Kansa method for the solution of a parabolic equation with an unknown spacewise-dependent coefficient subject to an extra measurement

• Published in: Computer physics communications Vol. 184; no. 3; pp. 582 - 595
• Main Authors: Parand, K., Rad, J.A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2013
• Subjects: Cartesian nodes; Coefficients; Computer simulation; Convergence; Errors; Exact solutions; Heat source; Inverse problem; Mathematical analysis; Mathematical models; Meshless methods; One-dimensional parabolic equation; Radial basis functions; Random noise; Specifications; Stability analysis; Unknown spacewise-dependent coefficient; Heat source; Radial basis functions; One-dimensional parabolic equation; Meshless methods; Stability analysis; Cartesian nodes; Inverse problem; Unknown spacewise-dependent coefficient
• ISSN: 0010-4655; 1879-2944
• DOI: 10.1016/j.cpc.2012.10.012

Parabolic partial differential equations with an unknown spacewise-dependent coefficient serve as models in many branches of physics and engineering. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional parabolic equation is presented by the method proposed by Kansa. The present numerical procedure is based on the product model of the space–time radial basis function (RBF), which was introduced by Myers et al. Using this method, a rapid convergent solution is produced which tends to the exact solution of the problem. The convergence of this scheme is accelerated when we use the Cartesian nodes as center nodes. The accuracy of the method is tested in terms of Error and RMS errors. Also, the stability of the technique is investigated by perturbing the additional specification data by increasing the amounts of random noise. The numerical results obtained show that the proposed method produces a convergent and stable solution.