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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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[1] Foundations of Potential Theory. Dover: New York, 1953. · Zbl 0053.07301
[2] . Methods of Mathematical Physics, vol. II, Partial Differential Equations. Wiley: New York, 1989.
[3] , . Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society: Providence, RI, 1986.
[4] Le probléme de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann: Paris, 1932.
[5] Martin, International Journal for Numerical Methods in Engineering 60 pp 1933– (2004)
[6] Reinhardt, SIAM Journal on Numerical Analysis 36 pp 890– (1999)
[7] Wei, SIAM Journal on Control and Optimization 42 pp 381– (2003)
[8] Cheng, Numerical Methods in Partial Differential Equations 19 pp 571– (2003)
[9] Kansa, Computers and Mathematics with Applications 19 pp 127– (1990) · Zbl 0692.76003
[10] Trefftz method. In Topics in Boundary Element Research, 1, Basic Principles and Applications, (ed.). Springer: Berlin, 1984; 225-253.
[11] Fairweather, Advances in Computational Mathematics 9 pp 69– (1998)
[12] Hon, Engineering Analysis with Boundary Elements 24 pp 599– (2000)
[13] Li, Communications in Numerical Methods in Engineering 20 pp 51– (2004)
[14] 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.
[15] Lesnic, Trends in Heat, Mass and Momentum Transfer 4 pp 37– (1998)
[16] Kozlov, Computational Mathematics and Mathematical Physics 31 pp 45– (1991)
[17] Franke, Mathematics of Computation 38 pp 181– (1982)
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