Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Taggar`GERF`
GERFSolve  |
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The following options can be given:
"BalanceConstant"  | Automatic | balance constant of the equation obtained using balancing principle | |
"RationalFunction" | 1 1+ # | the general exponential rational function used for solving | |
"WaveConstants" | { x y t | the wave constants used in wave transformation |
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"OutputMode" takes the following values:
"Solutions" | solutions in the form of u(x, y, ..., t) | |
"SolutionSets" | sets of dependent and independent variables used to obtain "Solutions" | |
All | both of the above |
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The balancing constant for wave transformation is calculated automatically for the given equation. However, if and when the package fails to calculate one, "BalanceConstant" be used as described above.
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accepts single nonlinear partial differential equations in integral or fractional order.
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For fractional order equations, may be used and the package regards it as the conformal derivative.
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The wave transformation is considered to be of the form . Custom wave transformation constants 's can be provided using "WaveConstants" as described above.
η=++...++t
x
1
x
1
x
2
x
2
x
n
x
n
t
x
i
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The coefficients in ansatz are taken to be 's for appropriate ranges of .
i
i
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For systems of equations, the th coefficient in the ansatz for is taken as .
m
k
k
m
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The number of unique equations and the number of unique dependent variables must be equal to each other.
Examples
(11)
Basic Examples
(2)
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]+
0
2ν
x
1+
x+t--ν
x
0
x
2
x
Plot the solution:
In[3]:=
Plot3D+/.{2,-1,-1},{x,-5,5},{t,-5,5}
0
1
1+
x+t-2-
x
1
2
0
x
1
x
0
1
x
Out[3]=
Obtain the solution sets as well:
In[4]:=
sols=
[burgers,u[x,t],"OutputMode"All]
 | GERFSolve |
Out[4]=
0,2ν,--ν,u[x,t]+
-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〚1〛;burgers/.uFunction[{x,t},pick〚1,2〛]
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]+(1+)+
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}
Scope
(4)
Options
(3)
Applications
(2)
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