## Forthcoming articles

### International Journal of Mathematical Modelling and Numerical Optimisation

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 International Journal of Mathematical Modelling and Numerical Optimisation (17 papers in press) Regular Issues Numerical analysis on thermal performance of a trapezoidal micro-channel heat sink using an improved version of the augmented -constraint method   by Lagouge Tartibu Abstract: This work proposes the use of an improved version of the augmented ɛ-constraint method (AUGMENCON2) for the analysis of the thermal performance of a micro-channel heat sink. In order to highlight the strength and the effectiveness of this new approach, a trapezoidal micro-channel heat sink has been considered. The geometrical configuration namely the micro-channel widths and depth are the main variables considered in this study. Surrogate models based on the Response Surface methodology have been adopted to approximate the thermal resistance and the pumping power which provide an indication of the thermal performance of the micro-channel heat sink. A two-objective Non-Linear Problem have been formulated and implemented within the General Algebraic Modelling System. Global Pareto optimal solutions has been computed using the proposed method. Despite being a relatively straightforward method, the AUGMENCON2 provides a reasonable level of accuracy and shortens the required computational time in comparison to two existing approaches. Keywords: Multi-objective optimization; ε-constraint method; heat sink; AUGMENCON. Frequency regulation of a power system integrated with renewables using a novel DE-DA optimized controller.   by Sayantan Sinha, Ranjan Kumar Mallick, Srikanta Patnaik Abstract: The proposed research paper mainly focuses on the automatic generation control of an interconnected power system integrated with solar power. The work has taken into consideration a two area power system consisting of a conventional thermal power source and a solar power plant in one area. The penetration of renewable sources give rise to deviations of frequency from their scheduled values. The AGC plays a very important role in minimizing the frequency deviations of the system by reducing the Area Control error to zero with the help of suitable controllers. A maiden attempt has been made to incorporate two degree of freedom proportional integral and derivative controller for AGC purposed. A novel attempt has been made to design a hybrid optimized Dragonfly algorithm- Differential Evolution technique for tuning the controller gains. The performance of the controller is also compared with the conventional PID controller under a system disturbance of 0.01 p.u when applied to area 1. The system performances are further analyzed with varying system parameters and different loading conditions. Comparisons with traditional PID controllers have also been done in terms of dynamic system parameters like settling time, maximum overshoot and minimum undershoot Keywords: AGC; Deregulated; benchmark; hybrid; PID; 2 DOF PID; renewables; two area. Numerical Analysis of the European and American Options with the SPH method.   by Abdelmjid Qadi El Idrissi, Boujemaa Achchab, Abdellahi Cheikh Maloum Abstract: In this paper, we propose a numerical method to solve the European and the American options by using the SPH method. Because its robustness and efficacy, this numerical method has been widely applied in the computation of partial differential equations particularly in fluid dynamic. To model these financial options, we use the Black Scholes equation. It is a mathematical model consisting of a set of partial differential equation supplemented by some boundary conditions. We evaluate the accuracy of our numerical method by giving some comparisons between the analytic solution and the numerical simulation. Keywords: Black Scholes equation; European option; American option; SPH Method. A Higher-Order Hybrid Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with Boundary and Weak Interior Layers   by Anirban Majumdar, Srinivasan Natesan Abstract: In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient and source term. Because of the positivity of the convection coefficient throughout the domain and the discontinuity of the convection coefficient and the source term at $x=xi$, the analytical solutions of these kind of problems exhibit a boundary layer near $x=0$ and a weak interior layer near $x=xi$. We discretize the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme, which is a combination of the central difference scheme in layer regions and the midpoint upwind scheme in outer regions. The implicit-Euler scheme is used for discretizing the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is $varepsilon$-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is $varepsilon$-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented. Keywords: Singularly Perturbed Convection-Diffusion Problem; Interior Layer; Piecewise-Uniform Shishkin Mesh; Finite Difference Scheme; Uniform Convergence. Reconstruction of an orthotropic thermal conductivity from nonlocal heat flux measurements   by Mousa Huntul, Mohammed Hussein, Daniel Lesnic, M.I. Ivanchov, N. Kinash Abstract: Raw materials are anisotropic and heterogeneous in nature, and recovering their conductivity is of utmost importance to the oil, aerospace and medical industries concerned with the identification of soils, reinforced fiber composites and organs. Due to the ill-posedness of the anisotropic inverse conductivity problem certain simplifications are required to make the model tracktable. Herein, we consider such a model reduction in which the conductivity tensor is orthotropic with the main diagonal components independent of one space variable. Then, the conductivity components can be taken outside the divergence operator and the inverse problem requires reconstructing one or two components of the orthotropic conductivity tensor of a two-dimensional rectangular conductor using initial and Dirichlet boundary conditions, as well as non-local heat flux over-specifications on two adjacent sides of the boundary. We prove the unique solvability of this inverse coefficient problem. Afterwards, numerical results indicate that accurate and stable solutions are obtained. Keywords: Inverse problem; Orthotropic thermal conductivity; Two-dimensional heat equation; Nonlinear optimization. Stability and Numerical Study of Theoretical Model of Zika Virus Transmission   by Maghnia Hamou Maamar, Leila Bouzid, Omar Belhamiti, Fethi Bin Muhammad Belgacem Abstract: In this paper, we examine the Zika virus transmission for human and mosquito populations. At the first time, a compartment model based on two populations, humans and mosquitoes, are proposed and analyzed quantitatively using the stability theory of the differential equations. In the second time, a nonhuman primate (monkey) is considered, we prove the influence of this second reservoir host on the spread of the disease. Numerical simulation of the models is implemented to investigate the effect of certain key parameters on the transmission of Zika virus. Keywords: Zika Virus Disease; Equilibrium Stability; Reproduction Number; Jacobi Multi-Wavelets Method. Community Detection in Complex Networks Using Multi-objective Bat Algorithm   by Iyad Abu-Doush Abstract: Community detection is the problem of identifying communities in which we aim to discover groups of nodes with high connectivity within the same group and with low connectivity outside the group. Community detection is considered to be a non-deterministic polynomial-time hard problem. Heuristic algorithms can be used to solve such a complex optimization problem. Bat Algorithm (BA) is a meta-heuristic optimization algorithm. The BA can be used to model a multi-objective optimization problem. In this paper, the Multi-objective Bat algorithm (MOBA) is adapted to model and solve the community detection problem. In order to evaluate the algorithm, four real-world datasets are used. The performance of the algorithm is compared with seven other methods from the literature. The comparison was in terms of two metrics to check the quality of the obtained community namely Modularity (Q) and Normalized Mutual Information (NMI). The results show that the proposed algorithm outperforms all algorithms in one dataset and that it is competitive in other cases. Keywords: Bat algorithm; Community detection; Multi-objective optimization; Multi-objective Bat algorithm. Algorithm for Generalized multi-objective set covering problem with an application in ecological conservation   by Lakmali Weerasena Abstract: The Set Covering problem (SCP) is one of the representative NP-hard combinatorial problems in which we are given a set of items and a collection of subsets of them. We find a sub-collection from the collection of subsets including each item in at least one subset in the sub-collection. In this paper we propose a generalization to the SCP: Find a sub-collection including each item in a given number of sets and introduce conflicting objective functions. We define the new problem as the generalized multi-objective SCP (GMOSCP). We propose a new algorithm to approximate the Pareto set of the GMOSCP. The GMOSCP is also a NP-hard combinatorial optimization problem. Thus, developing an algorithm to approximate the Pareto set of the GMOSCP is merited. In addition, GMOSCP has many real-life applications in fields such as scheduling, facility location, and ecological conservation. In particular, the reserve site selection problem in ecological conservation is a common field for its applications; therefore, the performances of the algorithm is verified using a real data set taken from the literature in ecological conservation. Conflicting objective functions are proposed to obtain Pareto sets, and several experiments have been conducted to show the performance of the proposed algorithm. Further, the results are compared to Pareto solutions. The results show that algorithm performs well and finds approximated solutions efficiently. Keywords: Multi-objective optimization; Algorithm; Approximation; Reserve site selection; Ecological Conservation. Magneto-electro-elastic analysis for two mode-III semi-permeable collinear cracks in piezo-electro-magnetic strip   by POOJA R.A.J. VERMA, Ravi Verma Abstract: A mode-III crack problem for piezo-electro-magnetic strip weakened by two transversely oriented semi-permeable collinear cracks under out-of-plane shear stress and in-plane electric displacement and magnetic induction is addressed. The problem is formulated and then solved using Fourier series method and Integral equation technique. Closed-form analyticrnexpressions are derived for various fracture parameters viz. crack sliding displacement, crack opening potential drop, crack opening induction drop, field intensity factors and energy release rates. The numerical case study is presented graphically for BaTiO3-CoFe2O4 piezo-electro-magnetic material to show the influence of inter-crack distance, volume fractionsrnand electro-magnetic crack-face boundary conditions on crack sliding displacement, crack opening potential drop, crack opening induction drop and energy release rates. Moreover, the effect of prescribed mechanical loadings as well as electro-magnetic loadings on energy release rates are also presented graphically. Results obtained are presented graphically,rnanalyzed and concluded. Keywords: Collinear cracks; Fourier series method; Inter-crack distance; Integral equation; Piezo-electro-magnetic strip; Semi-permeable crack; Volume fraction. Optimal multiple stopping under catastrophic event   by Noureddine Jilani Ben Naouara, Faouzi Trabelsi Abstract: In this paper, we introduce a new optimal multiple stopping times problem, where we assume each exercise right happens before the date of release of a catastrophic event modelled by a random variable $varsigma$. This catastrophe can be natural (i.g. earthquake, tsunami) or technological (e.g. nuclear event). Since a sudden catastrophe will have a direct influence on prices variation, especially those of underlying as well as option's prime, eventual catastrophic event will be modelled by the first time the underlying's price exceeds some large barrier. The originality of this paper comes from a mathematical model taking account of a nonlinear criteria of sum of the underlying stopped at stopping times of the holder's filtration information as well as a random number of exercise rights at sopping times involving prior to a catastrophic event. This will generalises the concept of Swing contracts, where the exercise rights number is only deterministic and finite. Keywords: Optimal multiple stopping; stopping times; catastrophic event; Dynamic programming; RCLL process; diffusion process; Markov process; Snell's envelope. The Integral-Differential and Integral Approach for the Estimation of the Classical Lennard-Jones and Biswas-Hamann Potentials.   by Samuel Surulere, Michael Shatalov, Andrew Mkolesia, Julius Ehigie Abstract: Many well-known semiempirical potential energy functions have been used to construct potential energy curves from the physical or chemical properties of atoms. In this study, we identify the Lennard-Jones and Biswas-Hamann potential parameters and use these to calculate and reconstruct potential energy curves using experimental data sets of gold atom. Two different approaches are studied in detail. The Lennard-Jones potential yielded complex conjugate eigenvalues for both approaches. Numerical estimates proved the considered approaches gives better approximations as constructed and reconstructed potential energy curves were almost graphically indistinguishable. Keywords: Potential parameters; Objective functions; Potential energy curves; Objective Least Squares function. Method of solving optimal design problems based on flexible tolerance strategy   by Larysa Korotka, Dmitriy Zelentsov Abstract: On solving problems of optimal design, structures working in aggressive external environments, a modified method is proposed, which is based on the flexible tolerance method. Based on the information about the degree of closeness of the point of the solution space to the local extremum point, which is received by the neural network controller, its parameters change. For this purpose, various matrices of neural network synapses, trained for different precisions of calculating the functions of constraints, are used. This strategy is used to modify the flexible tolerance method, based on the use of a neural network controller. As a criterion of the flexible tolerance, the error of calculating the constraint functions is used. Keywords: neural network regulator; flexible tolerance; optimal design; fuzzy variables. Mathematical Modelling of Piezoelectric Elastic Materials   by Boudjour Allaoua, Mohamed Dalah Abstract: We consider a quasistatic contact modelled with the regularized friction law for electro-elastic and the foundation is assumed to be electrically conductive. This regularization is obtained by replacing the function $j(.)$ by the function $j_{\rho}(.)$, where $\rho$ is a strictly positive parameter. The classical formulation for the antiplane problem is formulated as a time dependent of corresponding variational formulation and is solved by the Banach fixed-point theorem and classical results for variational inequalities. We provide a weak formulation of the contact problem in the form variational system in which the unknowns are the displacement and the stress fields, then we establish the existence of a unique weak solution to the model. Finally, we have given a convergence criterion of the solution as the paramater of regularization $\rho$ converges to zero. Keywords: Quasistatic contact; electro-elastic material; antiplane; regularised friction law; weak solution; variational formulation. Backtracking Immune Algorithm for Continuous Multi-objective Optimization   by Ahmed TCHVAGHA ZEINE Abstract: In this paper, a new Backtracking Immune Algorithm for Continuous Multi-objective Optimization (BIAMO) is proposed. It uses the update archive to sort the non-dominated solutions of the Pareto front and the mutation and crossover operators of the Backtracking Search Algorithm (BSA). Experimental results are produced for various benchmark problems and for a variety of engineering design problems. They show that, compared to the recent multi-objective optimization evolutionary algorithms, the proposed algorithm improves not only the convergence capacity but also preserves the diversity of the population. eleven benchmark problems and two engineering design problems have been solved and the obtained results were compared with other well-known optimizers. The obtained\r\nresults demonstrate that, the proposed algorithm requires less number of function evaluations and in most cases gives better results compared to other considered algorithms.\r\n Keywords: Multi-objective optimization; Evolutionary Algorithms; Backtracking Search; Hybrid recombination; Hybrid mutation. Study of Effects of Toxicants and Acidity on Oxygen-Dependent Aquatic Population: A Mathematical Model   by Preety Kalra, Shreya Tangri Abstract: The rapidly elevating contamination of water bodies is one of the present-day critical problems occurring worldwide. It is a well-determined fact that the direct discharge of household wastes, industrial effluents, chemicals, acids and other pollutants into water bodies along with eutrophication are degrading the water quality. Washing-off of land pollutants in water bodies coupled with acid rain further increase the toxicity and acidity of water. These phenomena are leading to decrease of dissolved oxygen in water which is endangering the growth and survival of populations residing in water. To investigate the stress of increasing toxicity and acidity of water bodies like ponds, lakes and rivers on the aquatic population due to decrease in dissolved oxygen, a non-linear mathematical model is proposed. The model is analysed and the stability conditions are derived in terms of the parameters. To substantiate the analytical results, numerical simulations are carried out using Matlab. Keywords: Aquatic Population; Toxicity; Acidity; Dissolved Oxygen; Model; Stability. A new fuzzy transportation algorithm for finding fuzzy optimal solution   by Farikhin Farikhin, Muhammad Sam'an Abstract: Based on the literature, many algorithms have been proposed to solve fuzzy transportation problems in real life. Among the existing algorithms, some do not use the ranking method to convert fuzzy numbers into crisp numbers, whereas some use ranking methods that fail to rank non-normal fuzzy numbers correctly and compensate with the use of areas. Therefore, in this paper, a new fuzzy transportation algorithm, i.e., the NNWC, NLC and NVA, is used to solve a fuzzy transportation problem. The ranking methods involving non-normal trapezoidal fuzzy numbers and triangular fuzzy numbers as well as a new ranking method using total integral value are used to solve case studies 1 and 2, and the results are compared with the results from existing methods. Because of the proposed method is a direct extension of a classical method, it is reasonable to apply it to real-life transportation problems. Keywords: fuzzy transportation problem; new fuzzy transportation algorithm;new total integral value; non-normal trapezoidal fuzzy numbers. A Metropolis within Gibbs algorithm for knowledge Discovery in Language Assessments   by Mengta Chung Abstract: The Reduced Reparameterized Unified Model (RRUM) has been frequently used in language assessment studies. The objective of this research is to advance an MCMC algorithm for the Bayesian RRUM. The algorithm starts with estimating correlated attributes. Using a saturated model and a binary decimal conversion, the algorithm transforms possible attribute patterns to a Multinomial distribution. Along with the likelihood of an attribute pattern, a Dirichlet distribution is used as the prior to sample from the posterior. The Dirichlet distribution is constructed using Gamma distributions. Correlated attributes of examinees are generated using the inverse transform sampling. Model parameters are estimated using the Metropolis within Gibbs sampler sequentially. Two simulation studies are conducted to evaluate the performance of the algorithm, and an empirical study uses the ECPE data obtained from the CDM R package are performed. Keywords: CDM; RUM; RRUM; Q-matrix; Bayesian; MCMC.