A simple method for solving Prandtl's integro-differential equation is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. Compared with known investigations, its advantages are that the representation of the exact solution is obtained in a reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. The final numerical experiments illustrate the method is efficient. Manuscript profile

In this work, an iterative method under the umbrella of inverse-free methods which do not rely on the calculation of the inverse matrix per loop is proposed for finding the maximal solution of a well-known nonlinear matrix equation (NME) in the form of Hermitian positive definite (HPD) matrices. The computational of the minimal solution is discussed as well. The iterative scheme is constructed based on methods for finding generalized matrix inverse. We illustrate some estimations for obtaining the solution, and its convergence. To ensure its validity and usefulness, some experiments are run which reveal the superiority of the proposed method. Manuscript profile