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The GM (1, N) model is an important prediction model among gray models, but the conventional GM (1, N) model shows big errors sometimes mainly due to the structural inconsistency between the gray differential equation for parameter estimation and the whitening equation for prediction. The paper derives the modified model of conventional gray differential equation based on the whitening equation, and the gray differential equation obtained in this way has the structure basically consistent with the structure of whitening equation. The paper estimates parameters with the gray differential equation of modified GM (1, N) model and gives three parameter estimation methods with different background values. The time response equation for conventional GM (1, N) model generally has no analytical solution, but the paper gives a specific solving method for the time response equation of modified model. The example proves that the modified GM (1, N) model given and the models built with the methods proposed improve the simulation and prediction accuracy significantly compared with the conventional model. [ABSTRACT FROM AUTHOR]

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order. [ABSTRACT FROM AUTHOR]

Student academic underachievement is a concern of paramount importance in Europe, where around 15% of the students in the last high school courses do not achieve the minimum knowledge academic requirement. In this paper, we propose a model based on a system of differential equations to study the dynamics of the students’ academic performance in the German region of the North Rhine-Westphalia. This approach is supported by the idea that both good and bad study habits, are a mixture of personal decisions and influence of classmates. This model allows us to forecast the student academic performance by means of confidence intervals over the next few years. [ABSTRACT FROM PUBLISHER]

The history match procedure in an oil reservoir is of paramount importance because good estimates of the reservoir parameters may lead to a better forecast of the production. The reservoir is said to be 'matched' when the discrepancies between the model predictions and the observations of the real reservoir are below a certain tolerance. The determination of the model parameters via history matching requires the minimisation of an objective function in a parameter space that may be populated by many local minima. In other words, more than one set of reservoir model parameters fits the observation. Moreover, even a solution associated to a given minimum may be unstable. In order to reduce the ambiguity, it is necessary to incorporate constraints in the reservoir parameters to be estimated. Here, we use the smoothness constraint to regularise this inverse problem using as an example a modified version of the PUNQ model. The PUNQ model is suitable for applying this constraint because the sediments were deposited in a deltaic costal plain environment where good lateral continuity of the sediments is expected. We show that it is possible to recover all important features of the spatial distribution of the permo-porosity fields in all layers of the PUNQ model. In particular, the directions and widths of the sand channels were recovered. [ABSTRACT FROM AUTHOR]