In this paper we study the existence of unique positive solutions for the following coupled system:where the integer number n > 3 and 1and k 1 , k 2 : R + → R + are continuous functions, D α 0 + and D β 0 + stand for the Riemann-Liouville derivatives. An illustrative example is given to show the effectiveness of theoretical results.

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples.

In this paper, an extended version of the method of minimizing an energy gap functional for determining the optimal source points in the method of fundamental solutions (MFS) is applied to the 3D Laplace operator subject to the Dirichlet and Neumann boundary conditions. As we know, the MFS is a more popular meshless method for solving boundary or initial-boundary value problems due to its simplicity and high accuracy. However, the accuracy of the MFS depends strongly on the distribution of the source points. Finally, some of the numerical experiments are carried out to express the simplicity and effectiveness of the presented method.

In this article, we deal with an inverse problem concerning the two-dimensional Laplace equation with local boundary conditions on a bounded region. In this problem, the goal is to reduce it into a system of Fredholm integral equations of the second kind involving kernels with weakly/or no singularities (Fredholm property) by considering some additional conditions on the parameters of the problem. Finally, the method is carried out on an example, to show the simplicity and efficiency of the method.

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