فهرست مطالب

Mathematical Modeling - Volume:10 Issue: 4, Autumn 2022

Journal of Mathematical Modeling
Volume:10 Issue: 4, Autumn 2022

  • تاریخ انتشار: 1401/10/19
  • تعداد عناوین: 12
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  • Chafik Allouch, Domingo Barrera, Mounaim Saou, Driss Sbibih, Mohamed Tahrichi * Pages 387-401
    In this paper, we propose collocation and Kantorovich methods based on spline quasi-interpolants defined on a bounded interval  to solve numerically a class of Fredholm integro-differential equations. We describe the computational aspects for calculating the approximate solutions and  give theoretical results corresponding to the convergence order of each method in terms of the degree of the considered spline quasi-interpolant. Finally, we provide some numerical tests that confirm the theoretical results and prove the efficiency of the proposed methods.
    Keywords: Integro-differential equations, quasi-interpolants, collocation method, Kantorovich method
  • Vahid Samadpour Khalifeh Mahaleh *, Reza Ezati Pages 403-415
    In this research, we investigate the fuzzy integral equations related to traffic flow. Using the Banach fixed point theorem, we prove the existence and uniqueness of the solution for such equations. Using the Picard iterative method, we obtain the upper bound for an accurate and approximate solution. Finally, we obtain an error estimation between the exact solution and the solution of the iterative method. Example shows the applicabilityof our results.
    Keywords: Fuzzy integral equations, traffic flow, iterative Method
  • Hadiseh Kamalgharibi, Akbar Hashemi Borzabadi *, Omid Solaymani Fard, Atefeh Solouk Mofrad, Mehdi Shafieian Pages 417-435
    DES, or drug-eluting stents, have the advantage of reducing restenosis rates relative to bare-metal stents. Modeling and simulation can be used to improve device performance. In this study, a general mathematical model for releasing a hydrophobic drug from a drug-eluting stent, DES, with a biostable coating is modeled. Most mathematical models allow the drug in the polymer to be released freely. This is suitable when the initial concentration of the drug in the polymer is less than the solubility, in which case the dissolution of the drug can be considered instantaneously. On the other hand, matrix devices can be loaded above solubility to provide zero-order release. to this end, we have equipped a model with a function that determines how the dissolution processes change with the dispersed phase discharge. The general model is analyzed with some limitations, and it is reduced to a new model that is consistent with previous studies. We examine the effects of initial drug loading and dissolution rate constant in numerically solving one of the new models, which is novel in DESs.
    Keywords: Mathematical model, drug eluting stent, biostable polymer, Dissolution, Diffusion
  • Ali Mesforush *, Stig Larsson Pages 437-452
    The Cahn-Hilliard equation is discretized by a Galerkin finite element  method based on continuous piecewise linear functions in space and discontinuous piecewise constant functions in time. A posteriori error estimates are proved by using the methodology of dual weighted residuals.
    Keywords: Cahn-Hilliard, Finite element, error estimate, a posteriori, dual weighted residuals
  • Nasser Akhoundi * Pages 453-461
    In this paper, the banded Toeplitz matrices generated by $f(\theta)=(2(1-\cos(\theta-\tilde{\theta})))^d$ are studied. The function $f$ is a real non-negative function with a zero of order $2d$ at $\tilde{\theta}$ and the generated matrices are ill-conditioned Hermitian positive definite. We show that these banded Toeplitz matrices are similar to the banded real symmetric positive definite Toeplitz matrices that are generated by $f(\theta)=(2(1-\cos(\theta)))^d$.  A fast direct solver is proposed to compute the inverse of these real matrices. Numerical experiments show that our proposed method is faster and more stable than the stable Levinson algorithm.
    Keywords: Toeplitz matrices, fast Toeplitz solver, Levinson algorithm
  • Kamal Rashedi *, Fatemeh Baharifard, Aydin Sarraf Pages 463-480
    In this paper, we consider the problem of approximating the displacement and the wave sink or source in a 1D wave equation from various measurements. First, the problem is recast as a certain hyperbolic equation. Then, we propose a Ritz approximation as the solution of the reformulated problem and apply the collocation method to convert the inverse problem to a system of linear equations. Since the problem is not well-posed, the numerical discretization of the problem may produce a system of equations that is not well-conditioned. Therefore, we apply the Tikhonov regularization method to obtain a stable solution. For the contaminated measurements, we take advantage of the mollification method in order to derive stable numerical derivatives. Several test examples are provided to show the effectiveness of the proposed technique for obtaining satisfactory results.
    Keywords: Inverse wave problem, Ritz collocation method, Tikhonov regularization, mollification technique
  • Mohammad Pourmohammadi Fallah, Maziar Salahi * Pages 481-494
    In (Zhang et al. An integrated strategy for a production planning and warehouse layout problem: modeling and solution approaches,  Omega 68  (2017) 85--94)    the authors have proposed a mixed-integer linear programming model for the integrated production planning and warehouse layout problem. To solve the model, they proposed a Lagrangian relax-and-fix heuristic that takes significant amount of time to stop with gaps above 5$\%$ for large-scale instances. Here, we present two heuristic algorithms to solve the problem. In the first one, we use a greedy approach by allocating  warehouse locations with less reservation  costs, and also less  transportation costs  from the production area to locations and from locations to the output point  to items with higher demands. Then a smaller model is solved. In the second heuristic, first we sort items in descending order according to the fraction of sum of the demands for that item in the time horizon plus the maximum demand for that item in the time horizon and  sum of all its demands in the time horizon. Then we categorize the sorted items into groups of 3, 4, or 5, and solve a small-scale optimization problem for each group, hoping to improve the solution of the first heuristic. Our preliminary numerical results show  the effectiveness of the proposed heuristics.
    Keywords: Capacitated lot-sizing, Warehouse layout, Mixed-integer linear programming, Heuristics algorithm
  • Mina Mortazavi, Mortaza Gachpazan *, Mahmood Amintoosi Pages 495-514
    One of the purposes of edge detection is to use methods that be able to process visual information according to human needs. Therefore, an edge detector is reliable when evaluated by measurement criteria before use in computer vision tools. These criteria compute the difference between the ground truth edge map (reference image) and the original image. In this study, we propose an improved Canny edge detection method based on the fractional-order operators to extract the ideal edge map. Then, by changing the hysteresis thresholds, the thin edges are obtained by filtering gradient calculations based on fractional-order masks. In addition, we employ common fractional-order derivative operators to extract the edge strength and enhance image edge contrast. The plotted curves of the edge detection criteria show that the obtained edge map of the proposed edge detection operator, which is considered to be the minimal rating of  measurement, is visually and quantitatively closer to ground truth.
    Keywords: Edge detection, hysteresis thresholds, Fractional derivatives, Canny method, edge map
  • Eshetu Derzie *, Justin B. Munyakazi, Tekle Dinka Pages 515-534
    We develop a robust uniformly convergent numerical scheme for singularly perturbed time dependent Burgers-Huxley partial differential equation. We first discretize the time derivative of the equation using the Crank-Nicolson finite difference method. Then, the resulting semi-discretized nonlinear ordinary differential equations are linearized using the quasilinearization technique, and finally, design a fitted operator upwind finite difference method to resolve the layer behavior of the solution in the spatial direction. Our analysis has shown that the presented method is second order parameter uniform convergent in time and first order in space. Numerical experiments are conducted to validate the theoretical results.
    Keywords: Singularly perturbed problem, Burgers-Huxley equation, Crank-Nicolson finite difference scheme, fitted operator method, parameter uniform convergence
  • Marzieh Dehghani-Madiseh * Pages 535-553
    In this paper, parametric Sylvester matrix equations whose elements are linear functions of interval parameters are considered. In contrast to deterministic problems, when a system of equations is derived from a stochastic model, its coefficients may depend on some parameters and so the parameterized system of equations appears. This work considers the parameterized Sylvester matrix equations and tries to propose some methods containing a direct method and two iterative methods to obtain outer estimations of the solution set.
    Keywords: Interval computation, matrix equation, Sylvester matrix equation, parameterized linear systems
  • Rooholah Abedian * Pages 555-567
    In this paper, a new fourth-order finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. A set of Z-type non-linear weights is constructed based on the $L_1$ norm, yielding improved WENO scheme with more accurate resolution. The Caputo fractional derivative of order $\alpha$ is split into a weakly singular integral and a classical second derivative. The classical Gauss-Jacobi quadrature is employed for solving the weakly singular integral. Also, a new WENO-type reconstruction methodology for approximating the second derivative is developed. Some benchmark examples are prepared to illustrate the efficiency, robustness, and good performance of this new finite difference WENO-Z scheme.
    Keywords: finite difference scheme, Fractional differential equations, WENO-Z scheme
  • Kanithi Jyothsna *, Vijaya Laxmi Pikkala, Vijaya Kumar Prathi Pages 569-585
    This paper addresses renewal input continuous  and  discrete time  queues with balking and vacation interruptions. An arriving client may join the system or  balk with some state-dependent probability. Whenever the server finds an empty system, he leaves for a working vacation. During working vacations, if there are clients to be served at a service completion instant, the server interrupts the  working vacation and switches to regular service period. The embedded Markov chain technique has been adopted for evaluating pre-arrival epoch probabilities and  supplementary variable approach is employed to evaluate arbitrary instant probabilities. Few performance characteristics and sojourn time distribution have also been demonstrated. Finally,  numerical investigations have been figured out to depict the impact of the model variables on the performance indices.
    Keywords: Renewal input, Balking, vacation interruption, embedded Markov chain, sojourn time