## Application of global optimization and radial basis functions to numerical solutions of weakly singular Volterra integral equations.(English)Zbl 0999.65150

Summary: A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the $$C^\infty$$ multiquadric (MQ) radial basis function expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than $$5\cdot 10^{-7}$$.

### MSC:

 65R20 Numerical methods for integral equations 45E05 Integral equations with kernels of Cauchy type
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### References:

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