A Two Step, Fourth Order, Nearly-Linear Method with Energy Preserving Properties

We introduce a family of fourth order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. Each method in the family may be viewed as a correction of a linear two-step method, where the correction term is O(h^5) (h is the stepsize of integration). The key tools the new methods are based upon are the line integral associated with a conservative vector field (such as the one defined by a Hamiltonian dynamical system) and its discretization obtained by the aid of a quadrature formula. Energy conservation is equivalent to the requirement that the quadrature is exact, which turns out to be always the case in the event that the Hamiltonian function is a polynomial and the degree of precision of the quadrature formula is high enough. The non-polynomial case is also discussed and a number of test problems are finally presented in order to compare the behavior of the new methods to the theoretical results.Comment: 14 pages, 4 figures, 2 table

Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

A homotopy perturbation transformation method (HPTM) which is based on homotopy perturbation method and Laplace transform is first applied to solve the approximate solution of the fractional nonlinear equations. The nonlinear terms can be easily handled by the use of He's polynomials. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm

METODE DIFERENSIAL NEWTON-COTES TERTUTUP UNTUK INTEGRASI LONG-TIME

Persamaan diferensial dengan integrasi long-time merupakan persamaan yang memerlukan periode waktu integrasi yang panjang untuk mendapatkan hasil solusinya. Hal ini menyebabkan akan sulit ditemukan penyelesaiannya apabila menggunakan cara analitik. Untuk mengatasi masalah tersebut, maka digunakan suatu metode numerik dengan sifat simplektik yang merupakan kharakteristik dari suatu sistem Hamiltonian. Metode diferensial Newton-Cotes tertutup akan dibuk- tikan bersifat simplektik dengan cara mengkonversikan metode tersebut kedalam matriks yang memiliki struktur simplektik sehingga dapat dinyatakan sebagai in- tegrator simplektik. Setelah dibuktikan bersifat simplektik, metode diferensial Newton-Cotes tertutup akan diterapkan dalam penyelesaian contoh soal dengan integrasi long-time. *********** Differential equations with long-time integration is an equation that re- quires a long integration time period to get the solution. It causes would be hard to find the solution in an analytical way. To overcome these problems, we used a nu- merical method that is symplectic which is characteristic of a Hamiltonian system. Closed Newton-Cotes differential methods will be symplectic evidenced by convert- ing the method into matrix which has symplectic structure so it can be expressed as symplectic integrators. After proven to be symplectic, Closed Newton-Cotes differential methods will be applied to solving the example problems with long-time integration

Geometric Pseudospectral Method on SE(3) for Rigid-Body Dynamics with Application to Aircraft

General pseudospectral method is extended to the special Euclidean group SE(3) by virtue of equivariant map for rigid-body dynamics of the aircraft. On SE(3), a complete left invariant rigid-body dynamics model of the aircraft in body-fixed frame is established, including configuration model and velocity model. For the left invariance of the configuration model, equivalent Lie algebra equation corresponding to the configuration equation is derived based on the left-trivialized tangent of local coordinate map, and the top eight orders truncated Magnus series expansion with its coefficients of the solution of the equivalent Lie algebra equation are given. A numerical method called geometric pseudospectral method is developed, which, respectively, computes configurations and velocities at the collocation points and the endpoint based on two different collocation strategies. Through numerical tests on a free-floating rigid-body dynamics compared with several same order classical methods in Euclidean space and Lie group, it is found that the proposed method has higher accuracy, satisfying computational efficiency, stable Lie group structural conservativeness. Finally, how to apply the previous discretization scheme to rigid-body dynamics simulation and control of the aircraft is illustrated

Numerical treatment of special second order ordinary differential equations: general and exponentially fitted methods

2010 - 2011The aim of this research is the construction and the analysis of new families of numerical methods for the integration of special second order Ordinary Differential Equations (ODEs). The modeling of continuous time dynamical systems using second order ODEs is widely used in many elds of applications, as celestial mechanics, seismology, molecular dynamics, or in the semidiscretisation of partial differential equations (which leads to high dimensional systems and stiffness). Although the numerical treatment of this problem has been widely discussed in the literature, the interest in this area is still vivid, because such equations generally exhibit typical problems (e.g. stiffness, metastability, periodicity, high oscillations), which must efficiently be overcome by using suitable numerical integrators. The purpose of this research is twofold: on the one hand to construct a general family of numerical methods for special second order ODEs of the type y00 = f(y(t)), in order to provide an unifying approach for the analysis of the properties of consistency, zero-stability and convergence; on the other hand to derive special purpose methods, that follow the oscillatory or periodic behaviour of the solution of the problem...[edited by author]X n. s