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Direct solution of ill-posed boundary value problems by radial basis function collocation method. (English) Zbl 1108.65059
Summary: Numerical solution of ill-posed boundary value problems normally requires iterative procedures. In a typical solution, the ill-posed problem is first converted to a well-posed one by assuming the missing boundary values. The new problem is solved by a conventional numerical technique and the solution is checked against the unused data. The problem is solved iteratively using optimization schemes until convergence is achieved. The present paper offers a different procedure. Using the radial basis function collocation method, we demonstrate that the solution of certain ill-posed problems can be accomplished without iteration. This method not only is efficient and accurate, but also circumvents the stability problem that can exist in the iterative method.

##### MSC:
 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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##### References:
  Foundations of Potential Theory. Dover: New York, 1953. · Zbl 0053.07301  . Methods of Mathematical Physics, vol. II, Partial Differential Equations. Wiley: New York, 1989.  , . Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society: Providence, RI, 1986.  Le probléme de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann: Paris, 1932.  Martin, International Journal for Numerical Methods in Engineering 60 pp 1933– (2004)  Reinhardt, SIAM Journal on Numerical Analysis 36 pp 890– (1999)  Wei, SIAM Journal on Control and Optimization 42 pp 381– (2003)  Cheng, Numerical Methods in Partial Differential Equations 19 pp 571– (2003)  Kansa, Computers and Mathematics with Applications 19 pp 127– (1990) · Zbl 0692.76003  Trefftz method. In Topics in Boundary Element Research, 1, Basic Principles and Applications, (ed.). Springer: Berlin, 1984; 225-253.  Fairweather, Advances in Computational Mathematics 9 pp 69– (1998)  Hon, Engineering Analysis with Boundary Elements 24 pp 599– (2000)  Li, Communications in Numerical Methods in Engineering 20 pp 51– (2004)  Boundary inverse problems in seepage and viscous fluid flows. In Computer Methods and Water Resources III, , , (eds). Computational Mechanics Publication: Southampton, Boston, 1995; 457-468.  Lesnic, Trends in Heat, Mass and Momentum Transfer 4 pp 37– (1998)  Kozlov, Computational Mathematics and Mathematical Physics 31 pp 45– (1991)  Franke, Mathematics of Computation 38 pp 181– (1982)
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