 2

2.2

problem number 14

0.276

0.821










446

chapter 2

2.2

problem number 15

0.724

1.048










447

chapter 2

2.2

problem number 16

1.586

0.658










448

chapter 2

2.2

problem number 17

0.987

0.828










449

chapter 2

2.2

problem number 18

0.276

0.108










450

chapter 2

2.2

problem number 19

0.299

0.793










451

chapter 2

2.2

problem number 20

0.24

1.996










452

chapter 2

2.2

problem number 21

0.205

2.307










453

chapter 2

2.2

problem number 22

0.347

1.529










454

chapter 2

2.2

problem number 23

0.442

2.915










455

chapter 2

2.2

problem number 24

0.321

0.734










456

chapter 2

2.2

problem number 25

(Timed out)

600.

(Timed out)

600.










457

chapter 2

2.2

problem number 26

170.56

45.839










458

chapter 2

2.2

problem number 27

101.608

3.194










459

chapter 2

2.2

problem number 28

0.529


solution contains RootOf

1.436










460

chapter 2

2.2

problem number 29

101.346

5.214










461

chapter 2

2.2

problem number 30

0.305

1.639










462

chapter 2

2.2

problem number 31, Hesse’s equation

236.35

10.262










463

chapter 2

2.3

problem number 1


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

0.504

3.677










464

chapter 2

2.3

problem number 2

0.133

0.278










465

chapter 2

2.3

problem number 3

0.27

2.572










466

chapter 2

2.3

problem number 4

0.262

5.678










467

chapter 2

2.3

problem number 5

0.916


Answer contains RootOf

0.536










468

chapter 2

2.3

problem number 6

0.227


Answer contains RootOf

0.263










469

chapter 2

2.3

problem number 7

0.234

0.239










470

chapter 2

2.3

problem number 8

0.263

4.681










471

chapter 2

2.3

problem number 9

(Timed out)

600.

0.102










472

chapter 2

2.4

problem number 1

0.035

0.214










473

chapter 2

2.4

problem number 2

0.345

1.645










474

chapter 2

2.4

problem number 3

0.239

0.843










475

chapter 2

2.4

problem number 4

0.647

6.135










476

chapter 2

2.4

problem number 5

0.533

(Timed out)

600.










477

chapter 2

2.4

problem number 6

2.024

1.328










478

chapter 2

2.4

problem number 7

1.838

65.123










479

chapter 2

2.5

problem number 1

0.085

0.583










480

chapter 2

2.5

problem number 2

0.302

1.028










481

chapter 2

2.5

problem number 3

1.418

0.404










482

chapter 2

2.5

problem number 4

19.884

4.145










483

chapter 2

2.5

problem number 5

1.06

4.072










484

chapter 2

2.5

problem number 6

1.814

9.467








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