Flows.jl



In [1]:

    
using Flows









    



INFO: Recompiling stale cache file /home/hofi/.julia/lib/v0.4/Flows.ji for module Flows.

TimeExpression

TimeVariable



In [2]:

    
t = TimeVariable("t")









    Out[2]:





$t$



In [3]:

    
typeof(t)









    Out[3]:





Flows.TimeVariable



In [4]:

    
@t_vars s r τ









    Out[4]:





(s,r,τ)



In [5]:

    
typeof(s)









    Out[5]:





Flows.TimeVariable

TimeLinearCombination



In [6]:

    
ex = TimeLinearCombination(t,1,s,-2,r,3)









    Out[6]:





$t-2s+3r$

The internal representation of a TimeLinearCombination is a list of (TimeExpression, coefficient)-pairs:



In [7]:

    
ex.terms









    Out[7]:





3-element Array{Tuple{Flows.TimeExpression,Real},1}:
 (t,1) 
 (s,-2)
 (r,3)

The operators +, -, * are overloaded. Therefore, the TimeLinearCombination ex from above can also be defined in the following way:



In [8]:

    
ex1 = t - 2s + 3r









    Out[8]:





$t-2s+3r$



In [9]:

    
ex == ex1









    Out[9]:





true



In [10]:

    
string(ex)









    Out[10]:





"t-2s+3r"



In [11]:

    
t-2ex+11r-2(r-s)









    Out[11]:





$-t+6s+3r$



In [12]:

    
ex-ex









    Out[12]:





$0$



In [13]:

    
string(ex-ex)









    Out[13]:





"t_zero"

t_zero

There is a predefined zero element t_zero which is implemented as an empty TimeLinearCombination:



In [14]:

    
ex-ex == t_zero









    Out[14]:





true



In [15]:

    
typeof(t_zero)









    Out[15]:





Flows.TimeLinearCombination



In [16]:

    
t_zero.terms









    Out[16]:





0-element Array{Tuple{Flows.TimeExpression,Real},1}

coefficient



In [17]:

    
ex









    Out[17]:





$t-2s+3r$



In [18]:

    
coefficient(ex,s)









    Out[18]:





-2

substitute



In [19]:

    
substitute(ex,s,ex)









    Out[19]:





$-t+4s-3r$



In [20]:

    
substitute(t,s,r)









    Out[20]:





$t$

SpaceExpression

Here 'space' is ot necessarily physical space, but abstract (Banach) space on which functions (i.e. operators) and flows of differential equations are defined.

SpaceVariable



In [21]:

    
u = SpaceVariable("u")









    Out[21]:





$u$



In [22]:

    
@x_vars v,w









    Out[22]:





(v,w)

AutonomousFunction



In [23]:

    
@funs F,G,H









    Out[23]:





(F,G,H)



In [24]:

    
typeof(G)









    Out[24]:





Flows.VectorFieldVariable

AutonomousFunctionExpression



In [25]:

    
ex = AutonomousFunctionExpression(F,u)









    Out[25]:





$F(u)$



In [26]:

    
ex1 = F(u)









    Out[26]:





$F(u)$



In [27]:

    
string(ex1)









    Out[27]:





"F(u)"



In [28]:

    
ex == ex1









    Out[28]:





true

AutonomousFunctionExpression involving differentials:



In [29]:

    
ex = F(u,v)









    Out[29]:





$F'(u)\cdot v$



In [30]:

    
string(ex)









    Out[30]:





"F(u,v)"



In [31]:

    
ex = F(u,v,w)









    Out[31]:





$F''(u)(v,w)$



In [32]:

    
string(ex)









    Out[32]:





"F(u,v,w)"

FlowExpression



In [33]:

    
ex = FlowExpression(F, t, u, 0) # (the last argument 0 indicates that it is a 0th derivative w.r.t. time)









    Out[33]:





$\mathcal{E}_{F}(t,u)$



In [34]:

    
ex1 = E(F,t,u)









    Out[34]:





$\mathcal{E}_{F}(t,u)$



In [35]:

    
ex == ex1









    Out[35]:





true



In [36]:

    
string(ex)









    Out[36]:





"E(F,t,u)"

FlowExpression involving differentials:



In [37]:

    
ex = E(F,t,u,v)









    Out[37]:





$\partial_{2}\mathcal{E}_{F}(t,u)\cdot v$



In [38]:

    
string(ex)









    Out[38]:





"E(F,t,u,v)"



In [39]:

    
ex = E(F,t,u,v,w)









    Out[39]:





$\partial_{2}^{2}\mathcal{E}_{F}(t,u)(v,w)$



In [40]:

    
string(ex)









    Out[40]:





"E(F,t,u,v,w)"

NonAutonomousFunction



In [41]:

    
@nonautonomous_funs(S)









    Out[41]:





(S,)

NonAutonomousFunctionExpression



In [42]:

    
ex = S(t,u)









    Out[42]:





$S(t,u)$



In [43]:

    
string(ex)









    Out[43]:





"S(t,u)"



In [44]:

    
ex = S(2,t,u,v)









    Out[44]:





$\partial_{1}^{2}\partial_{2}S(t,u)\cdot v$



In [45]:

    
string(ex)









    Out[45]:





"S(2,t,u,v)"

SpaceLinearCombinations



In [46]:

    
ex = -17E(F,t,u,v,w) + 2u + F(v,w)









    Out[46]:





$F'(v)\cdot w+2u-17\partial_{2}^{2}\mathcal{E}_{F}(t,u)(v,w)$



In [47]:

    
ex-ex









    Out[47]:





$0$



In [48]:

    
string(ex)









    Out[48]:





"F(v,w)+2u-17E(F,t,u,v,w)"

x_zero



In [49]:

    
ex-ex == x_zero









    Out[49]:





true



In [50]:

    
string(ex-ex)









    Out[50]:





"x_zero"

differential



In [51]:

    
ex = F(G(u)) + E(F,t,u,w)









    Out[51]:





$F(G(u))+\partial_{2}\mathcal{E}_{F}(t,u)\cdot w$



In [52]:

    
string(ex)









    Out[52]:





"F(G(u))+E(F,t,u,w)"

We form the differential of the expression ex with respect to the variable u and apply it to the expression H(v). Note that the differntial is a linear map which has to be applied to something ("the slots have to be filled").



In [53]:

    
differential(ex, u, H(v))









    Out[53]:





$F'(G(u))\cdot G'(u)\cdot H(v)+\partial_{2}^{2}\mathcal{E}_{F}(t,u)(w,H(v))$



In [54]:

    
ex = S(1,t-2s,2u+v)









    Out[54]:





$\partial_{1}S(t-2s,v+2u)$



In [55]:

    
expand(differential(ex, u, w))









    Out[55]:





$2\partial_{1}\partial_{2}S(t-2s,v+2u)\cdot w$

t_derivative



In [56]:

    
ex = E(F,t-2s,u+E(G,s,v))









    Out[56]:





$\mathcal{E}_{F}(t-2s,\mathcal{E}_{G}(s,v)+u)$



In [57]:

    
t_derivative(ex, s)









    Out[57]:





$-2F(\mathcal{E}_{F}(t-2s,\mathcal{E}_{G}(s,v)+u))+\partial_{2}\mathcal{E}_{F}(t-2s,\mathcal{E}_{G}(s,v)+u)\cdot G(\mathcal{E}_{G}(s,v))$



In [58]:

    
ex = S(2t,S(t,v))









    Out[58]:





$S(2t,S(t,v))$



In [59]:

    
t_derivative(ex, t)









    Out[59]:





$\partial_{2}S(2t,S(t,v))\cdot \partial_{1}S(t,v)+2\partial_{1}S(2t,S(t,v))$

expand



In [60]:

    
ex = E(F,t,u,2v+3w)









    Out[60]:





$\partial_{2}\mathcal{E}_{F}(t,u)\cdot (3w+2v)$



In [61]:

    
Flows.expand(ex)









    Out[61]:





$2\partial_{2}\mathcal{E}_{F}(t,u)\cdot v+3\partial_{2}\mathcal{E}_{F}(t,u)\cdot w$



In [62]:

    
ex = G(u, v+w, v+w)









    Out[62]:





$G''(u)(w+v,w+v)$



In [63]:

    
Flows.expand(ex)









    Out[63]:





$2G''(u)(w,v)+G''(u)(w,w)+G''(u)(v,v)$



In [64]:

    
ex = S(t,u,2v+3w)









    Out[64]:





$\partial_{2}S(t,u)\cdot (3w+2v)$



In [65]:

    
expand(ex)









    Out[65]:





$3\partial_{2}S(t,u)\cdot w+2\partial_{2}S(t,u)\cdot v$

reduce_order



In [66]:

    
ex = E(F,t,u,F(u))









    Out[66]:





$\partial_{2}\mathcal{E}_{F}(t,u)\cdot F(u)$



In [67]:

    
reduce_order(ex)









    Out[67]:





$F(\mathcal{E}_{F}(t,u))$



In [68]:

    
ex = E(F,t,u,F(u),v)









    Out[68]:





$\partial_{2}^{2}\mathcal{E}_{F}(t,u)(F(u),v)$



In [69]:

    
reduce_order(ex)









    Out[69]:





$-\partial_{2}\mathcal{E}_{F}(t,u)\cdot F'(u)\cdot v+F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}\mathcal{E}_{F}(t,u)\cdot v$



In [70]:

    
ex = E(F,t,u,F(u),v, w)









    Out[70]:





$\partial_{2}^{3}\mathcal{E}_{F}(t,u)(F(u),v,w)$



In [71]:

    
reduce_order(ex)









    Out[71]:





$-\partial_{2}^{2}\mathcal{E}_{F}(t,u)(F'(u)\cdot v,w)-\partial_{2}^{2}\mathcal{E}_{F}(t,u)(F'(u)\cdot w,v)+F'(\mathcal{E}_{F}(t,u))\cdot \partial_{2}^{2}\mathcal{E}_{F}(t,u)(v,w)-\partial_{2}\mathcal{E}_{F}(t,u)\cdot F''(u)(v,w)+F''(\mathcal{E}_{F}(t,u))(\partial_{2}\mathcal{E}_{F}(t,u)\cdot v,\partial_{2}\mathcal{E}_{F}(t,u)\cdot w)$

substitute



In [72]:

    
substitute(E(G,t,v), v, E(F,t,u))









    Out[72]:





$\mathcal{E}_{G}(t,\mathcal{E}_{F}(t,u))$

Commutator $[F,G]$:



In [73]:

    
C_FG = F(u,G(u))-G(u,F(u))









    Out[73]:





$-G'(u)\cdot F(u)+F'(u)\cdot G(u)$

Double commutator $[F,[F,G]]$ by substituting $G\to[F,G]$ in $[F,G]$:



In [74]:

    
expand(substitute(C_FG, G, C_FG, u))









    Out[74]:





$G'(u)\cdot F'(u)\cdot F(u)-2F'(u)\cdot G'(u)\cdot F(u)-F''(u)(G(u),F(u))+F'(u)\cdot F'(u)\cdot G(u)+G''(u)(F(u),F(u))$

commutator



In [84]:

    
ex=commutator(F,G,u)









    Out[84]:





$-G'(u)\cdot F(u)+F'(u)\cdot G(u)$



In [85]:

    
commutator(F,G,H,u)









    Out[85]:





$H'(u)\cdot G'(u)\cdot F(u)+F'(u)\cdot (G'(u)\cdot H(u)-H'(u)\cdot G(u))+H''(u)(G(u),F(u))-G''(u)(H(u),F(u))-G'(u)\cdot H'(u)\cdot F(u)$

Verify Jacobi identity $[F,[G,H]]+[H,[F,G]]+[G,[H,F]]=0$:



In [86]:

    
expand(commutator(F,G,H,u)+commutator(G,H,F,u)+commutator(H,F,G,u))









    Out[86]:





$0$



In [89]:

    
ex=expand(differential(ex,u,v))









    Out[89]:





$F''(u)(G(u),v)-G'(u)\cdot F'(u)\cdot v+F'(u)\cdot G'(u)\cdot v-G''(u)(F(u),v)$



In [90]:

    
expand(differential(ex,u,w))









    Out[90]:





$F''(u)(G'(u)\cdot v,w)+F'''(u)(G(u),v,w)-G''(u)(F'(u)\cdot v,w)+F''(u)(G'(u)\cdot w,v)+F'(u)\cdot G''(u)(w,v)-G''(u)(F'(u)\cdot w,v)-G'(u)\cdot F''(u)(v,w)-G'''(u)(F(u),v,w)$



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