Noniterative Localized and Space-Time Localized RBF Meshless Method to Solve the Ill-Posed and Inverse Problem

Mohammed Hamaidi,Fatima Ghafrani,Ahmed Naji,Mostafa Jourhmane

doi:10.1155/2020/5046286

Abstract

In many references, both the ill-posed and inverse boundary value problems are solved iteratively. The iterative procedures are based on firstly converting the problem into a well-posed one by assuming the missing boundary values. Then, the problem is solved by using either a developed numerical algorithm or a conventional optimization scheme. The convergence of the technique is achieved when the approximated solution is well compared to the unused data. In the present paper, we present a different way to solve an ill-posed problem by applying the localized and space-time localized radial basis function collocation method depending on the problem considered and avoiding the iterative procedure. We demonstrate that the solution of certain ill-posed and inverse problems can be accomplished without iterations. Three different problems have been investigated: problems with missing boundary condition and internal data, problems with overspecified boundary condition, and backward heat conduction problem (BHCP). It has been demonstrated that the presented method is efficient and accurate and overcomes the stability analysis that is required in iterative techniques.

Highlights

In contrast to the stationary and nonstationary direct boundary value problems, ill-posed problems are characterized by unknown boundary conditions on a part of the boundary
We should mention that the first problem is solved using the localized radial basis functions (RBFs) method and the second one by the space-time localized RBF method
The problem shows some difficulties to be solved using some comment techniques. We show that this problem will be overtaken by using the space-time localized RBF method to solve the problem in the domain 1⁄20, 1 × 1⁄20, tf

Summary

Introduction

In contrast to the stationary and nonstationary direct boundary value problems, ill-posed problems are characterized by unknown boundary conditions on a part of the boundary. An example is the problem of determining the temperature and the heat flux on the whole boundary or on its part, where the temperature and the heat flux are prescribed in selected points located inside the domain of the considered problem Another statement of ill-posed problems is the one referred to as the final boundary value problem or backward heat conduction problem (BHCP). For BHCP, we can mention the meshless method developed by Li et al [11] based on the RBF method for the nonhomogeneous backward heat conduction problem. For the second case, we solve a nonstationary backward heat conduction problem (BHCP) characterized by the final condition using the space-time localized RBF collocation method. Note that the two-dimensional nonstationary problems can be solved using the same approach

Mathematical Formulation of the Problems

Numerical Results and Discussions

Conclusion