A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems

In this paper a dissipative exponentially-fitted method for the numerical integration of the Schrödinger equation and related problems is developed. The method is called dissipative since is a nonsymmetric multistep method. An application to the the resonance problem of the radial Schrödinger equation and to other well known related problems indicates that the new method is more efficient than the corresponding classical dissipative method and other well known methods. Based on the new method and the method of Raptis and Cash a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schrödinger equation indicates the power of the new approach.     

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

Optimality of an error estimate of a one-step numerical integration method for certain initial value problems

In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.     

A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems

We examine a nonlinear explicit two-step method of fourth algebraic order and infinite phase-lag order for solving one-dimensional second-order linear periodic initial value problems (IVPs) of ordinary differential equations. Applying special vector arithmetic with respect to an analytic function, the method can be extended to be vector-applicable for multidimensional problems. Numerical results to illustrate the efficiency of the method are presented and sensitivity analysis indicates the validity of the method in the frequency regime.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

The numerical method of successive interpolations for two-point boundary value problems with deviating argument

A new numerical method for two-point boundary value problems associated to differential equations with deviating argument is obtained. The method uses the fixed point technique, the trapezoidal quadrature rule, and the cubic spline interpolation procedure. The convergence of the method is proved without smoothness conditions, the kernel function being Lipschitzian in each argument. The interpolation procedure is used only on the points where the argument is modified. A practical stopping criterion of the algorithm is obtained and the accuracy of the method is illustrated on some numerical examples of the pantograph type.     

On the use of spline functions of even degree for the numerical solution of the delay differential equations

One considers and investigates the notion of natural spline functions of even degree, satisfying given derivative-interpolating conditions on simple knots. By using such spline functions we shall develop some theory and algorithms for the numerical solution of a class of delay differential equations. It will be shown that such kind of spline functions are very suitable for the numerical treatment of delay differential equations with initial conditions.     

An efficient method for solving nonlinear singular initial value problems

In [M.M. Hosseini, Modified Adomain decomposition method for specific second order ordinary differential equations, Appl. Math. Comput. 186 (2007), pp. 117–123] an efficient modification of Adomian decomposition method has been proposed for solving some cases of ordinary differential equations. In this paper, this method is generalized to more cases. The proposed method can be applied to linear, nonlinear, singular and nonsingular problems. Here, it is focused on nonlinear singular initial value problems of ordinary differential equations. The scheme is tested for some examples and the obtained results demonstrate reliability and efficiency of the proposed method.     

A numerical method for solving singularly perturbed systems of nonlinear two-point boundary-value problems

In this paper, a numerical method is proposed to solve singularly perturbed systems of nonlinear two-point boundary-value problems. First, Newton's iteration is used to linearize such problems, reducing these to a sequence of linear singularly perturbed two-point boundary-value problems. Then,a difference scheme is applied to solve the linear systems. The difference scheme is accurate up to O(h ²). Test examples are included to demonstrate the efficiency of the method.     

A method for the numerical solution of some elliptic boundary value problems for a strip

A boundary integral equation for the numerical solution of a class of elliptic boundary value problems for a strip is derived. The equation should be particularly useful for the solution of an important class of problems governed by Laplace's equation and also for the solution of relevant problems in anisotropic thermostatics and elastostatics     

Sixth order non-dissipative C 1 spline collocation method for oscillatory ordinary initial value problems

In this paper we construct a global method, based on quintic C ¹-spline, for the integration of first order ordinary initial value problems (IVPs) including stiff equations and those possessing oscillatory solutions as well. The method will be shown to be of order six and in particular is A-stable. Attention is also paid for the phase error (or dispersion) and it is proved that the method is dispersive and has dispersion order six with small phase-lag (compared with the extant methods having the same order (cf. [7])). Moreover, the method may be regarded as a continuous extension of the closed four-panel Newton–Cotes formula (NC4) (typically it is a continuous extension of an implicit Runge–Kutta method). In additiona priori error estimates, in the uniform norm, together with illustrative test examples will also be presented.     

A novel numerical method on the resolution of the time-invariant systems response

An algorithm for evaluation of time response to a time-invariant system based on Haar wavelets is proposed. The system is assumed to be governed by a high order linear differential equation with constant coefficients. The main objective of this study is to convert a differential equation into an algebraic-form equation and provide an elegant approach for computer programming. By the method, the computation complexity can be greatly reduced or much simplified in calculate the time response of a time-invariant system. The accurate and fastness of the method have been demonstrated by the examples of integration of the stiff systems     

Direct solutions of nth-order IVPs by homotopy-perturbation method

In this paper, a class of linear and nonlinear nth-order initial value problems (IVPs) is considered. The solutions of these IVPs are obtained by the homotopy-perturbation method (HPM). The HPM can be considered as one of the new methods belonging to the general classification of perturbation methods. Generally, the HPM deals with exact solvers for linear differential equations and approximative solvers for nonlinear equations. Several test cases are chosen to demonstrate the efficiency of HPM.     

A new method for computing a solution of the Cauchy problem with polynomial data for the system of crystal optics

A new analytical method for solving an initial value problem (IVP) for the system of crystal optics with polynomial data and a polynomial inhomogeneous term is suggested. The found solution of the IVP is a polynomial. Theoretical and computational analysis of polynomial solutions and their comparison with non-polynomial solutions corresponding to smooth data are given. The applicability of polynomial solutions to physical processes is discussed. An implementation of this method has been made by symbolic computations in Maple 10.     

A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation

Based on the idea of the previous Obrechkoff's two-step method, a new kind of four-step numerical method with free parameters is developed for the second order initial-value problems with oscillation solutions. By using high-order derivatives and apropos first-order derivative formula, the new method has greatly improved the accuracy of the numerical solution. Although this is a multistep method, it still has a remarkably wide interval of periodicity, . The numerical test to the well known problem, the nonlinear undamped Duffing's equation forced by a harmonic function, shows that the new method gives the solution with four to five orders higher than those by the previous Obrechkoff's two-step method. The ultimate accuracy of the new method can reach about 5×¹⁰⁻¹³, which is much better than the one the previous method could. Furthermore, the new method shows the great superiority in efficiency due to a reasonable arrangement of the structure. To finish the same computational task, the new method can take only about 20% CPU time consumed by the previous method. By using the new method, one can find a better ‘exact’ solution to this problem, reducing the error tolerance of the one widely used method (¹⁰⁻¹¹), to below 10⁻¹⁴.