How to solve a non linear 3rd order differential equation

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Sumedh Soman 2020-7-4

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评论： darova 2020-7-5

Hi, I need to solve a 3rd order, non linear differential equation. I would preferably like a symbolic solution for the equation, as this equation is the input for another script. The equation is as follows:

where m is a constant (for my purposes, m =1)

f is a function of variable eta, and the initial conditions are as follows:

;

and

I tried using the code mentioned below, but it is unable to give me a symbolic solution. Can anybody please help me solve this problem?

m = 1;
syms f(eta)
Df = diff(f,eta);
D2f = diff(f,eta,2);
ode = diff(f,eta,3)+0.5*(m+1)*f*D2f+m*(1-Df^2) == 0;
cond1 = f(0) == 0;
cond2 = Df(0) == 0;
cond3 = D2f(10^100) == 1;
conds = [cond1 cond2 cond3];
uSol(eta) = dsolve(ode,conds)

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Sumedh Soman 2020-7-4

在 MATLAB Online 中打开

this is the solver code I used:

xmesh = linspace(0,100,5);
solinit = bvpinit(xmesh, @guess);
sol = bvp4c(@bvpfcn, @bcfcn, solinit);
plot(sol.x, sol.y, '-o');
function dydx = bvpfcn(x,y) % equation to solve
dydx = zeros(3,1);
dydx = [y(2) y(3) (-((1+1)/2)*(y(3)*y(1))-1*(1-(y(2))^2))];
end
function res = bcfcn(ya,yb) 
res = [ya(1) ya(2) yb(3)];
end
function g = guess(x) 
g = [x 1 0];
end