J.I. Ramos

On the variational iteration method and other iterative techniques for nonlinear differential equations

Abstract A variety of iterative methods for the solution of initial- and/or boundary-value problems in ordinary and partial differential equations is presented. These iterative procedures provide the solution or an approximation to it as a sequence of iterates. For initial-value problems, it is shown that these iterative procedures can be written in either an integral or differential form. The integral form is governed by a Volterra integral equation, whereas the differential one can be obtained from the Volterra representation by simply differentiation. It is also shown that integration by parts, variation of parameters, adjoint operators, Green’s functions and the method of weighted residuals provide the same Volterra integral equation and that this equation, in turn, can be written as that of the variational iteration method. It is, therefore, shown that the variational iteration method is nothing else by the Picard–Lindelof theory for initial-value problems in ordinary differential equations and Banach’s fixed-point theory for initial-value problems in partial differential equations, and the convergence of these iterative procedures is ensured provided that the resulting mapping is Lipschitz continuous and contractive. It is also shown that some of the iterative methods for initial-value problems presented here are special cases of the Bellman–Kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives, but such a condition is not required by the techniques presented in this paper. For boundary-value problems, it is shown that one may use the iterative procedures developed for initial-value problems but the resulting iterates may not satisfy the boundary conditions, and two new iterative methods governed by Fredholm integral equations are proposed. It is shown that the resulting iterates satisfy the boundary conditions if the first one does so. The iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of (differential) two- and three-level iterative techniques, harmonic balance procedures and standard and modified Linstedt–Poincare techniques. The method is also applied to two boundary-value problems.

Linearization techniques for singular initial-value problems of ordinary differential equations

Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions and globally smooth solutions. The accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non-homogeneous, linear and nonlinear Lane-Emden equations. It is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution. In fact, it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods. It is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size.

Linearization methods in classical and quantum mechanics

The applicability and accuracy of linearization methods for initial-value problems in ordinary differential equations are verified on examples that include the nonlinear Duffing equation, the Lane–Emden equation, and scattering length calculations. Linearization methods provide piecewise linear ordinary differential equations which can be easily integrated, and provide accurate answers even for hypersingular potentials, for which perturbation methods diverge. It is shown that the accuracy of linearization methods can be substantially improved by employing variable steps which adjust themselves to the solution.  2003 Elsevier Science B.V. All rights reserved.

Liquid curtains. I: Fluid mechanics

Abstract A mathematical model of liquid curtains which accounts for gravity, surface tension, pressure differences and nozzle exit geometry is presented. Analytical solutions are obtained in the absence of friction and compared with the results of numerical calculations; differences of at most three percent in the convergence length, are obtained between the numerical and analytical results even for nozzle exit angles of thirty degrees. It is shown that the convergence length is a monotonically increasing function of the Froude number, initial thickness to initial radius ratio, pressure difference and nozzle exit angle. The convergence length increases as the Weber number is increased. It is also shown that very small pressure differences between the gas enclosed by and the gas surrounding the liquid curtain are required to dramatically increase the convergence length. Pressure differences higher than a critical value which is related to the Froude and Weber numbers, are shown to result in flow divergence.

Magnetohydrodynamic channel flow study

A finite‐difference study of a steady, incompressible, viscous, magnetohydrodynamic (MHD) channel flow which has direct application to dc electromagnetic pumps is presented. The study involves the numerical solution of the coupled Navier–Stokes and Maxwell equations at low magnetic Reynolds numbers. It is shown that the axial velocity profiles have a characteristic M shape as the fluid approaches and passes the electrode. The electric potential varies almost linearly from the channel centerline to the channel wall. The current shows a steep gradient near the electrodes. Comparison between the finite‐difference solution and a quasi‐one‐dimensional approach are presented. The two‐dimensional numerical calculations predict a larger pressure rise, a smaller net current, and a smaller pump efficiency than the quasi‐one‐dimensional model.

Swirling flow in a research combustor

Measurements of turbulent, confined, swirling flows have been obtained by means of a two-color laser Doppler velocimeter in a research combustor and compared with other experimental data and numerical results obtained by means of two two-equation models of turbulence. The combustor consists of two confined, con- centric, swirling jets whose mass flow rates and swirl numbers can be controlled independently, and which can be used to study cold flow, premixed and non-premixed reactive flows, and two-phase flows. Results are reported for cold flow conditions under co- and counterswirl. It is shown that under both conditions a closed recirculation zone is created at the combustor centerline. This zone is characterized by the presence of a one-celled toroidal vortex, low tangential velocities, high turbulent intensities, and large dissipation rates of turbulence kinetic energy. The experimental data agree satisfactorily with the numerical results, but do not agree with other ex- perimental data under coswirl flow conditions. The reasons for the discrepancies are discussed. N important application of swirling flows lies in its use to provide flame stabilization and improved mixing in many combustion chambers.1 In order to achieve enhanced flame stabilization and better control of the mixing process, multiple coaxial swirling streams can be introduced in swirl combustors. Yetter and Gouldin2 carried out a series of experiments on a model combustor which uses two coaxial swirling jets. They found that the fluid mechanical aspects play an important role in determining the operation of the combustor; combustion under coswirl conditions (jets rotating in the same direction) was found to be significantly different from combustion under counterswirl conditions. Vu and Gouldin3 performed experiments in a model combustor composed of two confined coaxial swirling jets under nonreacting conditions. They found that a recirculation zone occurs only with counterswirl near the exit of the inner jet. The recirculation zone was in the form of a one-celled toroidal vortex having very low swirl velocities. Habib and Whitelaw4 performed similar experiments in confined coaxial jets although under weak swirl conditions, i.e., no recir- culation zone at the combustor centerline appeared in their experimental work. Gouldin et al. 5 performed experiments similar to those reported here and in Ref. 3 by means of a laser Doppler velocimeter (LDV) and found that a recir- culation zone is created at the combustor centerline only under counterswirl flow conditions. The results of Gouldin et al.5 and Vu and Gouldin3 indicate that, in the geometrical arrangement employed by these investigators, coswirl does not result in a recirculation zone under incompressible flow conditions. The results presented in this paper show that a recirculation does exist under both co- and counterswirl flow conditions. Experimental work on turbulent, confined, swirling flows has also been performed by Rhode,6 Rhode et al.,7 Gupta et al.,8 and Yoon and Lilley.9 Rhode6 and Rhode et al.7 studied swirling flows in a combustor provided with a sudden ex- pansion by means of a visualization technique in which neutrally buoyant helium filled soap bubbles were employed. The results of the visualization technique were compared with a two-equation model of turbulence and showed that the model predicts the gross features of the flowfield. Gupta et

On Linstedt–Poincaré techniques for the quintic Duffing equation

Abstract An artificial parameter method for the quintic Duffing equation is presented. The method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation, the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter, and results in linear ordinary differential equations at each order in the parameter. By imposing the nonsecularity condition at each order in the expansion, the method provides different approximations to both the solution and the frequency of oscillation. The method does not require that a small parameter be present in the governing equation, and its results are compared with those of Linstedt–Poincare, modified Linstedt–Poincare, harmonic balance and Galerkin techniques. It is shown that these four techniques predict the same first-order approximation to the frequency of oscillation as the artificial parameter method presented in this paper, and the latter introduces higher order corrections at second order, whereas similar corrections are introduced by the Linstedt–Poincare and modified Linstedt–Poincare methods at orders equal to and higher than three.

Linearization methods for reaction-diffusion equations: Multidimensional problems

Four types of linearization methods for the numerical solution of multidimensional reaction-diffusion equations are presented. The first two types are based on the discretization of the time variable, time linearization and approximate factorization. The first type also discretizes the spatial coordinates, results in block-tridiagonal matrices, and provides discrete solutions in space and time. The second type employs space linearization, yields linear, ordinary differential equations in space, and produces either piecewise continuous or piecewise differentiable solutions. The third type is based on the discretization of the spatial coordinates and time linearization, and yields continuous solutions in time. The fourth type uses time and space linearization, and results in a multidimensional, linear, elliptic equation whose solution by means of separation of variables provides continuous approximations in space and discrete in time. The fourth type also yields nine-point finite difference expressions compared with the five-point ones of the first three types.

Piecewise-linearized methods for initial-value problems

Piecewise-linearized methods for the solution of initial-value problems in ordinary differential equations are developed by approximating the right-hand-sides of the equations by means of a Taylor polynomial of degree one. The resulting approximation can be integrated analytically to obtain the solution in each interval and yields the exact solution for linear problems. Three adaptive methods based on the norm of the Jacobian matrix, maintaining constant the value of the approximation errors incurred by the linearization of the right-hand sides of the ordinary differential equations, and Richardsons extrapolation are developed. Numerical experiments with some nonstiff, first- and second-order, ordinary differential equations, indicate that the accuracy of piecewise-linearized methods is, in general, superior to those of the explicit, modified, second-order accurate Euler method and the implicit trapezoidal rule, but lower than that of the explicit, fourth-order accurate Runge-Kutta technique. It is also shown that piecewise-linearized methods do not exhibit computational (i.e., spurious) modes for the relaxation oscillations of the van der Pol oscillator, and, for those systems of equations which satisfy certain conservation principles, conserve more accurately the invariants than the trapezoidal rule. An error bound for piecewise-linearized methods is provided for ordinary differential equations whose right-hand-sides satisfy certain Lipschitz conditions.

Explicit finite difference methods for the EW and RLW equations

An extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal-width (EW) and regularized long-wave (RLW) equations is reported. Such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing. Two of the methods presented here make use of three-point, fourth-order accurate, finite difference formulae for the first- and second-order spatial derivatives. Two methods are based on the analytical solution of second-order ordinary differential equations which have locally exponential solutions, and the fourth technique is a standard finite difference scheme. A linear stability analysis of the four methods is presented. It is shown that, for the EW and RLW equations, a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations. The latter are reported to be more accurate than exponential techniques that employ second-order accurate approximations, and, these, in turn, are more accurate than the standard explicit method.