Symbolic function doesn't work

syms L alpha gamma Y2 Y3 Y4 Y5 Y6 x X dL T(L,x)
syms T(L,x)

M1 = 0;
M2 = 7;
Y1 = 1 + M2 / 10;
Y7 = 3 + M1 / 10;
xcp = [0 0.1*L 0.25*L 0.5*L 0.6*L 0.75*L L];
ycp = [Y1 Y2 Y3 Y4 Y5 Y6 Y7];
T0 = 5 + 1/L - 25/(L^2);
S = bezier_syms();

ode = diff(T,x) == (1+alpha*M1)*(T^2)*S(L,x,Y2,Y3,Y4,Y5,Y6) - 4*(2 + gamma*M2)*x*T;
derivL = diff(ode,L);
U = diff(T,L);
eqns = [derivL, diff(T,L) == dL];

eqns = matlabFunction(eqns);

sol = ode45(eqns,[0 5],diff(T0,L));

I get the error:

Warning: Function 'T' not verified to be a valid MATLAB function. 

For some reason it doesn't let me define the T function. Any ideas?

Edit:

function S = bezier_syms()
syms t x y s X Y2 Y3 Y4 Y5 Y6 L

M1 = 0;
M2 = 7;
Y1 = 1 + M2 / 10;
Y7 = 3 + M1 / 10;
xcp = [0 0.1*L 0.25*L 0.5*L 0.6*L 0.75*L L];
ycp = [Y1 Y2 Y3 Y4 Y5 Y6 Y7];

N = length(xcp) - 1;

% Calculate m matrix
m = zeros(N+1,N+1);
for i=0:N
    for j=0:N
        if i <= j
            m(i+1,j+1) = ((-1)^(j-i)) * nchoosek(N,j) * nchoosek(j,i);
        end
    end
end

% Calculate C
T = sym(zeros(N+1,1));
for i = 1:length(T)
    T(i) = t^(i-1);
end

C = m * T;

x = 0;
y = 0;
for i = 1:N+1
    x = x + xcp(i)*C(i);
    y = y + ycp(i)*C(i);
end

% Convert syms to functions
getx = matlabFunction(x);
gety = matlabFunction(y);

% Polynomial regression
regt = linspace(0,1,8);
n = length(regt);
xreg = sym(zeros(n,1));
Yreg = sym(zeros(n,1));

for i=1:n
    xreg(i) = getx(L,regt(i));
    Yreg(i) = gety(Y2,Y3,Y4,Y5,Y6,regt(i));
end

Xreg = sym(zeros(n,n));
Xreg(:,1) = ones(n,1);
for i=1:(n-1)
    Xreg(:,i+1) = xreg.^i;
end

coef = inv(Xreg' * Xreg) * Xreg' * Yreg; %#ok<MINV>
s = 0;
for i=1:n
    s = s + coef(i)*(X^(i-1));
end

S = matlabFunction(s);
end

[Star Strider]

I suspect the problem is ‘S’. We can’t run this because we don’t have the function.

Meanwhile:

syms T(L,x)
Eqn = diff(T,L) + diff(T,x)

produces:

Eqn(L, x) =
diff(T(L, x), L) + diff(T(L, x), x)

so ‘T’ itself is not the problem.

[John F]

@StarStrider I edited the question to include it

[Walter Roberson]

use odeFunction not matlabFunction when you are working with ode

[Star Strider]

The problem is likely that ‘T’ is a multivariate function, so you are attempting to have ode45 solve a partial differential equation. It is not designed to do that.

I usually use odeToVectorField before using matlabFunction for systems of equations. When I tried that here:

eqns1 = [derivL, diff(T,L) == dL];
eqns2 = odeToVectorField(eqns1);

the error was:

Error using odeToVectorField>mupadOdeToVectorField (line 160)
Symbolic ODEs must have exactly one independent variable.

You need to revisit this to understand what you want to do, and what you want the ordinary differential equation solvers to do.

I also suspect that ‘S’ needs to be defined and called in your code with all its arguments, rather that the way you are calling it now.

[John F]

@WalterRoberson I used something like:

eqns = odeFunction(eqns,[T(L,x),U(L,x)]);

but it doesn't work. I haven't used odeFunction before

[Walter Roberson]

eqns = [derivL, diff(T,L) == dL];

You cannot include boundary conditions in the equations. Please read the first example for odeFunction to see the work flow -- but also take into consideration that ode45 cannot be used for partial differential equations

[John F]

@StarStrider I had asked another question before this one here. Maybe that helps you understand what I'm trying to achieve

[Star Strider]

I posted an Answer to your other Question (that you cited in your Comment) just now.