# Solving symbolic equation using vpasolve

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Alessandro Sona on 19 Mar 2021
Commented: Walter Roberson on 21 Apr 2021
syms Del theta
assume(Del,'real')
% assume(theta == 0);
% assume(theta>0 & theta<2*pi)
alpha = 1;
rho_0 = 0.5;
rho = 0.75;
q = 2;
s = 0;
kappa_0 = 1;
kappa_p = 0;
kappa = kappa_0+kappa_p*(rho-rho_0);
kappa_t = kappa+rho*kappa_p;
delta_0 = 0;
delta_p = 0;
delta = delta_0+delta_p*(rho-rho_0);
delta_t = delta+rho*delta_p;
R_0 = 3;
Del_shift = Del*(rho-rho_0);
R = R_0 - Del_shift + rho*(cos(theta)-delta*sin(theta)^2);
eps = rho_0/R_0;
D = kappa*rho*(cos(theta)*(cos(theta)-Del-delta_t*sin(theta)^2)+kappa_t/kappa*sin(theta)*(sin(theta)+delta*sin(2*theta)));
D_t = (cos(theta)*(cos(theta)-Del-delta_t*sin(theta)^2)+kappa_t/kappa*sin(theta)*(sin(theta)+delta*sin(2*theta)));
C = (sin(theta)+delta*sin(2*theta))*(cos(theta)-Del-delta_t*sin(theta)^2)-kappa*kappa_t*sin(theta)*cos(theta);
A_0 = kappa_0^2*cos(theta)^2+(sin(theta)+delta_0*sin(2*theta))^2;
%dRdrho = diff(R, rho);
dRdrho = cos(theta)-delta_0*sin(theta)^2-2*rho*delta_p*sin(theta)^2-rho_0*delta_p*sin(theta)^2;
E = 1/(pi)* int(-D*dRdrho*R_0/R^2, theta, 0, 2*pi);
F = 1/(2*pi)* int(D^3/(R*rho_0^2*A_0), theta, 0, 2*pi);
G_1 = 1/(2*pi)* int(D^3*R/(R_0^2*rho_0^2*A_0), theta, 0, 2*pi);
CBd = diff(C/(D*R), theta);
CB = CBd * (D*R/rho_0/A_0) + 2*(1/rho_0+(kappa_0*kappa_p*cos(theta)^2+(sin(theta)+delta_0*sin(2*theta))*sin(2*theta*delta_p))/A_0);
G_2 = 1/(2*pi)* int(D*rho_0/R*CB, theta, 0, 2*pi);
H = 1/(2*pi)* int(D_t*R_0/R, theta, 0, 2*pi);
L = kappa_0*eps*H;
SB = -alpha/(2*L) + (s*L/eps-E-G_2/eps+alpha*G_1/(2*L^2))/(L^2/q^2+F/L);
Sol = vpasolve(2*Del*(SB) -alpha*eps/L*(Del*eps-1-Del^2) -(2*delta_0-Del)/kappa_0*H +(1+eps*Del)/(1-eps^2)*L == 0, Del);
I need to solve a symbolic equation (last line of my code), but my code just runs forever. This equation must be solved for one variable only (Del).
However, this equation requires that I first derive some quantities: these contain the symbolic variable theta. I am not sure how to handle this and why my code fails.
My hypotheses are:
• The symbolic solutions I want to obtain before putting together the final equation are too complicated for Matlab;
• I am not setting theta to any value so the equation cannot be solved (how can I do this before solving but after integrating?);
• There is just something completely wrong in how I wrote my code.
Any ideas?
Edit: I debugged my code and found that the lines where I integrate to compute E, F, G1... return very weird solutions. I do not know what the problem may be though.
Walter Roberson on 21 Apr 2021
for G_2,
Del = 9 exactly to Del = 15 (not included) is -infinity, but Del = 15 exactly is +infinity
It looks like it might be finite outside those bounds.

Divija Aleti on 21 Apr 2021
Hi Alessandro,
The "int" function cannot solve all integrals since symbolic integration is such a complicated task. It is also possible that no analytic or elementary closed-form solution exists.
In this case, the lines where you have to integrate to compute E,F,G1 etc., are returning unresolved integrals as no closed form solution exists. You can try approximating these integrals using numeric approximations.
Regards,
Divija

R2020a

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