system of differential equations

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I have a system of differential equations as follows
where k1,k2... are constants. I tried solving it by using odeToVectorField and matlabFunction then ode45. However, I had "number of indeterminates exceeds the number of ODEs" error. Is there another way to solve this?
%% Photoionization Regime
syms P(z) R(z) ne(z)
%Constants and initial conditions
l=8;
lambda0=775e-9;
n0=1;
n2=3e-23;
e=1.60217662e-19; %elementary charge
Pcr=lambda0^2/(2*pi*n0*n2); %critical power
hbar=1.0545718e-34; %h bar
epsilon0=8.85418782*1e-12;
mu0=4*pi*1e-7;
c=sqrt(1/epsilon0/mu0); %speed of light
k0=2*pi/lambda0; %wave number
w0=2*pi*c/lambda0; %angular frequency
nn=2.7e25; %neutral gas density
Uion=12.1*e;
% l=floor(Uion/hbar/w0+1);
sigmamp=6.4e-22; %cross section
Imp=hbar*w0^2/sigmamp; %photoionization intensity
re=2.8e-15; %classical electron radius
Pnl=lambda0^2/(2*pi*n2); %nonlinear focusing power
zf=2000; %z final
intnum=5000;
%Initial conditions
alpha0=-.7; %initial alpha
R0=1e-2; %initial spot size
P0=Pcr; %initial power
dRdz0=-2*alpha0/k0/R0;
PR0=R0^2/P0;
I0=2*P0/pi/R0^2; %initial intensity
dnedz0=2*pi*w0
%Differential equations
I=2*P/pi/R^2;
Wmp=2*pi*w0/factorial(l-1)*(I/Imp)^l;
dnedz=Wmp*nn/c;
dPdz=diff(P,1)==-c*pi/2*Uion*R^2/l*dnedz;
d2Rdz2=diff(R,2)==4/k0^2/R^3*(1-P/Pnl+2*pi*l/(l+1)^2*...
re*R^2*ne)-(l-1)/2/l/R^3*(R^2*diff(R^2/P*dPdz,1)+(l-1)/2/l...
*(R^2/P*dPdz)^2);
eqn=[d2Rdz2 dPdz dnedz];
v=odeToVectorField(eqn); %HERE I GET THE ERROR
  2 个评论
Ayberk Cakmak
Ayberk Cakmak 2020-5-26
I resubmitted with the code attached. Thanks in advance.

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darova
darova 2020-5-26
Here is the problem: there is second derivative of , but there is no expression for
Looks like 4 variables, but only 3 equations

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