27,250 research outputs found

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

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    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=0x = 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 Solution with a Subtraction Method to Improve Accuracy

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    This work addresses the solution to a Dirichlet boundary value problem for the Poisson equation in 1-D, d2u/dx2 = f using a numerical Fourier collocation approach. The order of accuracy of this approach can be increased by modifying f so the periodic extension of the right hand side is suffciently smooth. A proof for the order is given by Sköllermo. This work introduces a subtraction technique to modify the function\u27s right hand side to reduce the discontinuities or improve the smoothness of its periodic extension. This subtraction technique consists of cosine polynomials found by using boundary derivatives. We subtract cosine polynomials to match boundary values and derivatives of f. The derivatives need only be calculated numerically and approximately represent derivatives at the boundaries. Increasing the number of cosine polynomials in the subtraction technique increases the order of accuracy of the solution. The use of cosine polynomials matches well with the Fourier transform approach and is computationally efficient. Implementation of this technique results in a solution with variable accuracy depending on the number of collocation points and approximated boundary derivatives. Results show that the technique can be up to 14th order accurate

    Status of the differential transformation method

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    Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

    Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

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    In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O(p2d)\mathcal{O}(p^{2d}) storage and O(p3d)\mathcal{O}(p^{3d}) computational work, where pp is the degree of basis polynomials used, and dd is the spatial dimension. Our SVD-based tensor-product preconditioner requires O(pd+1)\mathcal{O}(p^{d+1}) storage, O(pd+1)\mathcal{O}(p^{d+1}) work in two spatial dimensions, and O(pd+2)\mathcal{O}(p^{d+2}) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in pp per degree of freedom in 2D, and reduce the computational complexity from O(p9)\mathcal{O}(p^9) to O(p5)\mathcal{O}(p^5) in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to p=15p=15. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees pp.Comment: 40 pages, 15 figure
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