General Lagrange Multiplier Research Articles

Application of variational iteration method to a general Riccati equation

In this paper, the variational iteration method (VIM) is applied to the solution of general Riccati differential equations. The equations under consideration includes one with variable coefficient and one in matrix form. In VIM, a correction functional is constructed by a general Lagrange multiplier which can be identified via a variational theory. The VIM yields an approximate solution in the form of a quickly convergent series. Comparisons with exact solution and the fourth-order RungeKutta method show that the VIM is a powerful method for the solution of nonlinear equations. The present paper may be a suitable and fruitful exercise for teaching and better understanding techniques in advanced undergraduate courses on classical mechanics.

Periodic solutions of Duffing equation with strong non-linearity

In this paper, the variational iteration procedure is described and used to give approximate periodic solutions for Duffing equation with strong non-linearity. A correction functional is constructed by a general Lagrange multiplier, which can be identified via the variational theory. The approximate period of motion obtained by using this method is compared with the exact period known in the literature.

Application of variational iteration method to the generalized Burgers–Huxley equation

In this paper, He’s variational iteration method (VIM) is applied to the generalized Burgers–Huxley equation. The VIM produces an approximate solution of the equation without any discretization. The VIM is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. Comparisons with the Adomian decomposition method (ADM) reveal that the VIM is very effective and convenient.

Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 3. Single-fluid-phase flow

This work is the third in a series of papers on the thermodynamically constrained averaging theory (TCAT) approach to modeling flow and transport phenomena in multiscale porous medium systems. Building upon the general TCAT framework and the mathematical foundation presented in previous works in this series, we demonstrate the TCAT approach for the case of single-fluid-phase flow. The formulated model is based upon conservation equations for mass, momentum, and energy and a general entropy inequality constraint, which is developed to guide model closure. A specific example of a closed model is derived under limiting assumptions using a linearization approach and these results are compared and contrasted with the traditional single-phase-flow model. Potential extensions to this work are discussed. Specific advancements in this work beyond previous averaging theory approaches to single-phase flow include use of macroscale thermodynamics that is averaged from the microscale, the use of derived equilibrium conditions to guide a flux–force pair approach to simplification, use of a general Lagrange multiplier approach to connect conservation equation constraints to the entropy inequality, and a focus on producing complete, closed models that are solvable.

Application of He's variational iteration method to nonlinear Jaulent–Miodek equations and comparing it with ADM

Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear Jaulent–Miodek, coupled KdV and coupled MKdV equations in this article. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with exact solutions.

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.

Variational iteration method—Some recent results and new interpretations

This paper is an elementary introduction to the concepts of variational iteration method. First, the main concepts in variational iteration method, such as general Lagrange multiplier, restricted variation, correction functional, are explained heuristically. Subsequently, the solution procedure is systematically addressed, in particular, for nonlinear oscillators. Particular attention is paid throughout the paper to give an intuitive grasp for the method. The main motivation is to put things together in a convenient form for later reference and systematic use.

Variational iteration method for solving generalized Burger–Fisher and Burger equations

Abstract We consider variational iteration method to investigate generalized Burger–Fisher and Burger equations. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. Comparison with Adomian decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method. Its remarkable accuracy is finally demonstrated in the study of some values of constants in generalized Burger–Fisher and Burger equations.

Application of He’s variational iteration method to Helmholtz equation

Abstract In this article, we implement a new analytical technique, He’s variational iteration method for solving the linear Helmholtz partial differential equation. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. The results compare well with those obtained by the Adomian’s decomposition method.

An iteration formulation for normalized diode characteristics

AbstractBy a general Lagrange multiplier, an iteration approach is proposed to solve the generalized normalized diode equation, by suitable choice of the initial trial‐function, one‐step iteration leads to a high accurate solution, which is valid for the whole solution domain. Copyright © 2004 John Wiley & Sons, Ltd.

Variational iteration method for autonomous ordinary differential systems

In this paper, a new iteration technique is proposed to solve autonomous ordinary differential systems. In this method, general Lagrange multipliers are introduced to construct correction functionals for the systems. The multipliers in the functionals can be identified by the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. Some examples are given. The results reveal that the method is very effective and convenient.

Variational iteration method – a kind of non-linear analytical technique: some examples

In this paper, a new kind of analytical technique for a non-linear problem called the variational iteration method is described and used to give approximate solutions for some well-known non-linear problems. In this method, the problems are initially approximated with possible unknowns. Then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Being different from the other non-linear analytical methods, such as perturbation methods, this method does not depend on small parameters, such that it can find wide application in non-linear problems without linearization or small perturbations. Comparison with Adomian’s decomposition method reveals that the approximate solutions obtained by the proposed method converge to its exact solution faster than those of Adomian’s method.

A new approach to nonlinear partial differential equations

In this paper, a novel method called variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. An analytical solution can be obtained from its trial-function with possible unknown constants, which can be identified by imposing the boundary conditions, by successively iteration.

Quasi-tangent vectors in flow-invariance and optimization problems on Banach manifolds

The aim of this paper is threefold: 1. (1) to introduce the notion of “quasi-tangent vector” to a subset S of a Banach manifold M and to extend the definitions (and some properties) of a generalized gradients (and tangent cones) from a Banach space E to M; 2. (2) to apply some of the above notions to the problem of flow-invariance of a closed subset of the manifold M (generalizing Nagumo-Brezis result); 3. (3) to extend some fundamental results in optimization (e.g., Clarke's general Lagrange multipliers rule in terms of generalized gradient from E to M. Points (1) and (3) are based on the results of Clarke, Rockafellar and Ursescu on tangent cones).

Optimal control of a system with intermediate conditions

Necessary optimality conditions are derived in the general problem of control with intermediate constraints on the trajectory. It is shown that many well-known results (see/1–11/, for example) are some realization or other of the general Lagrange multiplier rule. Statements, strenghthened in comparison with existing ones, of necessary optimality conditions are derived for two important cases (composite and discontinuous) of the general problem. The importance of certain conditions is verifed by an example.

General Lagrange multiplier theorems

A cone constraint is used to develop a general Lagrange multiplier theorem for normed linear spaces. Conditions for the payoff functional multiplier to be less than zero are given for Banach spaces. Sufficiency theorems involving Lagrange multipliers are developed for abstract programming problems. Generalizations of certain properties of convex functions will be used for optimization problems.

Solution of the Lorie-Savage and Similar Integer Programming Problems by the Generalized Lagrange Multiplier Method

A specific type of all-integer integer programming problem which seems to arise frequently in capital budgeting and other areas involves a zero-one restriction on the problem variables. Approaches to solutions to such problems by the General Lagrange Multiplier Method are discussed. This procedure, because of its relative simplicity, has certain advantages over other direct methods such as the use of integer programming algorithms, especially in areas such as capital budgeting where the constraints are usually not binding to the degree indicated by the problem statement. Throughout this paper, the Lorie-Savage problem in capital budgeting is used as a basis for the discussion.