restart;
read "C:/Program Files/Maple 2020/lib/ASP v4.6.3.txt";
DESOLVII_V5R5 (March 2011)(c), by Dr. K. T. Vu, Dr. J.
Carminati and Miss G. Jefferson
The authors kindly request that this software be referenced, if
it is used in work eventuating in a publication, by citing
the article:
K.T. Vu, G.F. Jefferson, J. Carminati, Finding generalised
symmetries of differential equations
using the MAPLE package DESOLVII,Comput. Phys. Commun. 183
(2012) 1044-1054.
-------------
ASP (November 2011), by Miss G. Jefferson and Dr. J. Carminati
The authors kindly request that this software be referenced, if
it is used in work eventuating in a publication, by citing
the article:
G.F. Jefferson, J. Carminati, ASP: Automated Symbolic
Computation of Approximate Symmetries
of Differential Equations, Comput. Phys. Comm. 184 (2013)
1045-1063.
[classify, comtab, defeqn, deteq_split, extgenerator, gendef,
genvec, icde_cons, liesolve, mod_eq, originalVar, pdesolv,
reduceVar, reduceVargen, symmetry, varchange]
ASP := _m2229977204928
with(ASP);
[ApproximateSymmetry, applygenerator, commutator]
with(desolv);
[classify, comtab, defeqn, deteq_split, extgenerator, gendef,
genvec, icde_cons, liesolve, mod_eq, originalVar, pdesolv,
reduceVar, reduceVargen, symmetry, varchange]
read "C:/Program Files/Maple 2020/lib/FracSym.v1.16.txt";
FracSym (April 2013), by Miss G. Jefferson and Dr. J. Carminati
The authors kindly request that this software be referenced, if
it is used in work eventuating in a publication, by citing:
G.F. Jefferson, J. Carminati, FracSym: Automated symbolic
computation of Lie symmetries
of fractional differential equations, Comput. Phys. Comm.
Submitted May 2013.
with(FracSym);
[Rfracdiff, TotalD, applyFracgen, evalTotalD, expandsum, fracDet,
fracGen, split]
Rfracdiff(u(x, t), t, alpha);
alpha
D[t ](u(x, t))
Rfracdiff(u(x, t) &* v(x, t), t, alpha);
infinity
-----
\
) (alpha - n) n
/ binomial(alpha, n) D[t ](u(x, t)) D[t ](v(x, t))
-----
n = 0
Rfracdiff(v(x, t) &* u(x, t), t, alpha);
infinity
-----
\
) (alpha - n) n
/ binomial(alpha, n) D[t ](v(x, t)) D[t ](u(x, t))
-----
n = 0
Rfracdiff(u(x, t) &* v(x, t), t, 2);
/ 2 \
| d | / d \ / d \
|---- u(x, t)| v(x, t) + 2 |--- u(x, t)| |--- v(x, t)|
| 2 | \ dt / \ dt /
\ dt /
/ 2 \
| d |
+ u(x, t) |---- v(x, t)|
| 2 |
\ dt /
TotalD(xi[x](x, y), x, 2);
2
D[x ](xi[x](x, y))
evalTotalD([%], [y], [x]);
[ / 2 \ / 2 \
[ 2 | d | | d |
[y_x |---- xi[x](x, y)| + 2 |------ xi[x](x, y)| y_x
[ | 2 | \ dy dx /
[ \ dy /
/ 2 \]
/ d \ | d |]
+ y_xx |--- xi[x](x, y)| + |---- xi[x](x, y)|]
\ dy / | 2 |]
\ dx /]
fde1 := Rfracdiff(u(x, t), t, alpha) = -u(x, t)*diff(u(x, t), x) - diff(u(x, t), x, x) - diff(u(x, t), x, x, x) - diff(u(x, t), x, x, x, x);
alpha / d \
fde1 := D[t ](u(x, t)) = -u(x, t) |--- u(x, t)|
\ dx /
/ 2 \ / 3 \ / 4 \
| d | | d | | d |
- |---- u(x, t)| - |---- u(x, t)| - |---- u(x, t)|
| 2 | | 3 | | 4 |
\ dx / \ dx / \ dx /
deteqs := fracDet([fde1], [u], [x, t], 2);
Intervals/values considered for the fractional derivative/s:
{0 < alpha, alpha < 1}
[
[
[[ 2
[[ d d
deteqs := [[---- eta[u](x, t, u), --- xi[t](x, t, u),
[[ 2 du
[[ du
d d d
--- xi[t](x, t, u), --- xi[t](x, t, u), --- xi[x](x, t, u),
du dx du
2 2
d d d
--- xi[x](x, t, u), ---- xi[t](x, t, u), ---- xi[t](x, t, u),
du 2 2
du du
2
d / d \
---- xi[t](x, t, u), alpha |--- xi[x](x, t, u)|,
2 \ dt /
du
2
/ d \ d
alpha |--- xi[x](x, t, u)|, ---- xi[t](x, t, u),
\ du / 2
du
2 2 3
d d d
---- xi[x](x, t, u), ---- xi[x](x, t, u), ---- xi[t](x, t, u),
2 2 3
du du du
3 3 4
d d d
---- xi[t](x, t, u), ---- xi[x](x, t, u), ---- xi[t](x, t, u),
3 3 4
du du du
4
d
---- xi[x](x, t, u),
4
du
/ 2 \
| d | / d \
-6 |---- xi[t](x, t, u)| - 3 |--- xi[t](x, t, u)|,
| 2 | \ dx /
\ dx /
/ d \ / d \
alpha |--- xi[t](x, t, u)| - 4 |--- xi[x](x, t, u)|,
\ dt / \ dx /
/ d \
|--- xi[t](x, t, u)| (alpha - 1),
\ du /
/ 2 \
/ d \ | d |
-3 |--- xi[t](x, t, u)| - 12 |------ xi[t](x, t, u)|,
\ du / \ dx du /
/ d \
alpha |--- xi[t](x, t, u)| (alpha - 1),
\ du /
/ d \
alpha |--- xi[x](x, t, u)| (alpha - 1),
\ du /
/ 2 \ / 3 \
| d | | d |
-3 |---- xi[t](x, t, u)| - 12 |------- xi[t](x, t, u)|,
| 2 | | 2 |
\ du / \ dx du /
/ 2 \
| d |
alpha |------ xi[t](x, t, u)| (alpha - 1),
\ du dt /
/ 2 \
| d |
alpha |------ xi[x](x, t, u)| (alpha - 1),
\ du dt /
/ 2 \
| d |
alpha |---- xi[t](x, t, u)| (alpha - 1),
| 2 |
\ du /
/ 2 \
| d |
alpha |---- xi[x](x, t, u)| (alpha - 1),
| 2 |
\ dt /
/ 2 \
| d |
alpha |---- xi[x](x, t, u)| (alpha - 1),
| 2 |
\ du /
/ 3 \ / 4 \
| d | | d |
-|---- xi[t](x, t, u)| - 4 |------- xi[t](x, t, u)|,
| 3 | | 3 |
\ du / \ dx du /
/ 2 \
/ d \ | d |
-|--- xi[t](x, t, u)| - 4 |------ xi[t](x, t, u)|
\ du / \ dx du /
/ d \ / d \
+ alpha |--- xi[t](x, t, u)|, -4 |--- xi[x](x, t, u)|
\ du / \ du /
/ 2 \ / 2 \
| d | | d |
+ 4 |---- eta[u](x, t, u)| - 16 |------ xi[x](x, t, u)|,
| 2 | \ dx du /
\ du /
/ 2 \
/ d \ | d |
-3 |--- xi[x](x, t, u)| + 3 |---- eta[u](x, t, u)|
\ du / | 2 |
\ du /
/ 2 \ / 3 \
| d | | d |
- 12 |------ xi[x](x, t, u)|, -4 |---- xi[t](x, t, u)|
\ dx du / | 3 |
\ dx /
/ 2 \
/ d \ | d |
- 2 |--- xi[t](x, t, u)| - 3 |---- xi[t](x, t, u)|,
\ dx / | 2 |
\ dx /
/ 2 \ / 3 \
| d | | d |
-6 |------ xi[t](x, t, u)| - 12 |------- xi[t](x, t, u)|
\ dx du / | 2 |
\ dx du /
/ d \
- 2 |--- xi[t](x, t, u)|,
\ du /
/ d \
alpha |--- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
\ du /
/ 2 \ / 3 \
| d | | d |
-6 |---- xi[x](x, t, u)| + 6 |---- eta[u](x, t, u)|
| 2 | | 3 |
\ du / \ du /
/ 3 \ / 3 \
| d | | d |
- 24 |------- xi[x](x, t, u)|, -3 |------- xi[t](x, t, u)|
| 2 | | 2 |
\ dx du / \ dx du /
/ 4 \ / 2 \
| d | | d | /
- 6 |-------- xi[t](x, t, u)| - |---- xi[t](x, t, u)|, alpha |
| 2 2 | | 2 | \
\ dx du / \ du /
d \ / d \
--- xi[t](x, t, u)| - 3 |--- xi[x](x, t, u)|
dt / \ dx /
/ 2 \ / 2 \
| d | | d |
+ 4 |------ eta[u](x, t, u)| - 6 |---- xi[x](x, t, u)|,
\ dx du / | 2 |
\ dx /
/ 2 \
| d |
alpha |------ xi[t](x, t, u)| (alpha - 1) (alpha - 2),
\ du dt /
/ 2 \
| d |
alpha |---- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
| 2 |
\ du /
/ 2 \ / 3 \
| d | | d |
-3 |------ xi[t](x, t, u)| - 6 |------- xi[t](x, t, u)|
\ dx du / | 2 |
\ dx du /
/
/ d \ / d \ |
- |--- xi[t](x, t, u)| + alpha |--- xi[t](x, t, u)|, alpha |
\ du / \ du / |
\
/ 2 \ / 2 \
| d | | d |
-alpha |---- xi[t](x, t, u)| + 2 |------ eta[u](x, t, u)|
| 2 | \ du dt /
\ dt /
/ 2 \\ / 3 \
| d || | d |
+ |---- xi[t](x, t, u)||, -|---- xi[x](x, t, u)|
| 2 || | 3 |
\ dt // \ du /
/ 4 \ / 4 \
| d | | d |
- 4 |------- xi[x](x, t, u)| + |---- eta[u](x, t, u)|,
| 3 | | 4 |
\ dx du / \ du /
/ 2 \
/ d \ | d |
-u |--- xi[t](x, t, u)| - |---- xi[t](x, t, u)|
\ dx / | 2 |
\ dx /
/ 3 \ / 4 \
| d | | d |
- |---- xi[t](x, t, u)| - |---- xi[t](x, t, u)|,
| 3 | | 4 |
\ dx / \ dx /
/ 3 \
| d |
alpha |------- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
| 2 |
\ du dt /
/ 3 \
| d |
alpha |------- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
| 2 |
\ du dt /
/ 3 \
| d |
alpha |---- xi[t](x, t, u)| (alpha - 1) (alpha - 2),
| 3 |
\ du /
/ 2 \
/ d \ | d |
-3 |--- xi[x](x, t, u)| + 3 |---- eta[u](x, t, u)|
\ du / | 2 |
\ du /
/ 2 \ / 3 \
| d | | d |
- 9 |------ xi[x](x, t, u)| + 12 |------- eta[u](x, t, u)|
\ dx du / | 2 |
\ dx du /
/ 3 \
| d | / d \
- 18 |------- xi[x](x, t, u)|, alpha |--- xi[t](x, t, u)| u
| 2 | \ du /
\ dx du /
/ 3 \ / 4 \
| d | | d |
- 3 |------- xi[t](x, t, u)| - 4 |------- xi[t](x, t, u)|
| 2 | | 3 |
\ dx du / \ dx du /
/ 2 \
| d | / d \ /
- 2 |------ xi[t](x, t, u)| - |--- xi[t](x, t, u)| u, alpha |
\ dx du / \ du / \
/ 3 \
d \ | d |
--- xi[t](x, t, u)| - 4 |---- xi[x](x, t, u)|
dt / | 3 |
\ dx /
/ 2 \
| d | / d \
- 3 |---- xi[x](x, t, u)| - 2 |--- xi[x](x, t, u)|
| 2 | \ dx /
\ dx /
/ 3 \ / 2 \
| d | | d |
+ 6 |------- eta[u](x, t, u)| + 3 |------ eta[u](x, t, u)|,
| 2 | \ dx du /
\ dx du /
/ 2 \ / 3 \
| d | | d |
-|---- xi[x](x, t, u)| + |---- eta[u](x, t, u)|
| 2 | | 3 |
\ du / \ du /
/ 3 \ / 4 \
| d | | d |
- 3 |------- xi[x](x, t, u)| + 4 |------- eta[u](x, t, u)|
| 2 | | 3 |
\ dx du / \ dx du /
/ 4 \ /
| d | |
- 6 |-------- xi[x](x, t, u)|, (alpha - 1) |
| 2 2 | |
\ dx du / \
/ 3 \ / 3 \
| d | | d |
-alpha |---- xi[t](x, t, u)| + 3 |------- eta[u](x, t, u)|
| 3 | | 2 |
\ dt / \ du dt /
/ 3 \\
| d || / d \
+ 2 |---- xi[t](x, t, u)|| alpha, -u |--- xi[x](x, t, u)|
| 3 || \ du /
\ dt //
/ 2 \ / 2 \
| d | | d |
+ |---- eta[u](x, t, u)| - 2 |------ xi[x](x, t, u)|
| 2 | \ dx du /
\ du /
/ 3 \ / 3 \
| d | | d |
+ 3 |------- eta[u](x, t, u)| - 3 |------- xi[x](x, t, u)|
| 2 | | 2 |
\ dx du / \ dx du /
/ 4 \ / 4 \
| d | | d |
- 4 |------- xi[x](x, t, u)| + 6 |-------- eta[u](x, t, u)|,
| 3 | | 2 2 |
\ dx du / \ dx du /
/ d \
-u |--- xi[x](x, t, u)| + eta[u](x, t, u)
\ dx /
/ 2 \
/ d \ | d |
+ alpha |--- xi[t](x, t, u)| u + 2 |------ eta[u](x, t, u)|
\ dt / \ dx du /
/ 2 \ / 3 \
| d | | d |
- |---- xi[x](x, t, u)| + 3 |------- eta[u](x, t, u)|
| 2 | | 2 |
\ dx / \ dx du /
/ 3 \ / 4 \
| d | | d |
- |---- xi[x](x, t, u)| + 4 |------- eta[u](x, t, u)|
| 3 | | 3 |
\ dx / \ dx du /
[
[
/ 4 \] [
| d |] [
- |---- xi[x](x, t, u)|], [xi[t](x, 0, u) = 0, (Diff(
| 4 |] [
\ dx /] [
/ d \
eta[u](x, t, u), t $ alpha)) + u |--- eta[u](x, t, u)|
\ dx /
/ / d \\
- u |Diff|--- eta[u](x, t, u), t $ alpha||
\ \ du //
/ 3 \ / 4 \
| d | | d |
+ |---- eta[u](x, t, u)| + |---- eta[u](x, t, u)|
| 3 | | 4 |
\ dx / \ dx /
/infinity
| -----
/ 2 \ | \
| d | | ) / 1 /
+ |---- eta[u](x, t, u)|, | / |- ----- |binomial(alpha, n)
| 2 | | ----- \ n + 1 \
\ dx / \ n = 3
/ (alpha - n) (n + 1)
|D[t ](u(x, t)) D[t ](xi[t](x, t, u)) alpha
\
(alpha - n) (n + 1)
- D[t ](u(x, t)) D[t ](xi[t](x, t, u)) n
(alpha - n) / d \ n
+ D[t ]|--- u(x, t)| D[t ](xi[x](x, t, u)) n
\ dx /
\ /Sum(
| |
| |
(alpha - n) / d \ n \\\| |
+ D[t ]|--- u(x, t)| D[t ](xi[x](x, t, u))|||| + |
\ dx / ///| |
/ \
/ / d \\
binomial(alpha, n) |Diff|--- eta[u](x, t, u), t $ n||
\ \ du //
(alpha - n) (u(x, t)), n = 3 .. infinity)\]
D[t ] |]
|]
|]
|],
|]
/]
]
]
]
]
[xi[x](x, t, u), xi[t](x, t, u), eta[u](x, t, u)], [x, t, u]]
]
]
sol1 := pdesolv(expand(deteqs[1]), deteqs[3], deteqs[4]);
Error, (in desolv/lderivx) cannot determine if this expression is true or false: 1 < x |C:/Program Files/Maple 2020/lib/ASP v4.6.3.txt:4312|
