Second-order Initial Value Problems Research Articles

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.

Symbolic derivation of order conditions for hybrid Numerov-type methods solving [formula omitted

Numerov-type ODE solvers are widely used for the numerical treatment of second-order initial value problems. In this work we present a powerful and efficient symbolic code in M ATHEMATICA for the derivation of their order conditions and principal truncation error terms. The relative tree theory for such order conditions is presented along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the symbolic code.

Positive Solutions for Singular Initial Value Problems with Sign Changing Nonlinearities Depending on y′

Using the theory of fixed point index, this paper presents the existence of positive solutions for the singular second-order initial value problems, where f(t,y,y′) may be singular at y=0 and y′=0, and f may change sign.

Embedded diagonally implicit Runge–Kutta–Nystrom 4(3) pair for solving special second-order IVPs

In this paper, third-order 3-stage diagonally implicit Runge–Kutta–Nystrom method embedded in fourth-order 4-stage for solving special second-order initial value problems is constructed. The method has the property of minimized local truncation error as well as the last row of the coefficient matrix is equal to the vector output. The stability of the method is investigated and a standard set of test problems are tested upon and comparisons on the numerical results are made when the same set of test problems are reduced to first-order system and solved using existing Runge–Kutta method. The results clearly shown the advantage and the efficiency of the new method.

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.

A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions

A new kind of trigonometrically fitted explicit Numerov-type method for the numerical integration of second-order initial value problems (IVPs) with oscillating or periodic solutions is presented in this paper. This new method is based on the original fifth-order method which is dispersion of order eight and dissipation of order five proposed in [J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006) 41–57]. Using trigonometrically fitting, we derive a more efficient method with higher accuracy for the numerical integration of second-order IVPs with oscillating solutions, and we analyze the stability, phase-lag(dispersion) and dissipation by the theory considered by Coleman and Ixaru (see [J.P. Coleman, L.Gr. Ixaru, P-stability and exponential-fitting methods for y″=f(x,y), IMA J. Numer. Anal. 16 (1996) 179–199]). Numerical experiments are carried out to show the efficiency of our new method in comparison with the methods proposed in [J.M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math. 187 (2006) 41–57] and the exponentially fitted fourth order RKN method derived in [J.M. Franco, Exponentially fitted explicit Runge–Kutta–Nyström methods, J. Comput. Appl. Math. 167 (2004) 1–19].

Structural dynamic responses analysis applying differential quadrature method

Unconditionally stable higher-order accurate time step integration algorithms based on the differential quadrature method (DQM) for second-order initial value problems were applied and the quadrature rules of DQM, computing of the weighting coefficients and choices of sampling grid points were discussed. Some numerical examples dealing with the heat transfer problem, the second-order differential equation of imposed vibration of linear single-degree-of-freedom systems and double-degree-of-freedom systems, the nonlinear move differential equation and a beam forced by a changing load were computed, respectively. The results indicated that the algorithm can produce highly accurate solutions with minimal time consumption, and that the system total energy can remain conservative in the numerical computation.

On the convergence of He's variational iteration method

In this work we will consider He's variational iteration method for solving second-order initial value problems. We will discuss the use of this approach for solving several important partial differential equations. This method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. This procedure is a powerful tool for solving the large amount of problems. Using the variational iteration method, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a sequence of functions which converges to the exact solution of the problem. Our emphasis will be on the convergence of the variational iteration method. In the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models.

Existence of positive solution to a singular second-order initial value problem with impulses

This paper deals with the existence of positive solutions to singular initial value problem with impulse effects. The right-hand side of the differential equation can be singular in its dependent variable.

The solution of the two-dimensional sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.

Special multistep methods based on numerical differentiation for solving the initial value problem

In this paper, methods based on numerical differentiation and their regions of absolute stability of some classes of special multistep methods are derived. Some of the methods given in Henrici [P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, 1962] and some methods derived in this paper are applied to solve a second-order initial value problem hence establishing the superiority of the methods derived in this paper.

Variable stepsize störmer-cowell methods

Störmer and Cowell methods are multistep codes for the numerical solution of second-order initial value problems where the first derivative does not appear explicitly. In this paper, we develop a procedure to obtain k-step Störmer and Cowell methods in their variable step size version, avoiding computation of the coefficients by recurrences or integrals. We offer a strategy for conveniently selecting the stepsize. Considering a pair of explicit and implicit formulae, these may be implemented in a predictor-corrector mode.

P-stable symmetric super-implicitmethods for periodic initial value problems

The idea of super-implicit methods (requiring not just past and present but also future values) was suggested by Fukushima recently. Here, we construct P-stable super-implicit methods for the solution of second-order initial value problems. The benefit of such methods is realized when using vector or parallel computers.

A HIGH-ACCURATE AND EFFICIENT OBRECHKOFF FIVE-STEP METHOD FOR UNDAMPED DUFFING'S EQUATION

In this paper, we present a five-step Obrechkoff method to improve the previous two-step one for a second-order initial-value problem with the oscillatory solution. We use a special structure to construct the iterative formula, in which the higher-even-order derivatives are placed at central four nodes, and show there existence of periodic solutions in it with a remarkably wide interval of periodicity, [Formula: see text]. By using a proper first-order derivative (FOD) formula to make this five-step method to have two advantages (a) a very high accuracy since the local truncation error (LTE) of both the main structure and the FOD formula are the same as O (h14); (b) a high efficiency because it avoids solving a polynomial equation with degree-nine by Picard iterative. By applying the new method to the well-known problem, the nonlinear Duffing's equation without damping, we can show that our numerical solution is four to five orders higher than the one by the previous Obrechkoff two-step method and it takes only 25% of CPU time required by the previous method to fulfil the same task. By using the new method, a better "exact" solution is found by fitting, whose error tolerance is below 5×10-15, than the one widely used in the lectures, whose error tolerance is below 10-11.

Variable stepsize implementation of multistep methods for [formula omitted

The Falkner method is a multistep scheme intended for the numerical solution of second-order initial value problems where the first derivative does appear explicitly. In this paper, we develop a procedure to obtain k-step Falkner methods (explicit and implicit) in their variable step-size versions, providing recurrence formulas to compute the coefficients efficiently. Considering a pair of explicit and implicit formulae, these may be implemented in predictor–corrector mode.

On the generation of P-stable exponentially fitted Runge–Kutta–Nyström methods by exponentially fitted Runge–Kutta methods

This paper provides an investigation of the stability properties of a family of exponentially fitted Runge–Kutta–Nyström (EFRKN) methods. P-stability is a very important property usually demanded for the numerical solution of stiff oscillatory second-order initial value problems. P-stable EFRKN methods with arbitrary high order are presented in this work. We have proved our results based on a symmetry argument.

Computational Quartic c3-spline and Quintic c2-spline algorithms for solving second-order initial value problems

In this paper we have written out two computational Quartic c 3-spline and Quintic c 2-spline algorithms and applied them for a few ordinary differential equations using Maple9. A comparison table of errors has been designed in each case to compare the implementation of the algorithms with each other. The algorithms are corresponding to articles. [J. Comput. Appl. Math. 75 (1996) 293 and 115 (2000) 495].

Dissipative trigonometrically-fitted methods for linear second-order IVPs with oscillating solution

A dissipative trigonometrically-fitted two-step explicit hybrid method is constructed in this paper. This method is based on a dissipative explicit two-step method developed recently by Tsitouras [1]. Numerical examples show that the procedure of trigonometrical fitting is an efficient way for one to produce numerical methods for the solution of second-order linear initial value problems (IVPs) with oscillating solutions.

New upper bounds for free linear and nonlinear vibration systems with applications of the differential calculus of norms

In this paper, a new concept is presented to derive upper bounds on the solution of initial value problems for free linear and nonlinear ordinary differential equations representing vibration problems. This concept is based on the idea to associate with the original problem of n degrees of freedom a scalar linear second-order initial value problem modelling an overdamped motion and to take its solution as an upper bound on the solution of the original problem. The best upper bound is computed by applying the differential calculus of norms developed by the author in earlier work.

Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations

A family of implicit methods based on intra-step Chebyshev interpolation has been developed to integrate oscillatory second-order initial value problems of the form y″(t)−2g y′(t)+(g 2+w 2)y(t)=f(t,y(t)) . The procedure integrates the homogeneous part exactly (in the absence of round-off errors). The Chebyshev approach uses stepsizes that are considerably larger than those typically used in Runge–Kutta or multistep methods. Computational overheads are comparable to those incurred by high-order conventional procedures. Chebyshev interpolation coupled with the exponential-fitted nature of the method substantially reduces local errors. Global error propagation rates are also reduced making these procedures good candidates to be used in long-term simulations of perturbed oscillatory systems with a dissipative term.