No 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.

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.