Painlevé Transcendents - 32.13 Reductions of Partial Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
32.13.E1 v t - 6 ⁢ v 2 ⁢ v x + v x ⁢ x ⁢ x = 0 subscript 𝑣 𝑡 6 superscript 𝑣 2 subscript 𝑣 𝑥 subscript 𝑣 𝑥 𝑥 𝑥 0 {\displaystyle{\displaystyle v_{t}-6v^{2}v_{x}+v_{xxx}=0}}
v_{t}-6v^{2}v_{x}+v_{xxx} = 0		 

v[t]- 6*(v)^(2)* v[x]+ v[x, x, x] = 0
 
Subscript[v, t]- 6*(v)^(2)* Subscript[v, x]+ Subscript[v, x, x, x] == 0
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex1 z = x ⁢ ( 3 ⁢ t ) - 1 / 3 𝑧 𝑥 superscript 3 𝑡 1 3 {\displaystyle{\displaystyle z=x(3t)^{-1/3}}}
z = x(3t)^{-1/3}		 

(x(+)*y*I) = (x(3*t))^(- 1/3)
 
(x[+]*y*I) == (x[3*t])^(- 1/3)
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex2 v ⁢ ( x , t ) = ( 3 ⁢ t ) - 1 / 3 ⁢ w ⁢ ( z ) 𝑣 𝑥 𝑡 superscript 3 𝑡 1 3 𝑤 𝑧 {\displaystyle{\displaystyle v(x,t)=(3t)^{-1/3}w(z)}}
v(x,t) = (3t)^{-1/3}w(z)		 

v(x , t) = (3*t)^(- 1/3)* w*((x + y*I))
 
v[x , t] == (3*t)^(- 1/3)* w*((x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
32.13.E3 u t + 6 ⁢ u ⁢ u x + u x ⁢ x ⁢ x = 0 subscript 𝑢 𝑡 6 𝑢 subscript 𝑢 𝑥 subscript 𝑢 𝑥 𝑥 𝑥 0 {\displaystyle{\displaystyle u_{t}+6uu_{x}+u_{xxx}=0}}
u_{t}+6uu_{x}+u_{xxx} = 0		 

u[t]+ 6*u*u[x]+ u[x, x, x] = 0
 
Subscript[u, t]+ 6*u*Subscript[u, x]+ Subscript[u, x, x, x] == 0
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex3 z = x ⁢ ( 3 ⁢ t ) - 1 / 3 𝑧 𝑥 superscript 3 𝑡 1 3 {\displaystyle{\displaystyle z=x(3t)^{-1/3}}}
z = x(3t)^{-1/3}		 

(x(+)*y*I) = (x(3*t))^(- 1/3)
 
(x[+]*y*I) == (x[3*t])^(- 1/3)
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex5 z = x + 3 ⁢ λ ⁢ t 2 𝑧 𝑥 3 𝜆 superscript 𝑡 2 {\displaystyle{\displaystyle z=x+3\lambda t^{2}}}
z = x+3\lambda t^{2}		 

(x + y*I) = x + 3*lambda*(t)^(2)
 
(x + y*I) == x + 3*\[Lambda]*(t)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex6 u ⁢ ( x , t ) = W ⁢ ( z ) - λ ⁢ t 𝑢 𝑥 𝑡 𝑊 𝑧 𝜆 𝑡 {\displaystyle{\displaystyle u(x,t)=W(z)-\lambda t}}
u(x,t) = W(z)-\lambda t		 

u(x , t) = W*((x + y*I))- lambda*t
 
u[x , t] == W*((x + y*I))- \[Lambda]*t
 Skipped - no semantic math Skipped - no semantic math - -
32.13.E6 u x ⁢ t = sin ⁡ u subscript 𝑢 𝑥 𝑡 𝑢 {\displaystyle{\displaystyle u_{xt}=\sin u}}
u_{xt} = \sin@@{u}		 

u[x, t] = sin(u)

Subscript[u, x, t] == Sin[u]
Failure Failure
Failed [300 / 300]
Result: .70450695e-2+.1624035369*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, u[x*t] = 1/2*3^(1/2)+1/2*I}


Result: -1.358980334+.5284289409*I
Test Values: {t = -3/2, u = 1/2*3^(1/2)+1/2*I, x = 3/2, u[x*t] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.007045069484300837, 0.16240353677712993]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[u, Times[t, x]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.3589803343001376, 0.5284289405615687]
Test Values: {Rule[t, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[Subscript[u, Times[t, x]], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
32.13#Ex7 z = x ⁢ t 𝑧 𝑥 𝑡 {\displaystyle{\displaystyle z=xt}}
z = xt		 

(x + y*I) = x*t
 
(x + y*I) == x*t
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex8 u ⁢ ( x , t ) = v ⁢ ( z ) 𝑢 𝑥 𝑡 𝑣 𝑧 {\displaystyle{\displaystyle u(x,t)=v(z)}}
u(x,t) = v(z)		 

u(x , t) = v*((x + y*I))
 
u[x , t] == v*((x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
32.13.E8 u t ⁢ t = u x ⁢ x - 6 ⁢ ( u 2 ) x ⁢ x + u x ⁢ x ⁢ x ⁢ x subscript 𝑢 𝑡 𝑡 subscript 𝑢 𝑥 𝑥 6 subscript superscript 𝑢 2 𝑥 𝑥 subscript 𝑢 𝑥 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle u_{tt}=u_{xx}-6(u^{2})_{xx}+u_{xxxx}}}
u_{tt} = u_{xx}-6(u^{2})_{xx}+u_{xxxx}		 

u[t, t] = u[x, x]- 6*(u)^(2)[x, x]+ u[x, x, x, x]
 
Subscript[u, t, t] == Subscript[u, x, x]- 6*Subscript[(u)^(2), x, x]+ Subscript[u, x, x, x, x]
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex9 z = x - c ⁢ t 𝑧 𝑥 𝑐 𝑡 {\displaystyle{\displaystyle z=x-ct}}
z = x-ct		 

(x + y*I) = x - c*t
 
(x + y*I) == x - c*t
 Skipped - no semantic math Skipped - no semantic math - -
32.13#Ex10 u ⁢ ( x , t ) = v ⁢ ( z ) 𝑢 𝑥 𝑡 𝑣 𝑧 {\displaystyle{\displaystyle u(x,t)=v(z)}}
u(x,t) = v(z)		 

u(x , t) = v*((x + y*I))
 
u[x , t] == v*((x + y*I))
 Skipped - no semantic math Skipped - no semantic math - -
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