1.4 Result lookup table on one web page

Miscellaneous PDE’s

Table 1.12: Miscellaneous PDE’s breakdownof results. Time in seconds










#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










1

General first order

Transport equation \(u_t+ u_x = 0\)

0.024

0.089

Yes










2

General first order

Transport equation \(u_t-3 u_x = 0\) IC \(u(0,x)=e^{-x^2}\). Peter Olver textbook, 2.2.2 (a)

0.004

0.04

Yes

Yes










3

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)

0.003

0.017

Yes

Yes










4

General first order

Transport equation \(u_t+u_x+\frac{1}{2}u = 0\) IC \(u(0,x)=\arctan (x)\). Peter Olver textbook, 2.2.2 (c)

0.003

0.055

Yes

Yes










5

General first order

Transport equation \(u_t-4u_x+u = 0\) IC \(u(0,x)=\frac{1}{1+x^2}\). Peter Olver textbook, 2.2.2 (d)

0.003

0.016

Yes

Yes










6

General first order

Transport equation \(u_t+2 u_x= \sin x\) IC \(u(0,x)=\sin x\). Peter Olver textbook, 2.2.5

0.049

0.049

Yes

Yes










7

General first order

Transport equation \(u_t+\frac{1}{1+x^2} u_x= 0\) IC \(u(x,0)=\frac{1}{1+(3+x)^2}\). Peter Olver textbook, page 27

0.014

0.073

Yes

Yes










8

General first order

Transport equation \(u_t-x u_x= 0\) IC \(u(x,0)=\frac{1}{1+x^2}\). Peter Olver textbook, problem 2.2.17

0.004

0.045

Yes

Yes










9

General first order

Transport equation \(u_t+(1-2 t) u_x= 0\) IC \(u(x,0)=\frac{1}{1+x^2}\). Peter Olver textbook, problem 2.2.29

0.013

0.055

Yes

Yes










10

General first order

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

0.011

0.065

Yes

Yes










11

General first order

\(3 u_x + 5 u_y = x\)

0.004

0.007

Yes










12

General first order

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

0.01

0.049

Yes










13

General first order

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

0.052

0.371

Yes










14

General first order

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

0.003

0.019

Yes










15

General first order

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

0.003

0.016

Yes










16

General first order

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

10.387


Solution contains unresolved invlaplace calls

0.243

Yes










17

General first order

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

0.027

0.037

Yes










18

General first order

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

0.023

0.05

Yes










19

General first order

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

0.083

0.03

Yes










20

General first order

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

0.012

0.196

Yes










21

General first order

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

0.009

0.611










22

General first order

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

0.016

0.013










23

General first order

Recover a function from its gradient vector

0.021

0.059










24

General first order

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

0.038

0.036

Yes










25

General first order

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

0.047

0.464










26

General first order

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

0.007

0.01

Yes










27

General first order

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

0.007

0.009

Yes










28

General first order

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

0.082

0.023

Yes










29

General first order

\((y-z) u_x + (z-x) u_y + (x-y) u_z = 0\) (Example 3.5.4 in Lokenath Debnath)

(Timed out)

600.

1.899

Yes










30

General first order

\(u(x+y) u_x+u(x-y) u_y=x^2+y^2\) (Example 3.5.5 in Lokenath Debnath)

0.361


Maple does not accept this form of Cauchy data

0.154

Yes










31

General first order

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

0.003

0.014










32

General first order

\((y-x)u_x+(u-x)u_y=x-y\) with \(u=0\) on \(x y=1\) Example 3.5.7 in Lokenath Debnath


Kernel error

120.019

(Timed out)

600.










33

General first order

\(y u_x+x u_y=u\) with \(u(x,0)=x^3\) and \(u(0,y)=y^3\) Example 3.5.8 in Lokenath Debnath

1.297

0.429










34

General first order

\(x u_x+y u_y=x e^{-u}\) with \(u=0\) on \(y=x^2\) Example 3.5.10 in Lokenath Debnath

0.1


Maple does not accept this form of Cauchy data

0.057










35

General first order

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


Kernel error

0.645

0.184










36

General first order

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

0.003

0.002










37

General first order

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

0.004

0.007










38

General first order

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

0.013

0.009










39

General first order

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

0.006

0.009










40

General first order

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

0.102

0.023










41

General first order

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

4.092

0.319










42

General first order

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

0.029

0.038










43

General first order

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

0.007

0.009










44

General first order

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

0.074

0.01










45

General first order

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

0.003

0.015










46

General first order

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

0.01

0.024










47

General first order

\(x u_x+y u_y=2 x y\) with \(u=2\) on \(y=x^2\). Problem 3.5(c) Lokenath Debnath

0.006


Maple does not accept this form of Cauchy data as I.C.

0.008










48

General first order

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

0.003

0.149










49

General first order

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

1.138

0.371










50

General first order

\(u_x+x u_y=(y-\frac{1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath

0.005

0.075










51

General first order

\(x u_x+y u_y=u+1\) with \(u=x^2\) on \(y=x^2\) Problem 3.5(g) Lokenath Debnath

0.007


Maple does not accept this form of Cauchy data as I.C.

0.01










52

General first order

\(u u_x - u u_y= u^2 + (x+y)^2\) with \(u(x,0)=1\) Problem 3.5(h) Lokenath Debnath

0.04

0.057










53

General first order

\(x u_x+(x+y)u_y=u+1\) with \(u(x,0)=x^2\) Problem 3.5(i) Lokenath Debnath

0.011

0.039










54

General first order

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

0.01

0.011










55

General first order

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

0.089

0.016










56

General first order

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

0.027

0.68










57

General first order

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

0.068

0.034










58

General first order

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

44.854

0.368










59

General first order

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

0.004

0.016










60

General first order

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

0.005

0.007










61

General first order

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

0.007

0.012










62

General first order

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

7.967

0.708










63

General first order

\(2 x u_x+(x+1) u_y=y\) with \(u(1,y)=2 y\). Problem 3.14(d) Lokenath Debnath

0.009

0.124










64

General first order

\(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath

0.01

0.033










65

General first order

\(y^2 u_x+(x y) u_y=x\) with \(u(x,1)=x^2\). Problem 3.14(f) Lokenath Debnath

0.016

0.037










66

General first order

\(x u_x+y u_y=x y\) with \(u=\frac{x^2}{2}\) at \(y=x\). Problem 3.14(g) Lokenath Debnath


Mathematica does not support this Cauchy data I.C.

0.007


Maple does not support this Cauchy data I.C.

0.009










67

General first order

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

0.018

0.02










68

General first order

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

30.116

20.194










69

General first order

\(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath

0.027

0.037










70

General first order

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

0.028

0.045










71

General first order

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

0.12

0.085

Yes










72

General first order

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

0.054

0.002

Yes










73

General first order

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

0.005

0.02

Yes










74

General first order

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

0.009

0.03

Yes










75

General first order

\(2 u_x + 3 u_y = 1\)

0.004

0.007

Yes










76

General first order

\(x u_t - t u_x = 0\)

0.009

0.013

Yes










77

General first order

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

0.003

0.008

Yes










78

General first order

\(u_x u_y = 1\)

0.002

0.013

Yes










79

General first order

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

1.074

0.556

Yes










80

Solved by factoring into two transport equations

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

0.514

0.082

Yes










81

Solved by factoring into two transport equations

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

1.554

0.246

Yes










82

Solved by factoring into two transport equations

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

0.467

2.034

Yes










83

Solved by factoring into two transport equations

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

0.392

1.312










84

Schrodinger PDE

pict

Logan textbook, page 30

0.377

0.487










85

Schrodinger PDE

pict

In a square, zero potential

0.259

2.934










86

Schrodinger PDE

pict

From Mathematica help pages

0.66

0.68










87

Schrodinger PDE

pict

From Mathematica help pages

0.005


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

3.897










88

Schrodinger PDE

pict

David Griffiths, page 47

35.841

1.52










89

Schrodinger PDE

pict

David Griffiths, page 47

0.011

0.782










90

Schrodinger PDE

pict

In a square

0.011

2.282










91

Beam PDE

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

0.206

0.229










92

Burger’s PDE

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


Implicit solution

0.027

0.019

Yes










93

Burger’s PDE

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

0.008

0.047

Yes










94

Burger’s PDE

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

0.026

0.064










95

Burger’s PDE

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

21.53

0.655










96

Burger’s PDE

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

24.232

0.788










97

Black Scholes PDE

classic Black Scholes model from finance, European call version

3.046

0.931










98

Black Scholes PDE

Boundary value problem for the Black Scholes equation

2.766

2.334










99

Korteweg-deVries PDE

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

0.029

0.177










100

Tricomi PDE

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

4.196

3.212










101

Tricomi PDE

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

0.005

1.635










102

Keldysh equation

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

0.004

1.7










103

Euler-Poisson-Darboux equation

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

0.007

0.082










104

Euler-Poisson-Darboux equation

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

0.006

0.26










105

Euler-Poisson-Darboux equation

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

1.963

3.855










106

Chaplygin’s equation

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

0.021

0.889










107

Cauchy Riemann PDE’s

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

0.005

0.13










108

Cauchy Riemann PDE’s

Cauchy Riemann PDE With extra term on right side

0.001

0.052










109

Hamilton-Jacobi PDE

Hamilton-Jacobi type PDE

0.007

0.157










110

Airy PDE

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

0.002

0.064

Yes










111

Nonlinear PDE’s

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

0.02

0.06










112

Nonlinear PDE’s

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

0.031

0.075










113

Nonlinear PDE’s

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

0.02

0.062










114

Nonlinear PDE’s

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

0.008

0.135










115

Nonlinear PDE’s

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

0.042

0.086










116

Nonlinear PDE’s

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

0.041

0.106










117

Nonlinear PDE’s

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

0.07


Answer in terms of RootOf.

0.352










118

Nonlinear PDE’s

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

0.174


Answer in terms of RootOf.

1.52










119

Nonlinear PDE’s

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

0.093

0.176










120

Nonlinear PDE’s

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

0.01

0.025










121

Nonlinear PDE’s

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

0.169


But still has unresolved ODE’s in solution

0.52










122

Nonlinear PDE’s

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

0.083


has RootOf

0.375










123

Nonlinear PDE’s

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

0.033

0.121










124

Nonlinear PDE’s

Fisher’s \(u_t = u(1-u)+u_{xx}\)

0.05

0.165










125

Nonlinear PDE’s

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

0.044


with RootOf

0.107










126

Nonlinear PDE’s

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

0.071

0.125










127

Nonlinear PDE’s

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

0.004

0.017










128

Nonlinear PDE’s

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

0.228

0.346










129

Nonlinear PDE’s

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

0.057

0.237










130

Nonlinear PDE’s

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

0.029

0.092










131

Nonlinear PDE’s

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

0.072

0.243










132

Nonlinear PDE’s

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

0.004

0.017










133

Nonlinear PDE’s

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

0.032

0.156










134

Nonlinear PDE’s

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

0.078


Has RootOf

0.127










135

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

0.187










136

Nonlinear PDE’s

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

0.007

0.013










137

Nonlinear PDE’s

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

0.007

0.014










138

Nonlinear PDE’s

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

0.006

0.011










139

Nonlinear PDE’s

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

0.06

0.339










140

Nonlinear PDE’s

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

0.043

0.472










141

more miscellaneous

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

0.031

0.023










142

more miscellaneous

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

0.012

0.513










143

more miscellaneous

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

3.551

2.448










144

more miscellaneous

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

0.413

5.663










145

more miscellaneous

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

0.271

0.426










146

more miscellaneous

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

0.003

0.621










147

more miscellaneous

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

0.003

0.616










148

more miscellaneous

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

0.003

2.179










149

more miscellaneous

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

0.004

0.194










150

more miscellaneous

Using integral transforms. \({x}^{2}{\frac{\partial ^{2}}{\partial{x}^{2}}}u \left ( x,y \right ) +x{\frac{\partial }{\partial x}}u \left ( x,y \right ) +{\frac{\partial ^{2}}{\partial{y}^{2}}}u \left ( x,y \right ) =0\)

0.982

2.129



















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










151

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

General initial conditions

30.383

1.527

Yes










152

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Specific initial condition

8.837

0.577

Yes

Yes










153

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

14.949

0.467

Yes

Yes










154

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (a)

0.416

2.448

Yes










155

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

IC hat function

18.465

1.688

Yes

Yes










156

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (b)

1.109

2.759

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

0.886


handled \(n=3\) case correctly.

5.885

Yes










158

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (d)

0.817

2.198

Yes










159

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

8.409

0.643

Yes










160

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

10.344

0.562

Yes

Yes










161

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (general case)

31.709

1.676

Yes










162

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (special case)

30.659

1.392

Yes

Yes










163

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

2.272

2.266

Yes










164

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

33.479

7.534

Yes

Yes










165

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Math 4567 Exam

18.088

1.585

Yes










166

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

With source

1.951

4.825










167

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

special initial condition

2.705

(Timed out)

600.

Yes










168

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction and Euler-Cauchy Sturm-Liouville

1.185

2.078










169

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Using growth form reaction term

21.792

0.531

Yes










170

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using logistic form for reaction term

1.614

3.782










171

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using Aleee form for reaction term

2.769

5.574










172

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Haberman 2.3.8

1.249

1.476

Yes










173

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

1.033

1.533

Yes










174

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

0.65

1.031










175

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

40.533

0.819










176

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (general case)

1.257

3.387

Yes










177

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (special case)

31.408

(Timed out)

600.

Yes

Yes










178

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.2 (general case)

0.366

1.772

Yes










179

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Left end zero, right end insulated, no source

0.324

1.086

Yes










180

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

One end insulated

30.734

1.54










181

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.7 (general case)

0.402

1.166

Yes










182

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

specific case

6.115

0.694

Yes

Yes










183

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (a)

0.549

3.674










184

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Both ends insulated, no source

13.565

1.255

Yes










185

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (b) (special case)

0.562

3.112

Yes










186

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (c) (special case)

0.846

6.312

Yes










187

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (d)

0.391

2.8










188

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

both ends insulated

19.042

3.996










189

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

30.572

0.796










190

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.25

0.989

7.672










191

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

external source

0.527

18.664










192

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction (general case)

28.869

0.726

Yes

Yes










193

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.23

1.329

26.237










194

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.21

0.374

2.572










195

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Euler-Cauchy Sturm-Liouville

1.665

(Timed out)

600.

Yes










196

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

31.058

1.952

Yes










197

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Mixed BC

1.006

3.45










198

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 8.2.1 (f)

0.52

3.463










199

Diffusion in 1D

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

pict

left end insulated (general case)

2.295

2.016

Yes










200

Diffusion in 1D

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

pict

left end insulated (special case)

5.676

1.126

Yes

Yes










201

Diffusion in 1D

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

pict

right end nonhomogeneous BC (general case)

32.608

0.957

Yes










202

Diffusion in 1D

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

pict

right end nonhomogeneous BC (special case)

0.009

2.862

Yes










203

Diffusion in 1D

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

pict

right end nonhomogeneous BC, special case

10.052

2.355

Yes

Yes










204

Diffusion in 1D

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

pict

convection heat loss

12.542

(Timed out)

600.

Yes










205

Diffusion in 1D

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

pict

nonhomogeneous BC

0.397

5.377










206

Diffusion in 1D

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

pict

nonhomogeneous BC

6.798

1.171










207

Diffusion in 1D

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

pict

nonhomogeneous BC

31.601

9.329

Yes










208

Diffusion in 1D

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

pict

left end BC depends on time (general case)

0.49

5.963

Yes










209

Diffusion in 1D

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

pict

left end BC depends on time (special case)

14.632

6.313

Yes

Yes










210

Diffusion in 1D

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

pict

left end BC depends on time (special case)

18.619

4.6

Yes

Yes










211

Diffusion in 1D

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

pict

Haberman 8.3.6 (special case)

30.966

16.03

Yes

Yes










212

Diffusion in 1D

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

pict

BC depends on time (special case)

7.153

5.204










213

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). IC given

0.492

5.799










214

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). No IC given

0.007

2.203

Yes










215

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

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

0.007

2.229

Yes










216

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends non-homogeneous BC (general case)

31.519

3.014

Yes










217

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

non-homogeneous BC (special case)

19.58

1.43

Yes










218

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.1 (special case)

19.845

1.498

Yes










219

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (general case)

31.438

3.576

Yes










220

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (special case)

30.815

2.017

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)

1.307

6.288

Yes










222

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (general case)

8.714

17.586

Yes










223

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (special case)

35.85

13.062

Yes

Yes










224

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends nonhomogeneous

30.804

11.831

Yes










225

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (a) (general case)

0.991

3.046

Yes










226

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (d) (general solution)

31.075

1.733

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)

1.194

(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)

30.834

5.136

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)

1.318

2.434

Yes










230

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

30.968

7.

Yes

Yes










231

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Pinchover and Rubinstein 6.17

1.967

1.941










232

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

nonhomogeneous BC

30.863

4.252










233

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.2. (a)

0.607

3.23

Yes










234

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.3

31.014

7.222










235

Diffusion in 1D

Semi-infinite domain

pict

left end constant (general case)

390.937

0.439

Yes










236

Diffusion in 1D

Semi-infinite domain

pict

left end constant (special case)


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

13.432

0.925

Yes

Yes










237

Diffusion in 1D

Semi-infinite domain

pict

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

0.56

1.139

Yes










238

Diffusion in 1D

Semi-infinite domain

pict

Left end function of time (special case)

133.902

1.431

Yes

Yes










239

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

390.726

0.51










240

Diffusion in 1D

Semi-infinite domain

pict

I.C. not zero


due to IC/BC not zero

0.008

0.599










241

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

394.948

0.55










242

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

38.713

1.905










243

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous B.C.

3.438

0.99










244

Diffusion in 1D

Semi-infinite domain

pict

Unit triangle I.C.

29.149

0.792










245

Diffusion in 1D

Semi-infinite domain

pict

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

0.008

0.596










246

Diffusion in 1D

Semi-infinite domain

pict

Diffusion with advection

0.828

1.413










247

Diffusion in 1D

Semi-infinite domain

pict

Practice exam problem

26.449

0.8










248

Diffusion in 1D

Infinite domain

pict

Inverse exponential I.C.

1.542

0.857

Yes










249

Diffusion in 1D

Infinite domain

pict

Advection term

0.51

0.824










250

Diffusion in 1D

Infinite domain

pict

UnitBox I.C.

1.974

1.225










251

Diffusion in 1D

Infinite domain

pict

No source

0.211

1.515










252

Diffusion in 1D

Infinite domain

pict

constant as source

7.7

1.086










253

Diffusion in 1D

Infinite domain

pict

No intial conditions

0.002

0.109

Yes










254

Diffusion in 1D

Infinite domain

pict

piecewise I.C.


due to i.c. not at zero

0.011

0.815

Yes










255

Diffusion in 1D

Infinite domain

pict

Practice exam problem

0.008

0.304










256

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

No source

0.003

1.75










257

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Internal source term

0.004

1.602










258

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Articolo 6.6.3

0.003

3.749










259

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

2.378


Cant get series solution

15.478

Yes










260

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

1.88


Do not understand Maple solution

6.748

Yes

Yes










261

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

2.27

1.276










262

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

0.914

0.312










263

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Haberman 8.3.5 (General solution)

2.176

1.565

Yes










264

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

1.734

1.463

Yes

Yes










265

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

1.74

1.765

Yes

Yes










266

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.1

0.004

1.387










267

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.2

0.004

6.82










268

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

0.515

Yes










269

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

1.213

Yes

Yes










270

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

1.144

Yes

Yes










271

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

2.447

Yes










272

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

2.788

Yes

Yes










273

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.004

2.141

Yes

Yes










274

Diffusion in 3D

Spherical coordinates

pict

No angle dependencies

2.195


Has unresolved Laplace integrals

1.151










275

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (a)

0.004

11.908

Yes

Yes










276

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (b)

0.003

6.707

Yes

Yes










277

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (c)

0.003

6.136










278

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (a)

0.004

1.939










279

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (b)

0.003

2.63










280

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (c)

0.004

1.612



















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










281

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge not

0.246

20.454

Yes










282

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, right edge not

0.713

1.493

Yes










283

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge has impulse

2.681

13.09

Yes










284

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (a)

0.198

15.391










285

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (b)

0.192

18.329










286

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (c)

0.405

22.601










287

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (d)

0.403

16.85










288

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (e)

0.796

22.019

Yes










289

Laplace in 2D

Cartesian coordinates

pict

Unit triangle B.C.

2.267

3.069










290

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.308

1.489

Yes










291

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.32

1.616

Yes










292

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.313

1.476

Yes










293

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

0.783

9.34

Yes










294

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.258

11.718










295

Laplace in 2D

Cartesian coordinates

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

0.045

1.641










296

Laplace in 2D

Cartesian coordinates

pict

One side homogeneous

0.585

18.027










297

Laplace in 2D

Cartesian coordinates

pict

In right half plane

19.451

1.513










298

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

0.829

6.872










299

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem Upper half

0.94

1.91










300

Laplace in 2D

Cartesian coordinates

pict

Right half-plane

20.57

1.968










301

Laplace in 2D

Cartesian coordinates

pict

First quadrant

15.696

18.134










302

Laplace in 2D

Cartesian coordinates

pict

Neumann problem upper half-plane

5.249


used convert(sol,Int).

3.617










303

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

1.257

1.789










304

Laplace in 2D

Cartesian coordinates

pict

Strip in upper half

0.369

3.974










305

Laplace in 2D

Cartesian coordinates

pict

in Rectangle, right edge at infinity

0.83

7.553










306

Laplace in 2D

Polar coordinates

pict

Laplace PDE inside quarter disk, Neumann BC at edge

2.091

0.992










307

Laplace in 2D

Polar coordinates

pict

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

149.331

5.08

Yes










308

Laplace in 2D

Polar coordinates

pict

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

0.016

0.339

Yes










309

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. General solution

0.134

1.492










310

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. Specific boundary conditions

0.041

1.51

Yes










311

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.5 (c)

2.194

0.695

Yes










312

Laplace in 2D

Polar coordinates

pict

semi-circle

2.064

0.67










313

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.8 (b)

0.011


But has unresolved Invfourier and Fourier calls

35.961

Yes










314

Laplace in 2D

Polar coordinates

pict

Circular annulus

2.316

3.298










315

Laplace in 2D

Polar coordinates

pict

Outside a disk

0.008

2.042










316

Laplace in 2D

Polar coordinates

pict

Outside a disk

0.038

1.556

Yes










317

Laplace in 3D

Spherical coordinates

In a sphere

0.021

1.758










318

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (a)

0.003

1.592










319

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (b)

0.003

1.779










320

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (c)

0.003

1.819










321

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (d)

0.003

1.919










322

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (e)

0.003

1.862










323

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (a)

0.003

2.144










324

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (b)

0.003

7.48










325

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (c)

0.003

9.012










326

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (d)

0.003

1.993










327

Poisson in 2D

Cartesian coordinates

pict

All boundaries at zero

0.01

23.384

Yes










328

Poisson in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

0.211

0.745










329

Poisson in 2D

Cartesian coordinates

pict

Poisson PDE in whole 2D plane

0.02

0.02

Yes










330

Helmholtz in 2D

Cartesian coordinates

pict

In rectangle

2.593

4.516










331

Helmholtz in 2D

Cartesian coordinates

pict

On whole plane


why? It solved earlier with BC?

0.004

0.039










332

Helmholtz in 2D

Cartesian coordinates

pict

Reduced Helmholtz Inside square

1.134

20.8










333

Helmholtz in 3D

Spherical coordinates

Chain reaction PDE

0.036


trivial solution

2.122

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










334

Wave PDE in 1D

Finite length string

pict

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


Surprising it could not solve this standard wave PDE on string

5.788

5.997

Yes










335

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital position zero (special case)

31.082

1.97

Yes

Yes










336

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital velocity zero (special case)

31.042

1.723

Yes

Yes










337

Wave PDE in 1D

Finite length string

pict

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

51.161

15.36

Yes

Yes










338

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

5.785

1.727

Yes










339

Wave PDE in 1D

Finite length string

pict

Logan book, page 28. Both ends fixed

0.422

0.375










340

Wave PDE in 1D

Finite length string

pict

non-zero initial velocity. Both ends fixed

27.558

1.625










341

Wave PDE in 1D

Finite length string

pict

Logan book page 149)

1.244

2.446










342

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (a)

1.242

10.801










343

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (b)

1.541

7.278

Yes










344

Wave PDE in 1D

Finite length string

pict

Both I.C. not zero

40.463

2.398










345

Wave PDE in 1D

Finite length string

pict

With constant source

0.965

6.548










346

Wave PDE in 1D

Finite length string

pict

Logan page 213

0.915

4.085










347

Wave PDE in 1D

Finite length string

pict

Telegraphy PDE

31.304


But \(n = 1\) should not be included.

3.083










348

Wave PDE in 1D

Finite length string

pict

Dispersion term present (general case)


Due to adding dispersion term

1.252

4.715

Yes










349

Wave PDE in 1D

Finite length string

pict

Dispersion term present


Due to adding dispersion term

31.593

20.208

Yes










350

Wave PDE in 1D

Finite length string

pict

Dispersion term present (specific case)


Due to adding dispersion term

31.468

3.76

Yes

Yes










351

Wave PDE in 1D

Finite length string

pict

non-zero initial position


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

31.137


Handled \(n=2\) case correctly

9.71










352

Wave PDE in 1D

Finite length string

pict

With source

30.99

10.826










353

Wave PDE in 1D

Finite length string

pict

Right end free (general case)

0.969

7.717

Yes










354

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (general case)

0.967

5.08

Yes










355

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (special case)

33.775

2.87

Yes

Yes










356

Wave PDE in 1D

Finite length string

pict

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

1.28

6.279

Yes










357

Wave PDE in 1D

Finite length string

pict

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

49.27

5.674

Yes

Yes










358

Wave PDE in 1D

Finite length string

pict

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

48.952

5.414

Yes

Yes










359

Wave PDE in 1D

Finite length string

pict

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

49.823

4.132

Yes

Yes










360

Wave PDE in 1D

Finite length string

pict

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

1.153

54.131

Yes










361

Wave PDE in 1D

Finite length string

pict

Right end oscillates

30.978

11.337










362

Wave PDE in 1D

Finite length string

pict

Perioidic B.C.

7.439

2.564










363

Wave PDE in 1D

Finite length string

pict

Mixed B.C.

21.322

3.43










364

Wave PDE in 1D

Finite length string

pict

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

0.963

4.239

Yes

Yes










365

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed, (general case)

2.14

3.623

Yes










366

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed with specific initial position

22.917

3.145

Yes










367

Wave PDE in 1D

Semi-infinite domain

pict

Logan page 115, left end fixed with source

0.183

2.443










368

Wave PDE in 1D

Semi-infinite domain

pict

Left moving boundary condition

7.585

2.458










369

Wave PDE in 1D

Semi-infinite domain

pict

moving Left end

17.273

1.045










370

Wave PDE in 1D

Semi-infinite domain

pict

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

0.01

0.33










371

Wave PDE in 1D

Semi-infinite domain

pict

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

10.107

0.543










372

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (general solution)

2.494

0.341










373

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (Special solution)

3.478

0.713

Yes

Yes










374

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed. zero initial velocity (Special solution)

2.6

0.536

Yes

Yes










375

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial position (general solution)

2.75

0.369










376

Wave PDE in 1D

Semi-infinite domain

pict

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

3.196

0.321










377

Wave PDE in 1D

Semi-infinite domain

pict

Left end free with source

0.182

0.246










378

Wave PDE in 1D

Infinite domain

pict

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

0.062

0.056










379

Wave PDE in 1D

Infinite domain

pict

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

0.005

0.195










380

Wave PDE in 1D

Infinite domain

pict

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

0.008

1.505










381

Wave PDE in 1D

Infinite domain

pict

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

0.048

0.084










382

Wave PDE in 1D

Infinite domain

pict

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

0.01

0.105










383

Wave PDE in 1D

Infinite domain

pict

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

0.03

0.166

Yes

Yes










384

Wave PDE in 1D

Infinite domain

pict

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

0.003

0.105










385

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.074

0.086










386

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.031

0.056










387

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

24.937

0.132










388

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.075

0.197

Yes

Yes










389

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.078

0.316

Yes

Yes










390

Wave PDE in 1D

Infinite domain

pict

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

0.134

0.087










391

Wave PDE in 1D

Infinite domain

pict

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

0.032

0.118

Yes

Yes










392

Wave PDE in 1D

Infinite domain

pict

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

0.095

0.116

Yes

Yes










393

Wave PDE in 1D

Infinite domain

pict

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

35.84

0.06

Yes










394

Wave PDE in 1D

Infinite domain

system of 2 inhomogeneous linear hyperbolic system with constant coefficients

0.279

0.359










395

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, General solution

0.028

184.836

Yes










396

Wave PDE in 2D

Cartesian coordinates

pict

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

0.023

2.036

Yes

Yes










397

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example

5.053

12.351

Yes

Yes










398

Wave PDE in 2D

Cartesian coordinates

pict

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

1.229

10.169

Yes

Yes










399

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed

0.997

5.191










400

Wave PDE in 2D

Cartesian coordinates

pict

All edges fixed (Haberman 8.5.5 (a)

0.003

2.079










401

Wave PDE in 2D

Cartesian coordinates

pict

2 edgs fixed, 2 free, zero initial velocity

0.003

9.615










402

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, general solution

0.1

(Timed out)

600.

Yes










403

Wave PDE in 2D

Cartesian coordinates

pict

With damping

0.003

18.699










404

Wave PDE in 2D

Cartesian coordinates

pict

On the whole plane

0.104

0.086










405

Wave PDE in 2D

Polar coordinates

pict

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

0.723


Has unresolved Invlaplace calls

0.785

Yes










406

Wave PDE in 2D

Polar coordinates

pict

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

31.224


Has unresolved Invlaplace calls

42.668










407

Wave PDE in 2D

Polar coordinates

pict

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

31.077


Has unresolved Invlaplace calls. How to get series solution?

6.716

Yes

Yes










408

Wave PDE in 2D

Polar coordinates

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

1.772

10.237










409

Wave PDE in 2D

Polar coordinates

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

1.954

3.56










410

Wave PDE in 2D

Polar coordinates

pict

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

0.004

1.901

Yes










411

Wave PDE in 2D

Polar coordinates

pict

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

0.004

4.776

Yes










412

Wave PDE in 2D

Polar coordinates

pict

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

0.004

1.127

Yes

Yes










413

Wave PDE in 2D

Polar coordinates

pict

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

0.005

1.689

Yes

Yes










414

Wave PDE in 2D

Polar coordinates

pict

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

0.027

0.398

Yes










415

Wave PDE in 3D

Spherical coordinates

pict

No I.C. no B.C.

0.031

3.537










416

Wave PDE in 3D

Cylindrical coordinates

pict

No I.C. no B.C.

0.003

0.268



















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










417

chapter 1

problem number 1

0.005

0.046










418

chapter 1

problem number 2

0.003

0.004










419

chapter 1

problem number 3

0.008

0.004

Yes










420

chapter 1

problem number 4

0.006

0.003










421

chapter 1

problem number 5

0.022

0.017

Yes










422

chapter 1

problem number 6

0.021

0.012










423

chapter 2

2.1

problem number 1

0.004

0.024

Yes










424

chapter 2

2.1

problem number 2

0.005

0.027

Yes










425

chapter 2

2.1

problem number 3

0.037

0.033

Yes










426

chapter 2

2.1

problem number 4

0.009

0.029

Yes










427

chapter 2

2.1

problem number 5

0.012

0.128

Yes










428

chapter 2

2.1

problem number 6

0.03

0.145

Yes










429

chapter 2

2.1

problem number 7

0.026

1.447

Yes










430

chapter 2

2.1

problem number 8

1.783

2.075










431

chapter 2

2.2

problem number 1

0.005

0.008










432

chapter 2

2.2

problem number 2

0.171

0.185










433

chapter 2

2.2

problem number 3

0.084

0.019










434

chapter 2

2.2

problem number 4

0.249

0.135










435

chapter 2

2.2

problem number 5

0.152

0.181










436

chapter 2

2.2

problem number 6

0.139

0.102










437

chapter 2

2.2

problem number 7

0.144

0.206










438

chapter 2

2.2

problem number 8

0.162

0.191










439

chapter 2

2.2

problem number 9

0.238

0.405










440

chapter 2

2.2

problem number 10

0.145

0.182










441

chapter 2

2.2

problem number 11

0.264

0.106










442

chapter 2

2.2

problem number 12

0.578

0.439










443

chapter 2

2.2

problem number 13

0.122

0.036










444

chapter 2

2.2

problem number 14

0.242

0.274










445

chapter 2

2.2

problem number 15

0.537

0.302










446

chapter 2

2.2

problem number 16

1.094

0.225










447

chapter 2

2.2

problem number 17

0.868

0.408










448

chapter 2

2.2

problem number 18

0.245

0.07










449

chapter 2

2.2

problem number 19

0.261

0.194










450

chapter 2

2.2

problem number 20

0.176

0.37










451

chapter 2

2.2

problem number 21

0.158

0.198










452

chapter 2

2.2

problem number 22

0.356

0.638










453

chapter 2

2.2

problem number 23

0.416

0.313










454

chapter 2

2.2

problem number 24

0.278

0.244










455

chapter 2

2.2

problem number 25

(Timed out)

600.

(Timed out)

600.










456

chapter 2

2.2

problem number 26

(Timed out)

600.

33.256










457

chapter 2

2.2

problem number 27

101.682

1.896










458

chapter 2

2.2

problem number 28

0.495


solution contains RootOf

0.147










459

chapter 2

2.2

problem number 29

101.287

1.356










460

chapter 2

2.2

problem number 30

0.237

0.24










461

chapter 2

2.2

problem number 31, Hesse’s equation

235.141

10.764










462

chapter 2

2.3

problem number 1


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

40.89

1.633










463

chapter 2

2.3

problem number 2

0.055

0.178










464

chapter 2

2.3

problem number 3

0.147

0.906










465

chapter 2

2.3

problem number 4

0.177

1.41










466

chapter 2

2.3

problem number 5

0.56


Answer contains RootOf

0.155










467

chapter 2

2.3

problem number 6

0.209


Answer contains RootOf

0.067










468

chapter 2

2.3

problem number 7

0.178

0.174










469

chapter 2

2.3

problem number 8

0.216

0.066










470

chapter 2

2.3

problem number 9

(Timed out)

600.

0.049










471

chapter 2

2.4

problem number 1

0.021

0.019










472

chapter 2

2.4

problem number 2

0.2

0.292










473

chapter 2

2.4

problem number 3

0.159

0.27










474

chapter 2

2.4

problem number 4

0.404

1.785










475

chapter 2

2.4

problem number 5

0.378

(Timed out)

600.










476

chapter 2

2.4

problem number 6

1.831

0.602










477

chapter 2

2.4

problem number 7

1.475

2.763










478

chapter 2

2.5

problem number 1