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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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[1] Hardy, R.L, Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 76, 1905-1915, (1971)
[2] Hardy, R.L, Theory and applications of the multiquadric-biharmonic method: 20 years of discovery, Computers math. applic., 19, 8/9, 163-208, (1990) · Zbl 0692.65003
[3] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite functions, Approx. theory applic., 4, 77-89, (1988) · Zbl 0703.41008
[4] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite functions, II, Math. comput., 54, 211-230, (1990) · Zbl 0859.41004
[5] Buhmann, M.D; Micchelli, C.A, Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Const. approx., 6, 12, 21-34, (1990) · Zbl 0682.41007
[6] Buhmann, M.D; Micchelli, C.A, Multiquadric interpolation improved, Computers math. applic., 24, 12, 21-26, (1992)
[7] Chui, C.K; Stoeckler, J; Ward, J.D, Analytic wavelets generated by radial functions, Adv. comput. math., 5, 95-123, (1996) · Zbl 0855.65145
[8] Micchelli, C.A, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Const. approx., 2, 11-22, (1986) · Zbl 0625.41005
[9] Kansa, E.J, Multiquadrics: A scattered data approximation scheme with applications to computational fluid dynamics: II. parabolic, hyperbolic, and elliptic partial differential equations, Computers math. applic., 19, 8/9, 146-161, (1990) · Zbl 0850.76048
[10] Makroglou, A, Radial basis functions in the numerical solution of Fredholm integral and integro-differential equations, (), 478-484, New Brunswick, NJ
[11] Baxter, B.J.C, The asymptotic cardinal function of the multiquadric, φ(r) = (r2 + c2)\(12\) as c → ∞, Computers math. applic., 24, 12, 1-6, (1992) · Zbl 0764.41016
[12] Madych, W.R, Miscellaneous error bounds for multiquadric and related interpolators, Computers math. applic., 24, 12, 121-138, (1992) · Zbl 0766.41003
[13] Kansa, E.J; Hon, Y.C, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Computers math. applic., 39, 7/8, 123-137, (2000) · Zbl 0955.65086
[14] Wong, S.M; Hon, Y.C; Li, T.S; Chung, S.L; Kansa, E.J, Multi-zone decomposition of time-dependent problems using the multiquadric scheme, Computers math. applic., 37, 8, 23-43, (1999) · Zbl 0951.76066
[15] Hon, Y.C, An efficient numerical scheme for Burgers’ equation, Appl. math. comput., 95, 37-50, (1998) · Zbl 0943.65101
[16] Wendland, H, Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree, Adv. comput. math., 4, 389-396, (1995) · Zbl 0838.41014
[17] Galperin, E.A; Zheng, Q, Solution and control of PDE via global optimization methods, Computers math. applic., 25, 10/11, 103-111, (1993) · Zbl 0794.35009
[18] Galperin, E.A; Kansa, E.J, On the solution of infinitely ill-conditioned weakly singular problems, Mathl. comput. modelling, 31, 13, 53-63, (2000) · Zbl 0955.65097
[19] Galperin, E.A, The cubic algorithm for optimization and control, (1990), NP Research Publ Montreal · Zbl 0781.90080
[20] Galperin, E.A, The fast cubic algorithm, Computers math. applic., 25, 10/11, 147-160, (1993) · Zbl 0784.90073
[21] Galperin, E.A, The alpha algorithm and application of the cubic algorithm in case of unknown Lipschitz constant, Computers math. applic., 25, 10/11, 71-78, (1993) · Zbl 0803.90112
[22] Ferrari, A; Galperin, E.A, Numerical experiments with one-dimensional adaptive cubic algorithm, Computers math. applic., 25, 10/11, 47-56, (1993) · Zbl 0776.90072
[23] Belykh, V.N, Algorithms without saturation in the problem of numerical integration, Soviet math. dokl., 39, 95-98, (1989) · Zbl 0702.41044
[24] Belykh, V.N, On the problem of numerical solution of Dirichlet problem by a harmonic single layer potential algorithm (without saturation), Russian acad. sci. dokl. math., 47, 252-256, (1993) · Zbl 0814.65122
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