1.4 Result lookup table on one web page

Miscellaneous PDE’s

Table 1.12Miscellaneous 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 \(u_t+ u_x = 0\)

0.029

0.124

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

0.059

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)

0.004

0.02

Yes

Yes










4

General first order

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

0.003

0.052

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)

0.004

0.018

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

0.05

0.105

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

0.182

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

0.051

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

0.018

0.076

Yes

Yes










10

General first order

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

0.012

0.065

Yes

Yes










11

General first order

\(3 u_x + 5 u_y = x\)

0.004

0.009

Yes










12

General first order

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

0.011

0.056

Yes










13

General first order

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

0.059

0.37

Yes










14

General first order

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

0.003

0.02

Yes










15

General first order

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

0.003

0.017

Yes










16

General first order

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

11.601


Solution contains unresolved invlaplace calls

0.405

Yes










17

General first order

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

0.028

0.046

Yes










18

General first order

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

0.024

0.048

Yes










19

General first order

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

0.096

0.042

Yes










20

General first order

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

0.025

0.321

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

0.682










22

General first order

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

0.029

0.013










23

General first order

Recover a function from its gradient vector

0.022

0.061










24

General first order

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

0.057

0.039

Yes










25

General first order

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

0.164

0.462










26

General first order

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

0.007

0.009

Yes










27

General first order

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

0.007

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

0.086

0.024

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.

1.901

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)

0.356

0.284

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

0.018










32

General first order

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

1.363

0.441










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

0.093

0.057










34

General first order

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

0.68

0.195










35

General first order

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

0.003

0.002










36

General first order

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

0.004

0.007










37

General first order

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

0.014

0.009










38

General first order

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

0.006

0.008










39

General first order

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

0.081

0.023










40

General first order

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

4.697

0.314










41

General first order

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

0.029

0.034










42

General first order

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

0.007

0.007










43

General first order

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

0.055

0.011










44

General first order

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

0.003

0.014










45

General first order

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

0.01

0.024










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

0.006

0.008










47

General first order

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

0.003

0.013










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

1.149

0.406










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

0.005

0.083










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

0.007

0.008










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

0.041

0.061










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

0.011

0.047










53

General first order

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

0.01

0.01










54

General first order

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

0.059

0.017










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

0.033

0.683










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

0.071

0.038










57

General first order

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

45.674

0.214










58

General first order

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

0.004

0.018










59

General first order

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

0.004

0.007










60

General first order

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

0.007

0.018










61

General first order

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

9.16

0.901










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

0.009

0.116










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

0.01

0.036










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

0.016

0.036










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

0.007

0.01










66

General first order

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

0.022

0.021










67

General first order

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

30.979

1.295










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

0.033

0.04










69

General first order

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

0.032

0.044










70

General first order

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

0.103

0.089

Yes










71

General first order

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

0.053

0.001

Yes










72

General first order

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

0.006

0.02

Yes










73

General first order

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

0.009

0.028

Yes










74

General first order

\(2 u_x + 3 u_y = 1\)

0.004

0.008

Yes










75

General first order

\(x u_t - t u_x = 0\)

0.01

0.01

Yes










76

General first order

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

0.004

0.009

Yes










77

General first order

\(u_x u_y = 1\)

0.002

0.015

Yes










78

General first order

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

1.108

0.23

Yes










79

Solved by factoring into two transport equations

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

0.009

0.081

Yes










80

Solved by factoring into two transport equations

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

0.009

0.245

Yes










81

Solved by factoring into two transport equations

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

0.009

1.926

Yes










82

Solved by factoring into two transport equations

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

0.013

1.645










83

Schrodinger PDE

pict

Logan textbook, page 30

0.391

0.393










84

Schrodinger PDE

pict

In a square, zero potential

0.852

3.302










85

Schrodinger PDE

pict

From Mathematica help pages

0.627

0.786










86

Schrodinger PDE

pict

From Mathematica help pages

0.005


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

9.647










87

Schrodinger PDE

pict

David Griffiths, page 47

37.402

1.725










88

Schrodinger PDE

pict

David Griffiths, page 47

0.565

0.913










89

Schrodinger PDE

pict

In a square

0.567

4.639










90

Beam PDE

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

0.212

0.235










91

Burger’s PDE

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


Implicit solution

0.026

0.025

Yes










92

Burger’s PDE

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

0.008

0.064

Yes










93

Burger’s PDE

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

0.03

0.077










94

Burger’s PDE

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

9.03

0.661










95

Burger’s PDE

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

36.079

0.727










96

Black Scholes PDE

classic Black Scholes model from finance, European call version

2.444

1.223










97

Black Scholes PDE

Boundary value problem for the Black Scholes equation

3.581

2.214










98

Korteweg-deVries PDE

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

0.03

0.191










99

Tricomi PDE

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

8.831

3.635










100

Tricomi PDE

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

0.01

1.559










101

Keldysh equation

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

0.007

1.803










102

Euler-Poisson-Darboux equation

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

0.01

0.091










103

Euler-Poisson-Darboux equation

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

0.009

0.424










104

Euler-Poisson-Darboux equation

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

1.838

4.568










105

Chaplygin’s equation

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

0.028

1.144










106

Cauchy Riemann PDE’s

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

0.009

0.168










107

Cauchy Riemann PDE’s

Cauchy Riemann PDE With extra term on right side

0.001

0.057










108

Hamilton-Jacobi PDE

Hamilton-Jacobi type PDE

0.007

0.186










109

Airy PDE

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

0.027

0.078

Yes










110

Nonlinear PDE’s

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

0.022

0.069










111

Nonlinear PDE’s

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

0.031

0.085










112

Nonlinear PDE’s

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

0.022

0.071










113

Nonlinear PDE’s

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

0.011

0.15










114

Nonlinear PDE’s

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

0.041

0.101










115

Nonlinear PDE’s

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

0.041

0.116










116

Nonlinear PDE’s

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

0.073


Answer in terms of RootOf.

0.648










117

Nonlinear PDE’s

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

0.182


Answer in terms of RootOf.

1.383










118

Nonlinear PDE’s

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

0.096

0.197










119

Nonlinear PDE’s

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

0.011

0.028










120

Nonlinear PDE’s

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

0.177


But still has unresolved ODE’s in solution

0.516










121

Nonlinear PDE’s

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

0.085


has RootOf

0.426










122

Nonlinear PDE’s

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

0.032

0.139










123

Nonlinear PDE’s

Fisher’s \(u_t = u(1-u)+u_{xx}\)

0.052

0.186










124

Nonlinear PDE’s

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

0.044


with RootOf

0.115










125

Nonlinear PDE’s

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

0.068

0.14










126

Nonlinear PDE’s

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

0.009

0.021










127

Nonlinear PDE’s

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

0.221

0.409










128

Nonlinear PDE’s

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

0.067

0.276










129

Nonlinear PDE’s

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

0.028

0.102










130

Nonlinear PDE’s

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

0.083

0.283










131

Nonlinear PDE’s

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

0.008

0.319










132

Nonlinear PDE’s

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

0.035

0.153










133

Nonlinear PDE’s

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

0.08


Has RootOf

0.158










134

Nonlinear PDE’s

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

0.085

0.14










135

Nonlinear PDE’s

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

0.01

0.014










136

Nonlinear PDE’s

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

0.009

0.014










137

Nonlinear PDE’s

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

0.009

0.012










138

Nonlinear PDE’s

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

0.066

0.427










139

Nonlinear PDE’s

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

0.044

0.118










140

more miscellaneous

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

0.033

0.038










141

more miscellaneous

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

26.96

0.707










142

more miscellaneous

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

7.356

3.055










143

more miscellaneous

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

1.27

16.98










144

more miscellaneous

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

0.224

0.569










145

more miscellaneous

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

2.618

0.735










146

more miscellaneous

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

2.998

0.744










147

more miscellaneous

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

1.94

2.332










148

more miscellaneous

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

0.02

0.319










149

more miscellaneous

Using integral transforms.

1.022

2.997



















Parabolic PDE’s (Diffusion)

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









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










150

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

General initial conditions

60.751

2.103

Yes










151

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Specific initial condition

9.824

0.916

Yes

Yes










152

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

14.917

1.108

Yes

Yes










153

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (a)

0.351

3.666

Yes










154

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

IC hat function

19.338

2.584

Yes

Yes










155

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (b)

0.863

3.852

Yes










156

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (c)


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

30.809


handled \(n=3\) case correctly.

9.375

Yes










157

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.3 (d)

499.335

38.43

Yes










158

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

17.557

1.255

Yes










159

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

domain from -1 to +1

15.594

1.259

Yes

Yes










160

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (general case)

68.416

2.5

Yes










161

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

with source that depends on space only (special case)

57.962

2.473

Yes

Yes










162

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

67.009

5.115

Yes










163

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

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

64.652

10.528

Yes

Yes










164

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Math 4567 Exam

17.974

2.245

Yes










165

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

With source

64.651

8.138










166

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

special initial condition

2.704

(Timed out)

600.

Yes










167

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction and Euler-Cauchy Sturm-Liouville

12.255

3.718










168

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Using growth form reaction term

81.603

1.02

Yes










169

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using logistic form for reaction term

1.648

4.908










170

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction, using Aleee form for reaction term

2.839

6.778










171

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Haberman 2.3.8

32.027

2.621

Yes










172

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

31.889

1.887

Yes










173

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

0.537

1.493










174

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction

39.849

1.266










175

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (general case)

30.778

1.681

Yes










176

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion convection (special case)

31.483

1.295

Yes

Yes










177

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.2 (general case)

30.45

2.846

Yes










178

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Left end zero, right end insulated, no source

30.554

4.609

Yes










179

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

One end insulated

30.898

2.701










180

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.3.7 (general case)

30.566

2.343

Yes










181

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

specific case

6.322

1.323

Yes

Yes










182

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (a)

(Timed out)

600.

34.522










183

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Both ends insulated, no source

14.057

3.831

Yes










184

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (b) (special case)

0.593

5.284

Yes










185

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (c) (special case)

31.19

11.313

Yes










186

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 2.4.1 (d)

0.39

4.62










187

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

both ends insulated

77.322

8.408










188

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

31.507

1.127










189

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.25

68.393

10.324










190

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

external source

47.067

24.241










191

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction (general case)

88.469

1.007

Yes

Yes










192

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.23

1.239

25.969










193

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Pinchover and Rubinstein 6.21

0.381

5.803










194

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Diffusion Reaction. Euler-Cauchy Sturm-Liouville

31.566

32.102

Yes










195

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

convection heat loss

30.888

2.48

Yes










196

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Mixed BC

31.407

5.308










197

Diffusion in 1D

Finite domain (bar), Both ends homogeneous BC

pict

Haberman 8.2.1 (f)

92.918

6.183










198

Diffusion in 1D

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

pict

left end insulated (general case)

32.209

3.295

Yes










199

Diffusion in 1D

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

pict

left end insulated (special case)

5.766

1.711

Yes

Yes










200

Diffusion in 1D

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

pict

right end nonhomogeneous BC (general case)

64.386

3.083

Yes










201

Diffusion in 1D

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

pict

right end nonhomogeneous BC (special case)

0.009

3.347

Yes










202

Diffusion in 1D

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

pict

right end nonhomogeneous BC, special case

11.58

3.084

Yes

Yes










203

Diffusion in 1D

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

pict

convection heat loss

20.874

(Timed out)

600.

Yes










204

Diffusion in 1D

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

pict

nonhomogeneous BC

0.422

10.994










205

Diffusion in 1D

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

pict

nonhomogeneous BC

66.959

1.681










206

Diffusion in 1D

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

pict

nonhomogeneous BC

66.762

14.425

Yes










207

Diffusion in 1D

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

pict

left end BC depends on time (general case)

80.609

11.675

Yes










208

Diffusion in 1D

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

pict

left end BC depends on time (special case)

80.316

11.947

Yes

Yes










209

Diffusion in 1D

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

pict

left end BC depends on time (special case)

85.399

9.148

Yes

Yes










210

Diffusion in 1D

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

pict

Haberman 8.3.6 (special case)

123.182

18.416

Yes

Yes










211

Diffusion in 1D

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

pict

BC depends on time (special case)

10.802

5.644










212

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). IC given

17.304

10.775










213

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

Periodic BC (general case). No IC given

0.013

1.554

Yes










214

Diffusion in 1D

Finite domain (bar), Periodic BC

pict

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

0.026

1.55

Yes










215

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends non-homogeneous BC (general case)

61.831

4.59

Yes










216

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

non-homogeneous BC (special case)

21.41

2.569

Yes










217

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.1 (special case)

19.417

2.634

Yes










218

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (general case)

80.332

5.211

Yes










219

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

With source that depends on space only (special case)

123.312

4.407

Yes

Yes










220

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

66.991

9.974

Yes










221

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (general case)

99.038

21.379

Yes










222

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Both ends depend on time (special case)

65.856

22.126

Yes

Yes










223

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

both ends nonhomogeneous

93.199

16.422

Yes










224

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (a) (general case)

37.261

5.292

Yes










225

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.1 (d) (general solution)

78.393

6.201

Yes










226

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

140.923

(Timed out)

600.

Yes










227

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

132.866

33.465

Yes

Yes










228

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

77.304

16.904

Yes










229

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

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

141.893

52.566

Yes

Yes










230

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Pinchover and Rubinstein 6.17

1.972

8.3










231

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

nonhomogeneous BC

139.915

18.675










232

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Haberman 8.2.2. (a)

119.345

28.079

Yes










233

Diffusion in 1D

Finite domain (bar), Both ends nonhomogeneous BC

pict

Articolo 8.4.3

34.029

15.807










234

Diffusion in 1D

Semi-infinite domain

pict

left end constant (general case)

60.035

1.84

Yes










235

Diffusion in 1D

Semi-infinite domain

pict

left end constant (special case)


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

34.512

2.836

Yes

Yes










236

Diffusion in 1D

Semi-infinite domain

pict

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

0.553

6.489

Yes










237

Diffusion in 1D

Semi-infinite domain

pict

Left end function of time (special case)

60.044

7.712

Yes

Yes










238

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.029

2.753










239

Diffusion in 1D

Semi-infinite domain

pict

I.C. not zero


due to IC/BC not zero

13.338

2.967










240

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.073

2.69










241

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous BC

60.07

7.536










242

Diffusion in 1D

Semi-infinite domain

pict

nonhomogeneous B.C.

60.034

2.384










243

Diffusion in 1D

Semi-infinite domain

pict

Unit triangle I.C.

61.013

3.709










244

Diffusion in 1D

Semi-infinite domain

pict

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

61.506

2.31










245

Diffusion in 1D

Semi-infinite domain

pict

Diffusion with advection

1.674

4.866










246

Diffusion in 1D

Semi-infinite domain

pict

Practice exam problem

60.118

3.146










247

Diffusion in 1D

Infinite domain

pict

Inverse exponential I.C.

0.753

3.941

Yes










248

Diffusion in 1D

Infinite domain

pict

Advection term

0.513

2.835










249

Diffusion in 1D

Infinite domain

pict

UnitBox I.C.

60.017

6.335










250

Diffusion in 1D

Infinite domain

pict

No source

0.384

6.612










251

Diffusion in 1D

Infinite domain

pict

constant as source

0.416

3.597










252

Diffusion in 1D

Infinite domain

pict

No intial conditions

0.002

0.707

Yes










253

Diffusion in 1D

Infinite domain

pict

piecewise I.C.


due to i.c. not at zero

51.763

7.01

Yes










254

Diffusion in 1D

Infinite domain

pict

Practice exam problem

0.066

1.631










255

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

No source

8.192

11.596










256

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Internal source term

16.973

11.128










257

Diffusion in 2D

Cartesian coordinates (Rectangle, Square)

pict

Articolo 6.6.3

12.205

17.917










258

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

0.926


Cant get series solution

53.213

Yes










259

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

2.87


Do not understand Maple solution

17.698

Yes

Yes










260

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

0.329

5.741










261

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

no \(\theta \) dependency

1.125

1.125










262

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Haberman 8.3.5 (General solution)

18.268

5.17

Yes










263

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

3.388

4.

Yes

Yes










264

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Specific example of the above

2.366

6.497

Yes

Yes










265

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.1

4.106

3.571










266

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

Articolo 6.9.2

4.525

42.538










267

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

326.327

1.684

Yes










268

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

8.153

4.289

Yes

Yes










269

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

9.657

4.13

Yes

Yes










270

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

6.323

10.47

Yes










271

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

19.174

7.694

Yes

Yes










272

Diffusion in 2D

Polar coordinates (disk, sector, annulus)

pict

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

(Timed out)

600.

7.392

Yes

Yes










273

Diffusion in 3D

Spherical coordinates

pict

No angle dependencies

61.069


Has unresolved Laplace integrals

3.78










274

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (a)

0.033

75.741

Yes

Yes










275

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (b)

0.028

26.434

Yes

Yes










276

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.4 (c)

0.029

26.128










277

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (a)

0.006

4.344










278

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (b)

0.006

4.188










279

Diffusion in 3D

Cylinderical coordinates

pict

Haberman 7.9.3 (c)

0.006

4.214



















Elliptic PDE’s (Laplace, Poisson, Helmholtz)

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









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










280

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge not

0.293

78.897

Yes










281

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, right edge not

0.674

7.527

Yes










282

Laplace in 2D

Cartesian coordinates

pict

Rectangle, 3 edges zero, buttom edge has impulse

2.443

25.953

Yes










283

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (a)

0.287

58.794










284

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (b)

0.277

72.414










285

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (c)

3.84

50.321










286

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (d)

3.146

76.114










287

Laplace in 2D

Cartesian coordinates

pict

Haberman 2.5.1 (e)

17.265

70.724

Yes










288

Laplace in 2D

Cartesian coordinates

pict

Unit triangle B.C.

0.948

19.002










289

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

2.19

7.02

Yes










290

Laplace in 2D

Cartesian coordinates

pict

Top edge at infinity

1.523

7.72

Yes










291

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

2.248

9.542

Yes










292

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

40.198

46.18

Yes










293

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

1.681

53.657










294

Laplace in 2D

Cartesian coordinates

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

0.043

3.918










295

Laplace in 2D

Cartesian coordinates

pict

One side homogeneous

0.566

67.987










296

Laplace in 2D

Cartesian coordinates

pict

In right half plane

21.874

4.062










297

Laplace in 2D

Cartesian coordinates

pict

Right edge at infinity

0.861

35.773










298

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem Upper half

1.339

5.267










299

Laplace in 2D

Cartesian coordinates

pict

Right half-plane

22.702

4.731










300

Laplace in 2D

Cartesian coordinates

pict

First quadrant

14.532

32.221










301

Laplace in 2D

Cartesian coordinates

pict

Neumann problem upper half-plane

4.536


used convert(sol,Int).

9.638










302

Laplace in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

1.167

10.724










303

Laplace in 2D

Cartesian coordinates

pict

Strip in upper half

60.25

12.8










304

Laplace in 2D

Cartesian coordinates

pict

in Rectangle, right edge at infinity

0.859

35.71










305

Laplace in 2D

Polar coordinates

pict

Laplace PDE inside quarter disk, Neumann BC at edge

2.708

2.815










306

Laplace in 2D

Polar coordinates

pict

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

143.09

13.167

Yes










307

Laplace in 2D

Polar coordinates

pict

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

0.01

1.92

Yes










308

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. General solution

0.593

9.977










309

Laplace in 2D

Polar coordinates

pict

Laplace inside disk. Specific boundary conditions

3.703

7.162

Yes










310

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.5 (c)

2.03

2.379

Yes










311

Laplace in 2D

Polar coordinates

pict

semi-circle

1.289

2.211










312

Laplace in 2D

Polar coordinates

pict

Haberman 2.5.8 (b)

8.536


But has unresolved Invfourier and Fourier calls

36.41

Yes










313

Laplace in 2D

Polar coordinates

pict

Circular annulus

2.22

1.924










314

Laplace in 2D

Polar coordinates

pict

Outside a disk

6.559

1.35










315

Laplace in 2D

Polar coordinates

pict

Outside a disk

4.247

1.549

Yes










316

Laplace in 3D

Spherical coordinates

In a sphere

0.028

0.63










317

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (a)

0.015

0.796










318

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (b)

0.013

0.722










319

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (c)

0.013

0.76










320

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (d)

0.013

0.818










321

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.1 (e)

0.013

0.878










322

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (a)

0.014

1.071










323

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (b)

0.013

5.71










324

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (c)

0.018

6.572










325

Laplace in 3D

Cylinderical coordinates

pict

Haberman 7.9.2 (d)

0.013

1.244










326

Poisson in 2D

Cartesian coordinates

pict

All boundaries at zero

0.013

20.204

Yes










327

Poisson in 2D

Cartesian coordinates

pict

Dirichlet problem in a rectangle

0.22

0.308










328

Poisson in 2D

Cartesian coordinates

pict

Poisson PDE in whole 2D plane

0.009

0.012

Yes










329

Helmholtz in 2D

Cartesian coordinates

pict

In rectangle

62.821

4.592










330

Helmholtz in 2D

Cartesian coordinates

pict

On whole plane


why? It solved earlier with BC?

0.004

0.062










331

Helmholtz in 2D

Cartesian coordinates

pict

Reduced Helmholtz Inside square

60.777

15.635










332

Helmholtz in 3D

Spherical coordinates

Chain reaction PDE

0.107


trivial solution

1.06

Yes



















Hyperbolic PDE’s (Wave)

Table 1.15Hyperbolic PDE’s (Wave) breakdown of results. Time in seconds









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










333

Wave PDE in 1D

Finite length string

pict

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

31.256

5.257

Yes










334

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital position zero (special case)

30.778

2.093

Yes

Yes










335

Wave PDE in 1D

Finite length string

pict

both ends fixed, inital velocity zero (special case)

28.551

1.545

Yes

Yes










336

Wave PDE in 1D

Finite length string

pict

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

30.489

13.026

Yes

Yes










337

Wave PDE in 1D

Finite length string

pict

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

4.061

3.058

Yes










338

Wave PDE in 1D

Finite length string

pict

Logan book, page 28. Both ends fixed

0.932

0.754










339

Wave PDE in 1D

Finite length string

pict

non-zero initial velocity. Both ends fixed

87.438

1.256










340

Wave PDE in 1D

Finite length string

pict

Logan book page 149)

18.954

2.06










341

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (a)

21.679

11.875










342

Wave PDE in 1D

Finite length string

pict

Haberman 8.5.2 (b)

44.734

7.368

Yes










343

Wave PDE in 1D

Finite length string

pict

Both I.C. not zero

39.364

2.855










344

Wave PDE in 1D

Finite length string

pict

With constant source

38.044

6.007










345

Wave PDE in 1D

Finite length string

pict

Logan page 213

13.653

4.996










346

Wave PDE in 1D

Finite length string

pict

Telegraphy PDE

86.112

2.658










347

Wave PDE in 1D

Finite length string

pict

Dispersion term present (general case)


Due to adding dispersion term

34.446

5.641

Yes










348

Wave PDE in 1D

Finite length string

pict

Dispersion term present


Due to adding dispersion term

91.097

17.417

Yes










349

Wave PDE in 1D

Finite length string

pict

Dispersion term present (specific case)


Due to adding dispersion term

93.203

4.549

Yes

Yes










350

Wave PDE in 1D

Finite length string

pict

non-zero initial position


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

31.018


Handled \(n=2\) case correctly

9.898










351

Wave PDE in 1D

Finite length string

pict

With source

38.874

15.175










352

Wave PDE in 1D

Finite length string

pict

Right end free (general case)

24.766

8.484

Yes










353

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (general case)

5.795

7.003

Yes










354

Wave PDE in 1D

Finite length string

pict

Right end free, zero initial velocity (special case)

94.767

4.104

Yes

Yes










355

Wave PDE in 1D

Finite length string

pict

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

49.358

8.853

Yes










356

Wave PDE in 1D

Finite length string

pict

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

112.752

5.928

Yes

Yes










357

Wave PDE in 1D

Finite length string

pict

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

112.322

7.007

Yes

Yes










358

Wave PDE in 1D

Finite length string

pict

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

114.153

6.029

Yes

Yes










359

Wave PDE in 1D

Finite length string

pict

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

1.199

60.707

Yes










360

Wave PDE in 1D

Finite length string

pict

Right end oscillates

95.279

9.051










361

Wave PDE in 1D

Finite length string

pict

Perioidic B.C.

7.374

1.578










362

Wave PDE in 1D

Finite length string

pict

Mixed B.C.

22.566

2.016










363

Wave PDE in 1D

Finite length string

pict

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

72.198

8.039

Yes

Yes










364

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed, (general case)

2.204

1.63

Yes










365

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed with specific initial position

13.27

2.888

Yes










366

Wave PDE in 1D

Semi-infinite domain

pict

Logan page 115, left end fixed with source

0.169

1.659










367

Wave PDE in 1D

Semi-infinite domain

pict

Left moving boundary condition

7.618

1.341










368

Wave PDE in 1D

Semi-infinite domain

pict

moving Left end

13.244

1.02










369

Wave PDE in 1D

Semi-infinite domain

pict

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

9.057

0.283










370

Wave PDE in 1D

Semi-infinite domain

pict

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

71.399

0.565










371

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (general solution)

5.943

0.347










372

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial velocity (Special solution)

3.816

0.73

Yes

Yes










373

Wave PDE in 1D

Semi-infinite domain

pict

Left end fixed. zero initial velocity (Special solution)

2.893

0.72

Yes

Yes










374

Wave PDE in 1D

Semi-infinite domain

pict

Left end free. zero initial position (general solution)

7.456

0.349










375

Wave PDE in 1D

Semi-infinite domain

pict

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

30.559

0.332










376

Wave PDE in 1D

Semi-infinite domain

pict

Left end free with source

0.156

0.253










377

Wave PDE in 1D

Infinite domain

pict

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

0.017

0.108










378

Wave PDE in 1D

Infinite domain

pict

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

0.004

0.178










379

Wave PDE in 1D

Infinite domain

pict

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

0.149

2.117










380

Wave PDE in 1D

Infinite domain

pict

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

0.002

0.095










381

Wave PDE in 1D

Infinite domain

pict

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

0.01

0.088










382

Wave PDE in 1D

Infinite domain

pict

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

0.028

0.132

Yes

Yes










383

Wave PDE in 1D

Infinite domain

pict

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

0.003

0.095










384

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.028

0.075










385

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

0.002

0.041










386

Wave PDE in 1D

Infinite domain

pict

Practice exam problem Math 5587

71.005

0.113










387

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.002

0.043

Yes

Yes










388

Wave PDE in 1D

Infinite domain

pict

zero initial velocity

0.004

0.766

Yes

Yes










389

Wave PDE in 1D

Infinite domain

pict

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

0.104

0.079










390

Wave PDE in 1D

Infinite domain

pict

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

0.003

0.073

Yes

Yes










391

Wave PDE in 1D

Infinite domain

pict

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

0.093

0.115

Yes

Yes










392

Wave PDE in 1D

Infinite domain

pict

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

0.472

0.125

Yes










393

Wave PDE in 1D

Infinite domain

system of 2 inhomogeneous linear hyperbolic system with constant coefficients

0.262

0.371










394

Wave PDE in 2D

Cartesian coordinates

pict

Rectangular membrane. Fixed on all edges, General solution

2.344

435.606

Yes










395

Wave PDE in 2D

Cartesian coordinates

pict

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

0.424

6.533

Yes

Yes










396

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, Specific example

35.568

12.471

Yes

Yes










397

Wave PDE in 2D

Cartesian coordinates

pict

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

9.778

13.637

Yes

Yes










398

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed

5.245

6.812










399

Wave PDE in 2D

Cartesian coordinates

pict

All edges fixed (Haberman 8.5.5 (a)

27.223

1.017










400

Wave PDE in 2D

Cartesian coordinates

pict

2 edgs fixed, 2 free, zero initial velocity

6.344

11.118










401

Wave PDE in 2D

Cartesian coordinates

pict

All 4 edges fixed, zero initial velocity, general solution

1.434

(Timed out)

600.

Yes










402

Wave PDE in 2D

Cartesian coordinates

pict

With damping

9.472

28.236










403

Wave PDE in 2D

Cartesian coordinates

pict

On the whole plane

0.133

0.161










404

Wave PDE in 2D

Polar coordinates

pict

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

31.99


Has unresolved Invlaplace calls

4.367

Yes










405

Wave PDE in 2D

Polar coordinates

pict

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

31.872


Has unresolved Invlaplace calls

51.951










406

Wave PDE in 2D

Polar coordinates

pict

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

31.175


Has unresolved Invlaplace calls. How to get series solution?

54.262

Yes

Yes










407

Wave PDE in 2D

Polar coordinates

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

1.746

16.58










408

Wave PDE in 2D

Polar coordinates

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

1.853

5.803










409

Wave PDE in 2D

Polar coordinates

pict

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

9.756

6.959

Yes










410

Wave PDE in 2D

Polar coordinates

pict

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

4.485

6.933

Yes










411

Wave PDE in 2D

Polar coordinates

pict

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

15.091

6.368

Yes

Yes










412

Wave PDE in 2D

Polar coordinates

pict

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

7.533

5.072

Yes

Yes










413

Wave PDE in 2D

Polar coordinates

pict

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

0.038

4.683

Yes










414

Wave PDE in 3D

Spherical coordinates

pict

No I.C. no B.C.

0.038

5.734










415

Wave PDE in 3D

Cylindrical coordinates

pict

No I.C. no B.C.

0.006

0.309



















Handbook of first order partial differential equations

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









#

PDE

description

Mathematica
Maple

hand solved?

Animated?





result

time

result

time










416

chapter 1

problem number 1

0.005

0.054










417

chapter 1

problem number 2

0.003

0.086










418

chapter 1

problem number 3

0.008

0.004

Yes










419

chapter 1

problem number 4

0.006

0.002










420

chapter 1

problem number 5

0.022

0.017

Yes










421

chapter 1

problem number 6

0.021

0.011










422

chapter 2

2.1

problem number 1

0.004

0.034

Yes










423

chapter 2

2.1

problem number 2

0.005

0.034

Yes










424

chapter 2

2.1

problem number 3

0.037

0.098

Yes










425

chapter 2

2.1

problem number 4

0.009

0.079

Yes










426

chapter 2

2.1

problem number 5

0.014

0.194

Yes










427

chapter 2

2.1

problem number 6

0.032

0.144

Yes










428

chapter 2

2.1

problem number 7

0.034

1.7

Yes










429

chapter 2

2.1

problem number 8

1.635

2.312










430

chapter 2

2.2

problem number 1

0.006

0.008










431

chapter 2

2.2

problem number 2

0.192

0.184










432

chapter 2

2.2

problem number 3

0.084

0.048










433

chapter 2

2.2

problem number 4

0.257

0.177










434

chapter 2

2.2

problem number 5

0.203

0.215










435

chapter 2

2.2

problem number 6

0.297

0.121










436

chapter 2

2.2

problem number 7

0.298

0.237










437

chapter 2

2.2

problem number 8

0.415

0.143










438

chapter 2

2.2

problem number 9

0.229

0.28










439

chapter 2

2.2

problem number 10

0.144

0.141










440

chapter 2

2.2

problem number 11

0.25

0.148










441

chapter 2

2.2

problem number 12

0.611

0.56










442

chapter 2

2.2

problem number 13

0.088

0.013










443

chapter 2

2.2

problem number 14

0.223

0.314










444

chapter 2

2.2

problem number 15

0.57

0.306










445

chapter 2

2.2

problem number 16

1.123

0.312










446

chapter 2

2.2

problem number 17

0.916

0.498










447

chapter 2

2.2

problem number 18

0.257

0.076










448

chapter 2

2.2

problem number 19

0.266

0.245










449

chapter 2

2.2

problem number 20

0.18

0.366










450

chapter 2

2.2

problem number 21

0.166

0.257










451

chapter 2

2.2

problem number 22

0.329

0.818










452

chapter 2

2.2

problem number 23

0.393

0.442










453

chapter 2

2.2

problem number 24

0.24

0.318










454

chapter 2

2.2

problem number 25

(Timed out)

600.

(Timed out)

600.










455

chapter 2

2.2

problem number 26

(Timed out)

600.

34.752










456

chapter 2

2.2

problem number 27

101.746

1.865










457

chapter 2

2.2

problem number 28

0.481


solution contains RootOf

0.234










458

chapter 2

2.2

problem number 29

101.391

1.419










459

chapter 2

2.2

problem number 30

0.242

0.362










460

chapter 2

2.2

problem number 31, Hesse’s equation

235.674

11.818










461

chapter 2

2.3

problem number 1


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

0.436

1.765










462

chapter 2

2.3

problem number 2

0.057

0.231










463

chapter 2

2.3

problem number 3

0.147

1.083










464

chapter 2

2.3

problem number 4

0.171

1.806










465

chapter 2

2.3

problem number 5

0.558


Answer contains RootOf

0.356










466

chapter 2

2.3

problem number 6

0.199


Answer contains RootOf

0.163










467

chapter 2

2.3

problem number 7

0.175

0.177










468

chapter 2

2.3

problem number 8

0.236

0.087










469

chapter 2

2.3

problem number 9

(Timed out)

600.

0.066










470

chapter 2

2.4

problem number 1

0.022

0.022










471

chapter 2

2.4

problem number 2

0.24

0.536










472

chapter 2

2.4

problem number 3

0.179

0.408










473

chapter 2

2.4

problem number 4

0.417

4.27










474

chapter 2

2.4

problem number 5

0.378

(Timed out)

600.










475

chapter 2

2.4

problem number 6

1.931

0.789










476

chapter 2

2.4

problem number 7

1.47

3.456










477

chapter 2

2.5

problem number 1

0.035

0.238










478

chapter 2

2.5

problem number 2

0.103

0.495










479

chapter 2

2.5

problem number 3

0.156

0.16