Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Taggar`GERF`
GERFSolve  |
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Details and Options
Examples
(8)
Basic Examples
(3)
This is the Burgers' equation in -dimensions:
(1+1)
In[1]:=
burgers=u[x,t]+u[x,t]u[x,t]-νu[x,t]0
∂
t
∂
x
∂
x,x
Out[1]=
(0,1)
u
(1,0)
u
(2,0)
u
Solve it using GERF method:
In[2]:=
 | GERFSolve |
Out[2]=
u[x,t],u[x,t]+,u[x,t]+
0
5
-1
2
0
0
2ν
x
1+
x+t--ν
x
0
x
2
x
Plot the solution:
In[3]:=
Plot3D+/.{2,-1,ν-1},{x,-5,5},{t,-5,5}
0
2ν
x
1+
x+t--ν
x
0
x
2
x
0
x
Out[3]=
Obtain the solution sets as well:
In[4]:=
sols=
[burgers,u[x,t],"OutputMode"All]
 | GERFSolve |
Out[4]=
{{0,0,(-+ν)},u[x,t]},{,0,0},u[x,t]+,0,2ν,--ν,u[x,t]+
-1
1
t
x
0
x
0
1
-1
t
x
5
-1
2
0
-1
1
x
t
0
x
2
x
0
2ν
x
1+
x+t--ν
x
0
x
2
x
Verify that the solutions are valid:
In[5]:=
pick=sols〚2〛;burgers/.uFunction[{x,t},pick〚2,2〛]
Out[5]=
True
Another example, Calogero–Bogoyavlenskii–Schiff equation in -dimensions:
(2+1)
In[1]:=
cbs=u[x,y,t]+4u[x,y,t]u[x,y,t]+2u[x,y,t]u[x,y,t]+u[x,y,t]0;
∂
x,t
∂
x
∂
x,y
∂
x,x
∂
y
∂
x,x,x,y
Solve it:
In[2]:=
sols=
[cbs,u[x,y,t]]
| GERFSolve |
Out[2]=
u[x,y,t],u[x,y,t]-,u[x,y,t]+(1+)+
0
0
2
x
1+
x+y-t
x
y
2
x
y
-1
1+
x
x
x
x
-1
0
Verification:
In[3]:=
Table[cbs/.uFunction[{x,y,t},pick〚i,2〛],{i,Length@sols}]
Out[3]=
{True,True,True}
Use a different general exponential rational function to solve an equation:
In[1]:=
| GERFSolve |
∂
t
∂
x
∂
x,x
Out[1]=
{u[x,t]+2νTan[x-t],u[x,t]-2νCot[x-t],u[x,t]-2νCot[x-t]+2νTan[x-t]}
0
x
x
0
x
0
x
0
x
x
0
x
0
x
x
x
x
0
x
In[2]:=
TrigToExp[Tan[x]]
Out[2]=
(-)
-x
x
-x
x
Generalizations & Extensions
(1)
Options
(3)
Applications
(1)
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