Variational iteration method for solving twelfth-order boundary-value problems using He’s polynomials

In this paper, we apply the variational iteration method using He’s polynomials (VIMHP) for solving the twelfth-order boundary-value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Technical note – On optimization approach for multidisk vertical allocation problems

Multidisk vertical allocation (MDVA) problems intend to find an allocation of relations to disks such that the expected query cost is minimized. Recently, Chang [European Journal of Operational Research 143 (2002) 210] modified Rotem et al.'s [IEEE Transactions on Knowledge and Data Engineering 5 (1993) 882] method for solving an MDVA problem using a smaller number of binary variables. Chang's method however is unable to treat MDVA problems with possible replication of relations. This paper proposes another method to solve MDVA problems, which is more effective than Rotem et al.'s and is able to treat replication problems insolvable by Chang's method.     

Modified Variational Iteration Method for Heat and Wave-Like Equations

In this paper, we apply the modified variational iteration method (MVIM) for solving the heat and wave-like equations. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Smoothing and newton's method for a discontinuous variational equation of stefan type ∗

An important way to treat discontinuous variational equations consists in its embedding into a family of continuous ones. In the present paper for this purpose a smooth penalty technique is applied. Convergence bounds are derived for the approximation of the original solution by solutions of the generated smoothed problems. The focus of the papers is to analyze a semi-discretization via Rothe's method and to establish Convergence bounds for Newton's method applied to it depending on discretization and smoothing parameters. The subproblems at each time level can be considered as nonsmooth convex variational problems of obstacle type     

Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.     

A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization

This paper proposes a primal-dual interior point method for solving large scale nonlinearly constrained optimization problems. To solve large scale problems, we use a trust region method that uses second derivatives of functions for minimizing the barrier-penalty function instead of line search strategies. Global convergence of the proposed method is proved under suitable assumptions. By carefully controlling parameters in the algorithm, superlinear convergence of the iteration is also proved. A nonmonotone strategy is adopted to avoid the Maratos effect as in the nonmonotone SQP method by Yamashita and Yabe. The method is implemented and tested with a variety of problems given by Hock and Schittkowskis book and by CUTE. The results of our numerical experiment show that the given method is efficient for solving large scale nonlinearly constrained optimization problems.Acknowledgement The authors would like to thank anonymous referees for their valuable comments to improve the paper.     

On stable least squares solution to the system of linear inequalities

The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author’s previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.      

Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters

To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers.In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods.     

Forecasting for the ordering and stock-holding of spare parts

A modern military organization like the UK's Royal Air Force is dependent on readily available spare parts for in-service aircraft in order to maximize operational capability. A large proportion of spare parts are known to have an intermittent or slow-moving demand pattern, presenting particular problems as far as forecasting and inventory control are concerned. In this paper, we use extensive demand and replenishment lead-time data to assess the practical value of forecasting models put forward in the literature for addressing these problems. We use an analytical method for classifying the consumable inventory into smooth, irregular, slow-moving and intermittent demand patterns. Recent forecasting developments are compared against more commonly used methods across the identified demand patterns. One recently developed method, a modification to Croston's method referred to as the approximation method, is observed to provide significant reductions in the value of the stock-holdings required to attain a specified service level for all demand patterns.     

Jacobian smoothing Brown’s method for NCP

Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented.     

Eulerian-Lagrangian localized adjoint method: The theoretical framework

This is the second of a sequence of papers devoted to applying the localized adjoint method (LAM), in space-time, to problems of advective-diffusive transport. We refer to the resulting methodology as the Eulerian-Lagrangian localized adjoint method (ELLAM). The ELLAM approach yields a general formulation that subsumes many specific methods based on combined Lagrangian and Eulerian approaches, so-called characteristic methods (CM). In the first paper of this series the emphasis was placed in the numerical implementation and a careful treatment of implementation of boundary conditions was presented for one-dimensional problems. The final ELLAM approximation was shown to possess the conservation of mass property, unlike typical characteristic methods. The emphasis of the present paper is on the theoretical aspects of the method. The theory, based on Herrera's algebraic theory of boundary value problems, is presented for advection-diffusion equations in both one-dimensional and multidimensional systems. This provides a generalized ELLAM formulation. The generality of the method is also demonstrated by a treatment of systems of equations as well as a derivation of mixed methods. © 1993 John Wiley & Sons, Inc.     

An application of Lemke's method to a class of Markov decision problems

This paper presents an application of Lemke's method to a class of Markov decision problems, appearing in the optimal stopping problems, and other well-known optimization problems. We consider a special case of the Markov decision problems with finitely many states, where the agent can choose one of the alternatives; getting a fixed reward immediately or paying the penalty for one term. We show that the problem can be reduced to a linear complementarity problem that can be solved by Lemke's method with the number of iterations less than the number of states. The reduced linear complementarity problem does not necessarily satisfy the copositive-plus condition. Nevertheless we show that the Lemke's method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended Lemke's method to compute a solution in the piecewise linear complementarity problem.     

Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.     

Parameter estimation based on interval-valued belief structures

Parameter estimation based on uncertain data represented as belief structures is one of the latest problems in the Dempster–Shafer theory. In this paper, a novel method is proposed for the parameter estimation in the case where belief structures are uncertain and represented as interval-valued belief structures. Within our proposed method, the maximization of likelihood criterion and minimization of estimated parameter’s uncertainty are taken into consideration simultaneously. As an illustration, the proposed method is employed to estimate parameters for deterministic and uncertain belief structures, which demonstrates its effectiveness and versatility.     

Mehar’s method for solving fuzzy sensitivity analysis problems with LR flat fuzzy numbers

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.     

Modified variational iteration method for solving Fisher’s equations

In this paper, we apply the modified variational iteration method (MVIM) for solving Fisher’s equations. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

A new scheme for the solution of reaction diffusion and wave propagation problems

In this paper, a robust numerical scheme is presented for the reaction diffusion and wave propagation problems. The present method is rather simple and straightforward. The Houbolt method is applied so as to convert both types of partial differential equations into an equivalent system of modified Helmholtz equations. The method of fundamental solutions is then combined with the method of particular solution to solve these new systems of equations. Next, based on the exponential decay of the fundamental solution to the modified Helmholtz equation, the dense matrix is converted into an equivalent sparse matrix. Finally, verification studies on the sensitivity of the method’s accuracy on the degree of sparseness and on the time step magnitude of the Houbolt method are carried out for four benchmark problems.     

Modified variational iteration method for solving fourth-order boundary value problems

In this paper, we apply the modified variational iteration method (MVIM) for solving the fourth-order boundary value problems. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.