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3

I can recommend collocation method based on Bernoulli wavelets described here. With this method we can solve problem for 8 and 16 collocation points (n=2 or n=3 in this code): Needs["DifferentialEquationsNDSolveProblems"]; Needs["DifferentialEquationsNDSolveUtilities"]; \ Get["NumericalDifferentialEquationAnalysis`"]; ue[t_]...

1

Not an answer for your desired collocation method, but extented comment. I don't know whether you already know the solution of this integral equation. It is x[t] = t Get this with two fold differentiation and then NDSolve. eq[t_] = x[t] == t - t^3/3 + t^4/6 + Integrate[s^3 1/x[s] , {s, 0, t}] + Integrate[-2 (-s + t) x[s]^2 , {s, 0, t}] eq[0] (* x[0] == ...

1

The first element of d is equal to exactly -1. This is because all of the elements u[i, ttt[[1]]] are zero: With[{l = 1}, Table[u[i, ttt[[l]]], {i, 1, 6}]] (* {0,0,0,0,0,0} *) For the same reason, the first element of dd is equal to -(S1[ttt[[1]]] + S2[ttt[[1]]] + f[ttt[[1]]])^2. It happens that S1[ttt[[1]]] and S2[ttt[[1]]] are zero as well, so this ...

4

I have looked into the paper and your code, and I think there are two main issues. I couldn't have deduced how you obtained the value $V_{dc}=100$, and I also think this is why your results are off. However, neither couldn't I have found the correct value in the paper ($\alpha_2 V^2 = 45, V = V_{ac} + V_{dc}, V_{ac}=0.01$ – this still leaves unknown $\alpha$...

6

There are two main problems in your code: A typo in your differential equations: phi -> psi Using the exact DSolve for such a complicated system instead of the numerical solver NDSolve. eqns = {a'[t] == gamma*q[t] - (b1 + b2 + delta1)*a[t] + b3*c[t] + b4*p[t] - F[a[t], c[t]]*a[t]*c[t] - G[a[t], p[t]]*a[t]*p[t], p'[t] == b2*a[t] - (b4 ...

0

Your nonlinear system is a good example for showing Hopf and Bogdanov-Takens bifurcations. With the following changes A=A0+y0;B=A0*y0; we obtain the equilibria: (*{{x -> 1/A0, y -> A0}, {x -> 1/y0, y -> y0}, {x -> (A0 + y0)/(A0 y0), y -> 0}}*) If we analyze the linearization at the first non-trivial equilibrium \left(\displaystyle\frac{...

1

All the objects involved should be defined before used like theta = {ctheta[1], ctheta[2], ctheta[3]}; omega = {comega[1], comega[2], comega[3]}; Omega = {cOmega[1], cOmega[2], cOmega[3]}; I2 = Table[i2[ctheta[1], ctheta[2], ctheta[3]][i, j], {i, 1, 3}, {j, 1, 3}]; Ip = Table[ip[ctheta[1], ctheta[2], ctheta[3]][i, j], {i, 1, 3}, {j, 1, 3}]; L = 1/2 theta....

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