The problem is to solve a system of nonlinear equations with a definite integral. $Q_i(x), i=1,2,3, x \in [-\text{max},\text{max}]$ \begin{align*} a_iQ_i''(x)-b_i(\vec{Q})-d_i\int_{-\text{max}}^{\text{max}} {\hat K Q_i dx'} = 0 \end{align*}
How can I solve the cases like $d_i=0.01,0.1$ or any other values. It appears to me Laplace transform doesn't help. I failed to turn it to a system of ODEs by differentiation, either. Or I missed something?
So the question is about dealing with the integral. Any suggestions or comments? Thanks in advance.
I use the following code to easily solve the case when all $d_i=0$. The parameters are set to their typical values for my need ($\alpha_1-\alpha_{2,3}>2w_1>0=w_{2,3}$), but one can tune. $b_i=\frac{2}{\alpha_i}(Q_i-w_i)-\frac{Q_i}{|\vec{Q}|}$ and $w_i$ is constant. The form of boundary condition is not to be changed.
The integral kernal $\hat K Q_i=\frac{Q_i(x)-Q_i(x')}{(x-x')^2}$ is singular, although somehow suppressed by the nominator. If not possible to solve this, adding some small cutoff $\frac{Q_i(x)-Q_i(x')}{(x-x')^2+\delta^2}$ is OK.
When all $d_i=0$, it seems that one of $Q_2,Q_3$ stays at zero. I wonder if this holds even when $d_i\neq0$.
ClearAll["Global`*"];
max = 5.0; accur = 15;
w1 = 0.01; w2 = 0; w3 = 0; w = {w1, w2, w3};
α1 = 1.5; α2 = 1.0; α3 = 0.8; α = {\
α1, α2, α3};
Qstart = {w1 - α1/2, 0.0, 0.0}; Qend = {w1 + α1/2, 0.0,
0.0};
M1 = 1/α1; M2 = 1/α2; M3 = 1/α3; M = {M1, M2, M3};
dV = {2/α1 (Q1[u] - w1) - Q1[u]/Sqrt[
Q1[u]^2 + Q2[u]^2 + Q3[u]^2],
2/α2 (Q2[u] - w2) - Q2[u]/Sqrt[Q1[u]^2 + Q2[u]^2 + Q3[u]^2],
2/α3 (Q3[u] - w3) - Q3[u]/Sqrt[
Q1[u]^2 + Q2[u]^2 + Q3[u]^2]};
s = NDSolve[{M1 Q1''[u] == dV[[1]], M2 Q2''[u] == dV[[2]],
M3 Q3''[u] == dV[[3]], Q1[-max] == Qstart[[1]],
Q2[-max] == Qstart[[2]], Q3[-max] == Qstart[[3]],
Q1[max] == Qend[[1]], Q2[max] == Qend[[2]],
Q3[max] == Qend[[3]]}, {Q1, Q2, Q3}, {u, -max, max},
AccuracyGoal -> accur];
QQ = First[{Q1, Q2, Q3} /. s];
Plot[{QQ[[1]][u], QQ[[2]][u], QQ[[3]][u]}, {u, -max, max},
PlotRange -> All, PlotLegends -> Automatic]