With the integral RBF method, the highest order derivatives of the unknown are approximated by a radial basis function network and then the derivative expressions are integrated to obtain the approximating expression for the unknown (original function). For example, a first derivative expression is integrated to yield an expression for the original function.

4. The use of the Integral RBFN least squares method to identify the yarn balloon shape

In this work the nonlinear ordinary equation (1) is rewritten as follows

where _ρ= {1+y'_x ²}^{1/ 2} and the second derivative is approximated and numerically integrated to obtain y' and y as follows:

Where and and t1 and t2 are the integral constant, (note: ρ is initially assumed to be a constant).

With a set of chosen collocation points x^p where p = 1 to n (n = m), the sum of the squares error associated with Eqs (1a) & (2a,b) is given by

where i denotes the i^th collocation point. By substituting equations (5abc) into equation (6) and applying the general linear least square principle, the resultant relationship between the m unknown weights w^j is defined. In order to solve this system of equations, the Picard type iterative method has been used to linearize this system. In this approach, at each iteration, the nonlinear term ρ of the equation (1a) is considered as a constant and determined from the result of the previous step (i.e. a²ρy of the step (k) is written by a²ρ^(k-1) y^k), and at the initial step (k =1), ρ is chosen as 1 (i.e.y'^{(k =0)} =0). The iteration procedure is completed when the convergence measure (CM) satisfies the following condition:

where tol is a tolerance and n is the number of collocation points. At this point the obtained set of weights w^j allows a final estimate of the function y.

5. Results and discussion

The identification of the balloon shape from the equations (1)-(2a,b) is very complex and has been carried out by many different researchers. Recently He, (2004) has applied the homotopy perturbation method to solve the problem and provides a point of reference for our analysis. In our method, the values used for the parameters in section 2 are as follows: h = 150mm; r = 27mm and a = 0.005mm-1. Using a range of coarse grids of collocation points (8, 10 and 12), the results obtained from the present method are in a very good agreement with the results obtained by He (2004). In fact, the CM was easily reached at tol = 10-5, the SSE difference between Hes results and the current results is less than 10-4 and the maximum difference between the two methods is less than 7%. More details of the method and further results (data tables and figures) will be presented at the conference because of the limitation of this extended abstract.

6. Conclusion

The Integral RBFN method was employed to identify the yarn balloon shape governed by a nonlinear differential equation. The results showed the method is reliable in determining the shape of the balloon using a coarse number of collocation points. This analysis has highlighted the prospect of the method for dealing with other complex problems in textile engineering.