m20230317摄动求解.nb
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chap2.nb
(**********************************************************************************)
(* Mathematica Code for the illustrative problem in chapter 2 of the book: *)
(* BEYOND BERTURBATION - INTRODUCTION TO THE HOMOTOPY ANALYSIS METHOD *)
(* by Shijun LIAO, Published by Chapman & Hall/CRC, Boca Raton, 2003 *)
(**********************************************************************************)
(* Governing equation: *)
(* V'(t) +V(t)*V(t) = 1, *)
(* *)
(* subject to the initial condition: *)
(* *)
(* V(0) = 0 *)
(* *)
(* Sept. 2005 *)
(* Shanghai Jiao Tong Uni *)
(* by SHIjun LIAO *)
(**********************************************************************************)
<<Calculus`Pade`;
<<Graphics`Graphics`;
(**********************************************************************************)
(* Which base functions would you like to use ? *)
(*--------------------------------------------------------------------------------*)
(* BASE = 1 corresponds to polynomial functions (2.50) *)
(* BASE = 2 corresponds to fractional functions (2.62) *)
(* BASE = 3 corresponds to exponential functions (2.70) *)
(* BASE = 4 corresponds to set of base functions (2.82) *)
(**********************************************************************************)
BASE = 1;
(**********************************************************************************)
(* Define chi[m] *)
(* This module gives the definition (2.42) of chi[m] *)
(**********************************************************************************)
chi[m_]:=If[m<=1,0,1];
(**********************************************************************************)
(* Define GetR[m] *)
(* This module gives R[m] defined by (2.43) *)
(**********************************************************************************)
GetR[m_]:=Module[{temp,n},
temp[1] = D[v[m-1],x];
temp[2] = Sum[v[n]*v[m-1-n],{n,0,m-1}];
temp[3] = temp[1] + temp[2] + chi[m] -1;
R[m] = temp[3]//Expand;
];
(**********************************************************************************)
(* Define GetRHS[m] *)
(* This module gets the whole right-hand side term of Eq. (2.39) *)
(**********************************************************************************)
GetRHS[m_]:=Module[{},
GetR[m];
RHS[m]= Expand[hbar*R[m]*H[x]];
];
(**********************************************************************************)
(* Define H *)
(* This module defines the auxiliary function H[t] *)
(**********************************************************************************)
H[x_]:=Module[{temp},
If[BASE == 1 || BASE == 4, temp = 1];
If[BASE == 2, temp = 1/x];
If[BASE == 3, temp = Exp[-x]];
Expand[temp]
];
(**********************************************************************************)
(* Define GetInitial *)
(* This module defines the initial guess of V(t) *)
(**********************************************************************************)
GetInitial = Module[{},
If[BASE == 1, v[0] = x];
If[BASE == 2, v[0] = 1-1/x];
If[BASE == 3 || BASE == 4, v[0] = 1-Exp[-x]];
Print[" initial guess = ",v[0]];
];
(**********************************************************************************)
(* Define L *)
(* This module defines the auxiliary linear operator (2.23) *)
(**********************************************************************************)
L[f_]:=Module[{},
If[BASE == 1, gamma[1] = 1; gamma[2] = 0];
If[BASE == 2, gamma[1] = x; gamma[2] = 1];
If[BASE == 3 || BASE == 4, gamma[1] = 1; gamma[2] = 1];
Expand[gamma[1]*D[f,x]+gamma[2]*f]
];
(**********************************************************************************)
(* Define Linv *)
(* This module defines the inverse of the auxiliary linear operator (2.23) *)
(* In other words, find solution u of the following equation: *)
(* L[u ] = f *)
(**********************************************************************************)
Linv[f_]:=Module[{temp,EQ,u,solution},
EQ = L[u[x]]-f;
temp = DSolve[EQ==0,u[x],x];
solution = temp[[1,1,2]]/.C[_]->0;
Expand[solution]
];
(**********************************************************************************)
(* the property of the inverse operator of L *)
(* Linv[f_+g_] := Linv[f] + Linv[g] *)
(**********************************************************************************)
Linv[p_Plus] := Map[Linv,p];
Linv[c_*f_] := c*Linv[f] /; FreeQ[c,x];
(**********************************************************************************)
(* Define GetSPECIAL *)
(* This module gives a special solution of (2.39) *)
(**********************************************************************************)
GetSPECIAL[m_]:=Module[{temp},
temp = Expand[RHS[m]];
SPECIAL = Linv[temp];
];
(**********************************************************************************)
(* Define GetCOMMON *)
(* This module gives a commoon solution of (2.39) *)
(**********************************************************************************)
GetCOMMON[m_]:=Module[{temp,u,solution},
temp = DSolve[L[u[x]]==0,u[x],x];
solution = temp[[1,1,2]];
COMMON = SPECIAL + chi[m]*v[m-1] + solution;
];
(**********************************************************************************)
(* Define Getv *)
(* This module gives the solution of (2.39) and (2.40) *)
(**********************************************************************************)
Getv[m_]:=Module[{temp,EQ,x0,res},
If[BASE == 1 || BASE == 3 || BASE == 4, x0 = 0];
If[BASE == 2, x0 = 1];
EQ = Expand[COMMON/.x->x0];
temp = Solve[EQ==0,C[1]];
res = COMMON/.temp[[1,1]];
v[m] = res//Expand;
];
(**********************************************************************************)
(* Define HP[F,m,n] *)
(* This module gives the [m,n] homotopy-Pade approximation of the series *)
(* F[k] = sum[f[i],{i,0,k}] *)
(* For details, please see section 2.3.7 *)
(**********************************************************************************)
hp[F_,m_,n_]:=Block[{i,k,dF,temp,q},
dF[0] = F[0];
For[k=1, k<=m+n, k=k+1, dF[k] = Expand[F[k] - F[k-1]] ];
temp = dF[0]+Sum[dF[i]*q^i,{i,1,m+n}];
Pade[temp,{q,0,m,n}]/.q->1
];
(**********************************************************************************)
(* Definition of the function (2.58) *)
(**********************************************************************************)
mu0[m_,n_,h_] := (-h)^n*Sum[Binomial[n-1+j,j]*(1+h)^j,{j,0, m-n}];
(**********************************************************************************)
(* Definition of the function (2.85) *)
(**********************************************************************************)
sigma0[m_,n_,k_,h_] := (mu0[m,n+k,h]+mu0[m,n+k-1,h])/2;
(**********************************************************************************)
(* Definition of the function (2.91) *)
(**********************************************************************************)
mu[m_,n_,alpha_,h_]:=If[n>m,0,(-h)^(n-alpha)
*(1+Sum[(-1)^j*Binomial[alpha-n,j]*(1+h)^j,{j,1,m-n}])];
(**********************************************************************************)
(* Definition of the function (2.96) *)
(**********************************************************************************)
sigma[m_,n_,k_,alpha_,h_]:=(mu[m,n+k,alpha,h]+mu[m,n+k-1,alpha,h])/2;
(**********************************************************************************)
(* Main code *)
(**********************************************************************************)
ham[m0_,m1_]:=Module[{temp,k,variable},
For[k=Max[1,m0],k<=m1,k=k+1,
Print[" k = ",k];
GetRHS[k];
GetSPECIAL[k];
GetCOMMON[k];
Getv[k];
If[BASE == 1 || BASE == 3 || BASE == 4, variable = t];
If[BASE == 2, variable = 1 + t ];
If[k == 1, V[0] = v[0]/.x->variable];
V[k] = Simplify[V[k-1] + v[k]/.x->variable];
Vtt0[k] = D[V[k],{t,2}]/.t->0//Expand;
Vttt0[k] = D[V[k],{t,3}]/.t->0//Expand;
If[PRN==1,
Print[" V''(0) = ",N[Vtt0[k],20], " delta = ",N[Vtt0[k]-chi[k]*Vtt0[k-1],20]];
Print[" V'''(0) = ",N[Vttt0[k],20]," delta = ",N[Vttt0[k]-chi[k]*Vttt0[k-1],20]];
];
];
Print[" Successful !"];
];
(**********************************************************************************)
(* Would you like to print more information to check the convergence ? *)
(* 1 correspondings to YES *)
(* 0 correspondings to NO *)
(**********************************************************************************)
PRN = 1;
(**********************************************************************************)
(* Parameters definitions *)
(**********************************************************************************)
hbar = h;
Print[ "The main code is ham[N_start,N_end]" ];
Print[ "BASE = ",BASE];
Print[ "Auxiliary function = ", H[x] ];
Print[ "Initial guess = ", v[0] ];
Print[ "PRN = ",PRN];
Print[ "hbar = ",hbar];
(* Get up-to 5th-order approximation *)
ham[1,5];
(* Get [1,1] homotopy-Pade approximation of V(t) *)
hp[V,1,1]//Simplify;
kdv-homotopy.m
(* ::Package:: *)
(************************************************************************)
(* This file was generated automatically by the Mathematica front end. *)
(* It contains Initialization cells from a Notebook file, which *)
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(* ".nb" instead of ".m". *)
(* *)
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(* *)
(* DO NOT EDIT THIS FILE. This entire file is regenerated *)
(* automatically each time the parent Notebook file is saved in the *)
(* Mathematica front end. Any changes you make to this file will be *)
(* overwritten. *)
(************************************************************************)
(*********************************************************************************************)
(* This ia a Mathematica code for Kdv equation *)
(* (1-lambda)ft+a*fttt+b*f*ft=0 *)
(*-------------------------------------------------------------------------------------------*)
(* Definition of global variables *)
(* lambda[0]=>initial guess of w0/(C0*k) *)
(* lambda[k]=>kth-order approximation of w/(C0*k) *)
(* f[0]=>initial guess of solution f *)
(* f[k]=>solution of kth-order deformation equation *)
(* F[k]=>kth-order approximation of solution *)
(* ft[k]=>D[f[k],t] *)
(* ftt[k]=>D[f[k].{t,2}] *)
(* fttt[k]=>D[f[k],{t,3}] *)
(* R[k]=>a term on the right hand side of the mth deformation equation *)
(* RHS[k]=>right hand side of the kth-order deformation equation *)
(* fSpecial=>a special solution of the kth-order deformation equation *)
(*********************************************************************************************)
(*********************************************************************************************)
(* Define GetR[k] *)
GetR[k_]:=Module[{temp,n},
temp[1]=ft[k-1];
temp[2]=Sum[lambda[n]*ft[k-1-n],{n,0,k-1}];
(*Print["lambda*f'=",temp[2]];*)
temp[3]=fttt[k-1];
(*Print["f*f'=",temp[3]];*)
temp[4]=Sum[f[n]*ft[k-1-n],{n,0,k-1}];
(*Print["fttt=",temp[4]];*)
temp[5]=TrigReduce[Expand[temp[1]]];
temp[6]=TrigReduce[Expand[temp[2]]];
temp[7]=TrigReduce[Expand[temp[3]]];
temp[8]=TrigReduce[Expand[temp[4]]];
temp[9]=temp[5]-temp[6]+a*temp[7]+b*temp[8];
R[k]=TrigReduce[Expand[temp[9]]];
(*Print["R=",R[k]];*)
];
(* Define GetRHS *)
GetRHS[k_]:=Module[{},
GetR[k];
RHS[k]=Expand[TrigReduce[hbar*R[k]]];
];
(* Define the axiliary linear operator L0 *)
L0[f_]:= Module[{},
Expand[D[f,{t,3}]+D[f,t]];
];
(* Define the inverse operator of the auxiliary linear operator L0 *)
L0inv[Sin[n_*t]]:=Cos[n*t]/(n^3-n);
L0inv[f_+g_]:=L0inv[f]+L0inv[g];
L0inv[c_*f_]:=c*L0inv[f]/;FreeQ[c,t];
L0inv[c_]:=c/;FreeQ[c,t];
L0inv[p_Plus]:=Map[L0inv,p];
(* Set initial guess of f[t] *)
Getf0:=Module[{},
f[0]=Cos[t];
];
(* Define the function chi_{k} *)
chi[k_]:=If[k<=1,0,1];
(* Define Getft *)
Getft[k_]:=Module[{},
ft[k]=Expand[D[f[k],t]];
];
(* Define Getftt *)
Getftt[k_]:=Module[{},
ftt[k]=Expand[D[f[k],{t,2}]];
];
(* Define Getfttt *)
Getfttt[k_]:=Module[{},
fttt[k]=Expand[D[f[k],{t,3}]];
];
(* Define GetfAll *)
GetfAll[k_]:=Module[{},
Getft[k];
Getftt[k];
Getfttt[k];
];
(* Define lumbda[0] *)
GetLambda0:=Module[{temp,eq},
GetfAll[0];
GetR[1];
temp[1]=Expand[R[1]];
temp[2]=TrigReduce[temp[1]];
(*Print["R[1]=",temp[2]];*)
eq=Coefficient0[temp[2],Sin[t]];
(*Print["eq=",eq];*)
temp[0]=Solve[eq==0,lambda[0]];
temp[1]=lambda[0]/.temp[0];
temp[2]=temp[1][[1]];
lambda[0]=Simplify[temp[2]];
(*Print["lambda[0]=",lambda[0]];*)
];
(* Define lambda[k] *)
GetLambda[k_]:=Module[{temp,eq},
If[k==1,Print["lambda[0] must be known"]];
If[k>1,
GetfAll[k-1];
GetRHS[k];
temp[0]=Expand[RHS[k]];
(*Print["RHS=",temp[0]];*)
temp[1]=Expand[temp[0]];
temp[2]=TrigReduce[temp[1]];
eq=Expand[Coefficient[temp[2],Sin[t]]];
(*Print["Eq=",eq];*)
temp[0]=First[lambda[k-1]/.Solve[eq==0,lambda[k-1]]];
lambda[k-1]=Simplify[temp[0]];
];
(*Print["lambda=",lambda[k-1]];*)
];
(* Define CheckRHS[k] *)
CheckRHS[k_]:=Module[{temp,C1},
temp[0]=Expand[RHS[k]];
C1=Coefficient[temp[0],Sin[t]];
temp[1]=temp[0]-C1*Sin[t];
RHS[k]=Expand[temp[1]];
temp[0]=RHS[k]/.Sin[t]->0;
RHS[k]=Expand[temp[0]];
];
(* Define GetfSpecial *)
GetfSpecial[k_]:=Module[{temp},
temp[0]=Expand[RHS[k]];
fSpecial=L0inv[temp[0]];
(*Print["fSpecial=",fSpecial];*)
];
(* Define Getf *)
(*Getf[k_]:=Module[{eq,temp,n,C2},*)
(*temp[0]=fSpecial+C2*Cos[t];*)
(*eq[1]=temp[0]/.t->0;*)
(*eq[2]=temp[0]/.t->Pi;*)
(*eq[3]=Expand[eq[1]];*)
(*eq[4]=Expand[eq[2]];*)
(*temp[1]=First[C2/.Solve[eq[3]-eq[4]==0,C2]];*)
(*C2=Expand[temp[1]];*)
(*Print["C2=",C2];*)
(*temp[2]=Expand[fSpecial+C2*Cos[t]];*)
(*f[k]=temp[2]+chi[k]*f[k-1];*)
(*Print["f=",f[k]];*)
(*];*)
(* Define Getf *)
Getf[k_]:=Module[{eq,temp,n,C2},
temp[0]=fSpecial;
temp[1]=fSpecial+chi[k]*f[k-1];
eq[1]=temp[1]/.t->0;
eq[2]=temp[1]/.t->Pi;
(*eq[3]=Expand[eq[1]];*)
(*eq[4]=Expand[eq[2]];*)
(*temp[1]=First[C2/.Solve[eq[3]-eq[4]==0,C2]];*)
C2=(eq[2]-eq[1])/2;
(*Print["C2=",C2];*)
f[k]=Expand[temp[1]+C2*Cos[t]];
(*f[k]=temp[2]+chi[k]*f[k-1];*)
(*Print["f=",f[k]];*)
];
(* Define GetFig *)
GetFig[m0_,m1_]:=Module[{k,temp},
For[k=m0,k<=m1,k=k+1,
temp[1]=lambda[k-1];
temp[2]=Sqrt[a*6];
fig[k]=Show[exact,ParametricPlot[{temp[1],temp[2]},{}]];
];
];
(* Main Code *)
main[m0_,m1_]:=Module[{temp,k,n},
For[k=Max[1,m0],k<=m1,k=k+1,
Print["k=",k];
GetfAll[k-1];
GetRHS[k];
GetLambda[k];
CheckRHS[k];
GetfSpecial[k];
Getf[k];
LAMBDA[k-1]=Expand[Sum[lambda[n],{n,0,k-1}]];
Print["LAMBDA=",LAMBDA[k-1]//N];
F[k]=F[k-1]+f[k];
(* Print["F=",F[k]];*)
delta1[k-1]=LAMBDA[k-1]-LAMBDA[k-2];
Print["delta1=",delta1[k-1]//N];
delta2[k]=F[k]-F[k-1]/.t->0;
Print["delta2=",delta2[k]//N];
];
(*k=k-1;*)
f1=F[k]/.t->0;
f2=F[k]/.t->Pi;
(*Print["f1=",f1];*)
(*Print["f2=",f2];*)
h=f1-f2;
(*F[k]=F[k]-h;*)
Print["h=",h];
(*makeplotfile["E:\wave_kdv\Result1\F_30_8.dat",F[k],t,0,10,10,0,0];*)
(*makeplotfile["E:\wave_kdv\Result1\LAMBDA_30_3.dat",LAMBDA[k-1],c,0,1.5,100,0,0];*)
(*makeplotfile["E:\wave_kdv\Result1\LAMBDAb_30_3.dat",LAMBDA[k-1],d,0,0.7,100,0,0];*)
(*F[k]>>"E:\wave_kdv\Result\F_30_10.nb";*)
(*LAMBDA[k-1]>>"E:\My Research\Result_2\LAMBDA_20.nb"*)
cpu[];
Print["Successful !"];
];
(* set the initial guess *)
hbar=-1;
Getf0;
F[0]=f[0];
(*Z[0]=lambda[0];*)
lambda[0]=1-a;
(*c=kH,a=gamma;d=A/H;b=3A/2H*)
c=1;
d=3/10;
a=c^2/6;
b=(3/2)*d;
Print["kH=",c];
Print["A/H=",d];
$RecursionLimit=40000;
初始解的选取.nb
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线性算子的选取-小练习1.nb
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