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The purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.

The small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.

The proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.

The use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.

To validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.

The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions.

The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method.

Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem.

This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.

A new approach to national economy models is proposed using the decomposition method. Reasons are discussed justifying the approach and consequences.

The purpose of this paper is to reduce the computational burden and improve the precision of the parameter optimization in the convection-diffusion equation, a new algorithm, the refined gray-encoded evolution algorithm (RGEA), is proposed.

In the new algorithm, the differential evolution algorithm (DEA) is introduced to refine the solutions and to improve the search efficiency in the evolution process; the rapid cycle operation is also introduced to accelerate the convergence rate. The authors apply this algorithm to parameter optimization in convection-diffusion equations.

Two cases for parameter optimization in convection-diffusion equations are studied by using the new algorithm. The results indicate that the sum of absolute errors by the RGEA decreases from 74.14 to 99.29 percent and from 99.32 to 99.98 percent, respectively, compared to those by the gray-encoded genetic algorithm (GGA) and the DEA. And the RGEA has a faster convergent speed than does the GGA or DEA.

A more complete convergence analysis of the method is under investigation. The authors will also explore the possibility of adapting the method to identify the initial condition and boundary condition in high-dimension convection-diffusion equations.

This paper will have an important impact on the applications of the parameter optimization in the field of environmental flow analysis.

This paper will have an important significance for a sustainable social development.

The authors establish a new RGEA algorithm for parameter optimization in solving convection-diffusion equations. The application results make a valuable contribution to the parameter optimization in the field of environmental flow analysis.

The Debye theory of dielectric relaxation as corrected for inertial effects has as yet been only considered in the linear approximation. There, the rise and decay transients are identical. Here a method recently developed for the treatment of a rotator in a periodic potential is applied to calculate the transient behaviour when the linear approximation is discarded. The Kramers equation for the problem is expanded in a set of orthogonal functions which lead to a set of linear differential difference equations giving the relaxation behaviour. It is shown that the Mori formalism for the problem leads to the same set of differential difference equations as the Kramers equation.

Investigates the global stability of the zero solution of an impulsive system of differential‐difference equations with variable impulsive perturbations. By means of piecewise continuous functions which are analogues of Lyapunov’s functions, and of the comparison principle, sufficient conditions for global stability of the zero solution of the systems considered are found.

The main objective of this paper is to study the optimal system for series systems with mixed standby (including cold standby, warm standby and hot standby) components.

The paper deals with the reliability and availability characteristics of four different series system configurations. The failure time of the operative, hot standby and warm standby are assumed to be exponentially distributed with parameters λ, λ, and α respectively. The repair time distribution of each server is also exponentially distributed with parameter μ.

The mean time to failure, MTTF_i, and the steady‐state availability A_i(∞) for four configurations are examined and comparisons made. For all four configurations, the configurations are ranked based on: MTTF_i, A_i(∞), and C_i/B_i where B_i is either MTTF_i or A_i(∞). Obviously, the system with height MTTF_i and A_i(∞), do not need frequent maintenance, i.e. less maintenance.

Numerical results for the cost/benefit measure have been obtained for all configurations. It is interesting to note first that the optimal configuration using the cost/MTTF_i measure is configuration 4. Next the optimal configuration using the cost/A_i(∞) measure is configuration 2.

The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.

In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.

In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.

The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.

In an earlier paper (Turkyilmazoglu, 2011a), the author introduced a new optimal variational iteration method. The idea was to insert a parameter into the classical variational iteration formula in an aim to prevent divergence or to accelerate the slow convergence property of the classical approach. The purpose of this paper is to approve the superiority of the proposed method over the traditional one on several physical problems treated before by the classical variational iteration method.

A sufficient condition theorem with an upper bound for the error is also presented to further justify the convergence of the new variational iteration method.

The optimal variational iteration method is found to be useful for heat and fluid flow problems.

The optimal variational iteration method is shown to be convergent under sufficient conditions. A novel approach to obtain the optimal convergence parameter is introduced.