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Abstract This article presents the new version of the backward substitution method (BSM) for simulating transfer in anisotropic and inhomogeneous media governed by linear and fully nonlinear advection–diffusion-reaction equations (ADREs). The key idea of the method is to formulate a general analytical expression of the solution in the form of the series over a basis system which satisfies the boundary conditions with any choice of the free parameters. The radial basis functions (RBFs) of the different types are used to generate the basis system for expressing the solution. Then the expression is substituted into the ADRE under consideration and the free parameters are determined by the collocation inside the solution domain. As a result we separate the approximation of the boundary conditions and the approximation of the PDE inside the solution domain. This approach leads to an important improvement of the accuracy of the approximate solution and can be easily extended onto irregular domain problems. Furthermore, the proposed method is extended to general fully nonlinear ADREs in combination with the quasilinearization technique. Some numerical results and comparisons are provided to justify the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents a new meshless in space numerical method – the method of approximate fundamental solutions (MAFS) for solving problems with moving boundaries of the Stefan type. This technique first suggested for solving the Stefan problems in plane geometry is now extended onto the cases of inward and outward spherical problems. Computed results for the inward and outward spherical problems are numerically evaluated for comparison with the results obtained by other authors. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary and the results are in a good agreement with the previous studies. [Copyright &y& Elsevier]

Abstract: The paper presents a new meshless numerical technique for solving one-dimensional problems with moving boundaries including the Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) firstly suggested for solving elliptic problems and for heat equations in domains with fixed boundaries. The numerical examples are presented and the results are compared with analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for a region that initially has zero thickness. [Copyright &y& Elsevier]