A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Spectral Methods for Numerical Relativity. The Initial Data Problem

Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an elliptic partial differential equation) for a black hole spacetime with angular momentum and for a black hole spacetime superposed with gravitational radiation. In PSC, an approximate solution, generally expressed as a sum over a set of orthogonal basis functions (e.g., Chebyshev polynomials), is substituted into the exact system of equations and the residual minimized. For systems with analytic solutions the approximate solutions converge upon the exact solution exponentially as the number of basis functions is increased. Consequently, PSC has a high computational efficiency: for solutions of even modest accuracy we find that PSC is substantially more efficient, as measured by either execution time or memory required, than finite differencing; furthermore, these savings increase rapidly with increasing accuracy. The solution provided by PSC is an analytic function given everywhere; consequently, no interpolation operators need to be defined to determine the function values at intermediate points and no special arrangements need to be made to evaluate the solution or its derivatives on the boundaries. Since the practice of numerical relativity by finite differencing has been, and continues to be, hampered by both high computational resource demands and the difficulty of formulating acceptable finite difference alternatives to the analytic boundary conditions, PSC should be further pursued as an alternative way of formulating the computational problem of finding numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR

Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email: [email protected] or [email protected]

Inverse heat conduction problems by using particular solutions

Based on the method of fundamental solutions, we develop in this paper a new computational method to solve two-dimensional transient heat conduction inverse problems. The main idea is to use particular solutions as radial basis functions (PSRBF) for approximation of the solutions to the inverse heat conduction problems. The heat conduction equations are first analyzed in the Laplace transformed domain and the Durbin inversion method is then used to determine the solutions in the time domain. Least-square and singular value decomposition (SVD) techniques are adopted to solve the ill-conditioned linear system of algebraic equations obtained from the proposed PSRBF method. To demonstrate the effectiveness and simplicity of this approach, several numerical examples are given with satisfactory accuracy and stability.Peer reviewe