Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Taggar`GERF`
GERFSolve  |
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Details and Options
Examples
(7)
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]++,u[x,t]+++
0
0
(+)
-x-t
1
0
1
x-t
1
0
1
1
-+
-x-t
1
0
1
x-t
1
0
1
(-+)
-x-t
1
0
1
x-t
1
0
1
-1
-x-t
1
0
1
x-t
1
0
1
0
(-+)
-x-t
1
0
1
x-t
1
0
1
-1
-x-t
1
0
1
x-t
1
0
1
(+)
-x-t
1
0
1
x-t
1
0
1
-1
-+
-x-t
1
0
1
x-t
1
0
1
0
Obtain the solution sets as well:
In[3]:=
sols=
[burgers,u[x,t],"OutputMode""Both"]
 | GERFSolve |
Out[3]=
{{0,0},u[x,t]},ν,0,-,u[x,t]+,ν,0,-,u[x,t]++,ν,,-,u[x,t]+++
-1
1
0
1
2
1
-1
2
0
1
0
(+)
-x-t
1
0
1
x-t
1
0
1
1
-+
-x-t
1
0
1
x-t
1
0
1
-1
2
1
1
2
0
1
(-+)
-x-t
1
0
1
x-t
1
0
1
-1
-x-t
1
0
1
x-t
1
0
1
0
-1
2
1
1
-1
2
0
1
(-+)
-x-t
1
0
1
x-t
1
0
1
-1
-x-t
1
0
1
x-t
1
0
1
(+)
-x-t
1
0
1
x-t
1
0
1
-1
-+
-x-t
1
0
1
x-t
1
0
1
0
Verify that the solutions are valid:
In[4]:=
pick=sols〚2〛;burgers/.uFunction[{x,t},pick〚2,2〛]
Out[4]=
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]+++,u[x,y,t]+++,u[x,y,t]+,u[x,y,t]++,u[x,y,t]+++,u[x,y,t]+++,u[x,y,t]+++
(-+)
-y+t
2
3
y+t
2
3
-1
-y+t
2
3
y+t
2
3
0
(+)
-y+t
2
3
y+t
2
3
1
-+
-y+t
2
3
y+t
2
3
0
(-+)
-y
2
y
2
-1
-y
2
y
2
0
(+)
-y
2
y
2
1
-+
-y
2
y
2
(-+)
-x
1
x
1
-1
-x
1
x
1
0
(+)
-x
1
x
1
1
-+
-x
1
x
1
0
2+
-x+y+4t
1
2
2
1
2
x+y+4t
1
2
2
1
2
1
-+
-x+y+4t
1
2
2
1
2
x+y+4t
1
2
2
1
2
0
2-+
-x+y+4t
1
2
2
1
2
x+y+4t
1
2
2
1
2
1
-x+y+4t
1
2
2
1
2
x+y+4t
1
2
2
1
2
0
2-+
-x+y+16t
1
2
2
1
2
x+y+16t
1
2
2
1
2
1
-x+y+16t
1
2
2
1
2
x+y+16t
1
2
2
1
2
2+
-x+y+16t
1
2
2
1
2
x+y+16t
1
2
2
1
2
1
-+
-x+y+16t
1
2
2
1
2
x+y+16t
1
2
2
1
2
(-+)
-y
2
y
2
-1
-y
2
y
2
(+)
-y
2
y
2
-1
-+
-y
2
y
2
0
(-+)
-x
1
x
1
-1
-x
1
x
1
(+)
-x
1
x
1
-1
-+
-x
1
x
1
0
Verification:
In[3]:=
Table[cbs/.uFunction[{x,y,t},pick〚i,2〛],{i,Length@sols}]
Out[3]=
{True,True,True,True,True,True,True,True,True}
Use some other values for coefficients and exponents in the ansatz to obtain different solutions:
In[1]:=
burgers=u[x,t]+u[x,t]u[x,t]-νu[x,t]0;
[burgers,u[x,t],"AnsatzCoefficients"{1,-I,1,-I},"AnsatzExponents"{1,1,1,-1}]
∂
t
∂
x
∂
x,x
| GERFSolve |
Out[1]=
u[x,t],u[x,t]++,u[x,t]+-,u[x,t]+,u[x,t]1++
0
-1
0
1
0
(1-)
x--t(1+)+
1
1
2
2
0
1
1
1
1
x--t(1+)+
1
1
2
2
0
1
1
1
-x+-t(1+)+
1
1
2
2
0
1
1
1
0
1
2
-1
0
Scope
(3)
Applications
(1)
""