In[]:=
CompoundExpression[
]
​​deploy
Tue 16 May 2023 10:53:51

Approximating solution of
∂
∂t
fhf+hdihf

Solution has a simple form in Laplace Domain in terms of
-1
tan
(
s
)
​No closed form in time domain , hence try asymptotic approximations​Related questions:- “Simplifying arctan expression using Puiseux series” math.SE post​ Lutz recommended
x=2
2
s
replacement, Goncalo provided expression from
-1
tan
(x)
≃x equivalence - “Solving ....” math.SE post​ DinosaurEgg gives
-1
tan
(
s
)
solution by some algebra on the self-consistent equation- “When to use Series vs Asymptotic” mathematica.SE post​​Parent notebook:​forum-mean-field-equations.nb​​

Asymptotic expansions of ArcTan expression

In[]:=
ClearAll["Global`*"];​​expr=
1
s
2
+
2
s
π-2ArcTan
s
2

;​​SF=StringForm;​​series[order_]:=Normal@Series[expr,{s,0,order}];​​asymp[order_]:=Asymptotic[expr,{s,0,order}];​​{approxSeries,approxAsymp}=Table[{i,#[i]},{i,0,3}]&/@{series,asymp};​​​​visualize[approx_,label_]:=(​​Print[TableForm[approx,TableHeadings->{{},{"order","expr"}}]];​​LogLogPlot@@{{expr}~Join~(Last/@approx),{s,0.,.2},​​PlotLegends->{"true"}~Join~(SF["order ``",First@#]&/@approx),​​PlotStyle->{{Thick,Opacity[.5]},Automatic,Dashed,Dotted,DotDashed},​​PlotLabel->label}​​)​​Quiet[visualize[approxSeries,"Series"]]​​Quiet[visualize[approxAsymp,"Asymptotic"]]
Asymptotic
:Approximation order specification 0 should be a positive integer or Infinity.
order
expr
0
1
4
(-4-
2
π
)+
π
2
s
1
1
4
(-4-
2
π
)+
π
2
s
+
π(8+
2
π
)
s
8
2
+
1
96
(-32-36
2
π
-3
4
π
)s
2
1
4
(-4-
2
π
)+
π
2
s
+
π(8+
2
π
)
s
8
2
+
1
96
(-32-36
2
π
-3
4
π
)s+
π(112+48
2
π
+3
4
π
)
3/2
s
192
2
+
(-512-1200
2
π
-300
4
π
-15
6
π
)
2
s
3840
3
1
4
(-4-
2
π
)+
π
2
s
+
π(8+
2
π
)
s
8
2
+
π-
2
3π
-
3π
4
2
-
3
π
16
2
s
2
+
π
7
12
+
2
π
4
+
4
π
64
3/2
s
2
+
π-
2
2
15π
-
5π
8
2
-
5
3
π
32
2
-
5
π
128
2
2
s
2
+
π
3
10
+
13
2
π
48
+
3
4
π
64
+
6
π
512
5/2
s
2
+
π-
29
315
2
π
-
33π
80
2
-
5
3
π
24
2
-
7
5
π
256
2
-
7
π
1024
2
3
s
2
Out[]=
true
order 0
order 1
order 2
order 3
order
expr
0
Asymptotic
1
s
2
+
2
s
π-2ArcTan
s
2

,{s,0,0}
1
π
2
s
2
1
4
(-4-
2
π
)+
π
2
s
+
π(8+
2
π
)
s
8
2
+
1
96
(-32-36
2
π
-3
4
π
)s+
π(112+48
2
π
+3
4
π
)
3/2
s
192
2
+
(-512-1200
2
π
-300
4
π
-15
6
π
)
2
s
3840
3
1
4
(-4-
2
π
)+
π
2
s
+
π(8+
2
π
)
s
8
2
+
π-
2
3π
-
3π
4
2
-
3
π
16
2
s
2
+
π
7
12
+
2
π
4
+
4
π
64
3/2
s
2
+
π-
2
2
15π
-
5π
8
2
-
5
3
π
32
2
-
5
π
128
2
2
s
2
+
π
3
10
+
13
2
π
48
+
3
4
π
64
+
6
π
512
5/2
s
2
+
π-
29
315
2
π
-
33π
80
2
-
5
3
π
24
2
-
7
5
π
256
2
-
7
π
1024
2
3
s
2
Out[]=
true
order 0
order 1
order 2
order 3

Do
x=2
2
s
replacement

In[]:=
ClearAll["Global`*"];​​$Assumptions={x>0};​​​​expr=Simplify
1
s
2
+
2
s
π-2ArcTan
s
2

/.s->2
2
x
;​​SF=StringForm;​​series[order_]:=Normal@Series[expr,{x,0,order}];​​asymp[order_]:=Asymptotic[expr,{x,0,order}];​​{approxSeries,approxAsymp}=Table[{i,#[i]},{i,0,3}]&/@{series,asymp};​​​​visualize[approx_,label_]:=(​​Print[TableForm[approx,TableHeadings->{{},{"order","expr"}}]];​​LogLogPlot@@{{expr}~Join~(Last/@approx),{x,0.,.2},​​PlotLegends->{"true"}~Join~(SF["order ``",First@#]&/@approx),​​PlotStyle->{{Thick,Opacity[.5]},Automatic,Dashed,Dotted,DotDashed},​​PlotLabel->label}​​)​​Quiet[visualize[approxSeries,"Series"]]​​Quiet[visualize[approxAsymp,"Asymptotic"]]
Asymptotic
:Approximation order specification 0 should be a positive integer or Infinity.
order
expr
0
-1-
2
π
4
+
π
2x
1
-1-
2
π
4
+
π
2x
+
1
8
(8π+
3
π
)x
2
-1-
2
π
4
+
π
2x
+
1
8
(8π+
3
π
)x+
1
48
(-32-36
2
π
-3
4
π
)
2
x
3
-1-
2
π
4
+
π
2x
+
1
8
(8π+
3
π
)x+
1
48
(-32-36
2
π
-3
4
π
)
2
x
+
1
96
(112π+48
3
π
+3
5
π
)
3
x

Connection to differential equation

Check that this equation is solution to differential equation as in
https://math.stackexchange.com/a/4694070/998
​