A novel meshless method for fully nonlinear advection-diffusion-reaction problems to model transfer in anisotropic media

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.