Journal of Mathematical Modeling
https://jmm.guilan.ac.ir/
Journal of Mathematical Modelingendaily1Sat, 01 Jul 2023 00:00:00 +0430Sat, 01 Jul 2023 00:00:00 +0430A mixed algorithm for smooth global optimization
https://jmm.guilan.ac.ir/article_6201.html
This paper presents a covering algorithm for solving bound-constrained global minimization problems with a differentiable cost function. In the proposed algorithm, we suggest to explore the feasible domain using a one-dimensional global search algorithm through a number of parametric curves that are relatively spread and simultaneously scan the search space. To accelerate the corresponding algorithm, we incorporate a multivariate quasi-Newton local search algorithm to spot the lowest regions. &nbsp;The proposed algorithm converges in a finite number of iterations to an $\varepsilon$-approximation of the global minimum. The performance of the algorithm is demonstrated through numerical experiments on some typical test functions.A new approach for solving constrained matrix games with fuzzy constraints and fuzzy payoffs
https://jmm.guilan.ac.ir/article_6684.html
The main purpose of this study is to construct a new approach for solving a constrained matrix game where the payoffs and the constraints are LR-fuzzy numbers. The method that we propose here is based on chance constraints and on the concept of a comparison of fuzzy numbers. First, we formulate the fuzzy constraints of each player as chance constraints with respect to the possibility measure. According to a ranking function $\mathcal{R}$, a crisp constrained matrix game is obtained. Then, we introduce the concept of $\mathcal{R}$-saddle point equilibrium. Using results on ordering fuzzy numbers, sufficient existence conditions of this concept are provided. The problem of computing this solution is reduced to a &nbsp;pair of primal-dual linear programs. To illustrate the proposed method, an example of the market competition game is given.Pricing American option under exponential Levy Jump-diffusion model using Random Forest instead of least square regression
https://jmm.guilan.ac.ir/article_6300.html
In this paper, we aim to propose a new hybrid version of the Longstaff and Schwartz algorithm under the exponential Levy Jump-diffusion model using Random Forest regression. For this purpose, we will build the evolution of the option price according to the number of paths. Further, we will show how this approach numerically depicts the convergence of the option price towards an equilibrium price when the number of simulated trajectories tends to a large number. In the second stage, we will compare this hybrid model with the classical model of the Longstaff and Schwartz algorithm (as a benchmark widely used by practitioners in pricing American options) in terms of computation time, numerical stability and accuracy. At the end of this paper, we will test both approaches on the Microsoft share &ldquo;MSFT&rdquo; as an example of a real market.&nbsp;Strategies for disease diagnosis by machine learning techniques
https://jmm.guilan.ac.ir/article_6650.html
Machine learning (ML) techniques have become a point of interest in medical research. To predict the existence of a specified disease, two methods K-Nearest Neighbors (KNN) and logistic regression can be used, which are based on distance and probability, respectively. These methods have their problems, which leads us to use the ideas of both methods to improve the prediction of disease outcomes. For this sake, first, the data is transformed into another space based on logistic regression. Next, the features are weighted according to their importance in this space. Then, we introduce a new distance function to predict disease outcomes based on the neighborhood radius. Lastly, to decrease the CPU time, we present a partitioning criterion for the data.Investigation and solving of initial-boundary value problem including fourth order PDE by contour integral and asymptotic methods
https://jmm.guilan.ac.ir/article_6251.html
In this paper, we consider a fourth order mixed partial differential equation with some initial and boundary conditions which is unsolvable by classical methods such as Fourier, Fourier-Bircove and Laplace Transformation methods. For this problem we will apply the contour integral and asymptotic methods. The convergence of the appeared integrals, existence and uniqueness of solution, satisfying the solution and holding the given initial &nbsp;and boundary conditions are stablished by complex analysis theory and related contour integrals. Finally, the form of analytic &nbsp;and approximate solutions are given due to different cases of eigenvalues distributions in the &nbsp;complex plane.Persistence in mean and extinction of a hybrid stochastic delay Gompertz model with Levy jumps
https://jmm.guilan.ac.ir/article_6628.html
This paper deals with a stochastic delay Gompertz model under regime switching with Levy jumps. Firstly, the existence of a unique global positive solution has been derived. Secondly, sufficient conditions for extinction and persistence in mean are obtained. Finally, an example is given to illustrate our main results.The results in this paper indicate that Levy jumps noise, the white noise and switching noise have certain effects on the properties of the model.A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices
https://jmm.guilan.ac.ir/article_6207.html
In this paper, we propose a new approach to solve weakly singular fractional delay integro-differential equations. In the proposed approach, we apply &nbsp;the operational matrices of fractional integration and delay function based on the shifted Chebyshev polynomials to approximate the solution of the considered equation. By approximating the fractional derivative of the unknown function as well as the unknown function in terms of the shifted Chebyshev polynomials and substituting these approximations into the original equation, we obtain a system of nonlinear algebraic equations. We present the convergence analysis of the proposed method. Finally, to show the accuracy and validity of the proposed method, we give some numerical examples.An efficient approach for solving the fractional model of the human T-cell lymphotropic virus I by the spectral method
https://jmm.guilan.ac.ir/article_6647.html
This paper aims to present a new and efficient numerical method to approximate the solution of the fractional model of human T-cell lymphotropic virus I (HTLV-I) infection $CD4^+T$-cells. The approximate solution of the model is obtained using the shifted Chebyshev collocation spectral method. This model relates to the class of nonlinear ordinary differential equations. The proposed algorithm reduces the Caputo sense fractional model to a system of nonlinear algebraic equations that can be solved numerically. The convergence of the proposed method is investigated. The graphical result is compared with existing numerical methods reported in the literature to indicate the efficiency and reliability of the presented method.Application of S-Boxes based on the chaotic Hindmarsh-Rose system for image encryption
https://jmm.guilan.ac.ir/article_6278.html
The substitution box (S-Box) is a critical component in symmetric cipher algorithms. In this paper, we choose the Hindmarsh-Rose system to generate chaotic S-Boxes. We propose two S-Boxes based on the rotation algorithm relative to the rows (or columns) and the other based on the Zigzag transformation. The performance of the new S-Boxes is evaluated by bijective, nonlinearity, strict avalanche criterion (SAC), output bits independence criterion (BIC), differential approximation probability, linear approximation probability, and algebraic degree. The analysis results show that the presented S-Boxes have suitable cryptographic properties. Also, an image encryption algorithm based on two proposed S-Boxes, and a chaotic Hindmarsh-Rose system are presented. Experimental results show the recommended method has attained good security, and the suggested plan has potent resistance to different attacks.On the inverse eigenvalue problem for a specific symmetric matrix
https://jmm.guilan.ac.ir/article_6649.html
The aim of the current paper is to study a partially described inverse eigenvalue problem of &nbsp;a specific symmetric &nbsp;matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by &nbsp;the &nbsp;minimal eigenvalue of all &nbsp;leading principal submatrices &nbsp;and eigenpair $(\lambda_2^{(n)},x)$ such that $ \lambda_2^{(n)}$ is the maximal eigenvalue of the required matrix. We investigate &nbsp;conditions for the solvability of the problem, and finally an algorithm and &nbsp;its numerical results are presented.Symmetric-diagonal reductions as preprocessing for symmetric positive definite generalized eigenvalue solvers
https://jmm.guilan.ac.ir/article_6374.html
We discuss &nbsp;some potential advantages of the &nbsp;orthogonal symmetric-diagonal reduction in &nbsp;two main versions of the Schur-QR method &nbsp;for symmetric positive definite &nbsp;generalized eigenvalue problems. We also advise and use the appropriate reductions &nbsp;as preprocessing on &nbsp;the solvers, mainly &nbsp;the Cholesky-QR method, of the &nbsp;considered &nbsp;problems. We discuss numerical stability of the &nbsp;methods via providing upper bound for backward error of the computed eigenpairs and via investigating two kinds of &nbsp;scaled residual errors. We also propose &nbsp;and apply &nbsp;two kinds of symmetrizing &nbsp;which &nbsp;improve &nbsp;the stability and the performance &nbsp;of the methods. Numerical experiments show that the &nbsp;implemented versions of the Schur-QR method and the preprocessed versions of the Cholesky-QR &nbsp;method are &nbsp;usually more stable than the Cholesky-QR method.&nbsp;Taylor's formula for general quantum calculus
https://jmm.guilan.ac.ir/article_6629.html
Let $I\subseteq\mathbb{R}$ be an interval and $\beta\colon I\to I$ a strictly increasing continuous function with a unique fixed point $s_0\in I$ satisfying $(t-s_0)(\beta(t)-t)\le 0$ for all $t\in I$. Hamza et al. introduced the general quantum difference operator $D_{\beta}$ by $D_{\beta}f(t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t}$ if $t\ne s_0$ and $D_{\beta}f(t):=f'(s_0)$ if $t=s_0$. &nbsp; In this paper, we establish results concerning Taylor's formula associated with $D_{\beta}$. For this, we define two types of monomials and then present our main results. The obtained results are new in the literature and are useful for further research in the field.A new version of augmented self-scaling BFGS method
https://jmm.guilan.ac.ir/article_6533.html
A new version of the augmented self-scaling memoryless BFGS quasi-Newton update, &nbsp;proposed in [Appl. Numer. Math. 167, &nbsp;187--201, &nbsp;(2021)], &nbsp;is suggested for unconstrained optimization problems. To use the corresponding scaled parameter, &nbsp;the clustering of the eigenvalues of the approximate Hessian matrix about one point is applied with three approaches. The first and second approaches are based on the trace and the determinant of the matrix. The third approach is based on minimizing the measure function. The sufficient descent property is guaranteed for uniformly convex functions, &nbsp;and the global convergence of the proposed algorithm is proved both for the uniformly convex and general nonlinear objective functions, &nbsp;separately. Numerical experiments on a set of test functions of the CUTEr collection show that the proposed method is robust. In addition, &nbsp;the proposed algorithm is effectively applied to the salt and pepper noise elimination problem.On determining radius in nonmonotone trust-region approaches
https://jmm.guilan.ac.ir/article_6704.html
This paper proposes two effective nonmonotone trust-region frameworks for solving nonlinear unconstrained optimization problems while provide a new effective policy to update the trust-region radius. Conventional nonmonotone trust-region algorithms apply a specific nonmonotone ratio to accept &nbsp;new trial step and update the trust-region radius. This paper recommends using the nonmonotone ratio only as an acceptance criterion for a new trial step. In contrast, the monotone ratio or a hybrid of monotone and nonmonotone ratios is proposed as a criterion for updating the trust-region radius. We investigate the global convergence to first- and second-order stationary points for the proposed approaches under certain classical assumptions. &nbsp;Initial numerical results indicate that the proposed methods significantly enhance the performance of nonmonotone trust-region methods.Eigenvalue problem with fractional differential operator: Chebyshev cardinal spectral method
https://jmm.guilan.ac.ir/article_6683.html
In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on Chebyshev cardinal polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the Chebyshev cardinal functions are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.Stability analysis of fractional-order predator-prey model with anti-predator behaviour and prey refuge
https://jmm.guilan.ac.ir/article_6766.html
This article investigates a fractional-order predator-prey model incorporating prey refuge and anti-predator behaviour on predator species. For our proposed model, we prove the existence, uniqueness, non-negativity and boundedness of solutions. Further, all biologically possible equilibrium points and their stability analysis for the proposed system are carried out with the linearization process. Moreover, by using an appropriate Lyapunov function, the global stability of the co-existence equilibrium point is studied. Finally, we provide numerical simulations to demonstrate how the theoretical approach &nbsp;is &nbsp;consistent.Semi-algebraic mode analysis for multigrid method on regular rectangular and triangular grids
https://jmm.guilan.ac.ir/article_6831.html
In this work, a Semi-Algebraic Mode Analysis (SAMA) technique for multigrid waveform relaxation method applied to the finite element discretization on rectangular and regular triangular grids in two dimensions and cubic and triangular prism elements in three dimensions for the heat equation is proposed. &nbsp;For all the studied cases especially for the general triangular prism element, both the stiffness and mass stencils are introduced comprehensively. Moreover, several numerical examples are included to illustrate the efficiency of the convergence estimates. Studying this analysis for the finite element method is more involved and more general than that finite-difference discretization since the mass matrix must be considered. The proposed analysis results are a very useful tool to study the behavior of the multigrid waveform relaxation method depending on the parameters of the problem. &nbsp;&nbsp;Caputo fractional-time of a modified Cahn-Hilliard equation for the inpainting of binary images
https://jmm.guilan.ac.ir/article_6569.html
In this work, we present a new version of the Cahn-Hilliard equation to deal with binary image inpainting. The proposed model is unique due to its memory effect ability implemented by the time fractional derivative. Also, this model has a new diffusion term that gives a topological reconnection and a well sharpness of edges and corners. We give an existence result with some numerical tests implemented by the convexity splitting to show the efficiency of the proposed model.Picard iterative approach for $\psi-$Hilfer fractional differential problem
https://jmm.guilan.ac.ir/article_6839.html
&nbsp;In present work, we discuss local existance and uniqueness of solution to the $\psi-$Hilfer fractional differential problem. By using the Picard successive approximations, we construct a computable iterative scheme uniformly approximating solution. Two illustrative examples are given to support our findings.Stability and bifurcation of stochastic chemostat model
https://jmm.guilan.ac.ir/article_6637.html
The main purpose of this paper is to study dynamics of stochastic chemostat model. In this order, Taylor expansions, polar coordinate transformation and stochastic averaging method are our main tools. The stability and bifurcation of the stochastic chemostat model are considered. Some theorems provide sufficient conditions to investigate &nbsp;stochastic stability, $D$-bifurcation and $P$-bifurcation of the &nbsp;model. As a final point, to show the effects of the &nbsp;noise intensity and illustrate our theoretical results, some numerical simulations are presented.A fitted operator method of line scheme for solving two-parameter singularly perturbed parabolic convection-diffusion problems with time delay
https://jmm.guilan.ac.ir/article_6601.html
This paper presents a parameter-uniform numerical scheme for the solution of two-parameter singularly perturbed parabolic convection-diffusion problems with a delay in time. The continuous problem is semi-discretized using the Crank-Nicolson finite difference method in the temporal direction. The resulting differential equation is then discretized on a uniform mesh using the fitted operator finite difference method of line scheme. The method is shown to be accurate in $ O(\left(\Delta t \right)^{2} &nbsp;+ N^{-2}) $, where $ N $ is the number of mesh points in spatial discretization and $ \Delta t $ is the mesh length in temporal discretization. The parameter-uniform convergence of the method is shown by establishing the theoretical error bounds. Finally, the numerical results of the test problems validate the theoretical error bounds.Numerical solution of system of nonlinear Fredholm integro-differential equations using CAS wavelets
https://jmm.guilan.ac.ir/article_6879.html
&nbsp;In this paper, we use the CAS wavelets as basis functions to numerically solve a system of nonlinear Fredholm integro-differential equations. To simplify the problem, we transform the system into a system of algebraic equations using the collocation method and operational matrices. We show the convergence of the presented method and then demonstrate its high accuracy with several illustrative examples. This approach is particularly effective for equations that admit periodic functions because the employed basis CAS functions are inherently periodic. Throughout our numerical examples, we observe that this method provides exact solutions for equations with trigonometric functions at a lower computational cost when compared to other methods.Synchronization of the chaotic fractional-order multi-agent systems under partial contraction theory
https://jmm.guilan.ac.ir/article_6842.html
In this paper, a new synchronization criterion for leader-follower fractional-order chaotic systems using partial contraction theory under an undirected fixed graph is presented. Without analyzing the stability of the error system, first the condition of partial contraction theory for the synchronization of fractional systems is explained, and then the input control vector is designed to apply the condition. An important feature of this control method is the rapid convergence of all agents into a common state. Finally, numerical examples with corresponding simulations are presented to demonstrate the efficiency and performance of the stated method in controlling fractional-order systems. The simulation results show the appropriate design of the proposed control input. &nbsp; &nbsp; &nbsp; &nbsp;Robust computational technique for a class of singularly perturbed nonlinear differential equations with Robin boundary conditions
https://jmm.guilan.ac.ir/article_6648.html
In this article, a class of singularly perturbed nonlinear differential equations with Robin boundary conditions is considered. A numerical method consists of the classical finite difference operator over a Shishkin mesh with two-mesh algorithm is constructed to solve the problems. The method is proved to be first order convergent uniformly with respect to the perturbation parameter. Experiments are carried out for two different types of Robin boundary conditions and Neumann boundary conditions as a special case of Robin boundary conditions.An investigation into the optimal control of the horizontal and vertical incidence of communicable infectious diseases in society
https://jmm.guilan.ac.ir/article_6843.html
This article aims at proposing and developing a three-component mathematical model for susceptible, infected and recovered $(SIR)$ population, under the control of vaccination of the susceptible population and drug therapy (antivirus) of the infected population (patient) in case of an infectious disease. The infectious disease under study can be transmitted through direct contact with an infected person (horizontal transmission) and from parent to child (vertical transmission). We investigate the basic reproduction number of the mathematical model, the existence and local asymptotic stability of both the disease free and endemic equilibrium. Using Pontryagin's minimum principle, we investigate the conditions of reducing the susceptible and infected population and increasing the recovered population based on the use of these two controllers in society. A numerical simulation of the optimal control problem shows, using &nbsp;both controllers is much more effective and leads to a rapid increase in the recovered population and prevents the disease from spreading and becoming an epidemic inthe society.Numerical simulation for unsteady Helmholtz problems of anisotropic FGMs
https://jmm.guilan.ac.ir/article_6854.html
The unsteady Helmholtz type equation of anisotropic functionally graded materials (FGMs) is considered in this study. The study is to find numerical solutions to initial boundary value problems governed by the equation. A combined Laplace and boundary element method is used to solve the problems. The analysis derives a boundary-only integral equation that is used to compute the numerical solutions. The analysis also results in another class of anisotropic FGMs of applications. Some problems are considered. The numerical solutions obtained are accurate and consistent.New algorithms to estimate the real roots of differentiable functions and polynomials on a closed finite interval
https://jmm.guilan.ac.ir/article_6948.html
&nbsp;We propose an algorithm that estimates the real roots of differentiable functions on closed intervals. Then, we extend this algorithm to real differentiable functions that are dominated by a polynomial. For each starting point, our method converges to the nearest root to the right or left hand side of that point. Our algorithm can look for missed roots as well and theoretically it misses no root. Furthermore, we do not find the roots by randomly chosen initial guesses. The iterated sequences in our algorithms converge linearly. Therefore, the rate of convergence can be accelerated considerably to make it comparable to Newton-Raphson and other high-speed methods. We have illustrated our algorithms with some concrete examples. Finally, the pseudo-codes of the related algorithms are presented at the end of this manuscript.Fuzzy approximating functions and its application in solving fuzzy multi-choice linear programming models
https://jmm.guilan.ac.ir/article_6964.html
This article considers a particular type of fuzzy multi-choice linear programming (FMCLP) model in which there are several choices for the fuzzy parameters on the right-hand side (RHS) of problem constraints. We first construct the fuzzy polynomials to solve this model using the fuzzy multi-choice parameters on the RHS of constraints. We construct the fuzzy polynomials by approximating fuzzy functions, including the binary variable approach, Lagrange, and Newton's interpolating polynomials. Also, we use the least squares approach to construct the approximating fuzzy polynomial. Then we solve the resulting model. Finally, we will examine the above techniques in numerical examples.Designing and evaluating an optimal budget plan for parallel network systems through DEA methodology
https://jmm.guilan.ac.ir/article_6979.html
For an optimal system design (OSD), data envelopment analysis (DEA) treats companies as black boxes disregarding their internal processes. Considering the effect of these processes into account companies can upgrade their internal mechanisms of optimal budgeting allocation. The internal processes can be defined as series and parallel networks or a combination of them. In the literature, DEA is utilized as an approach for OSD in order to determine the optimal budget for a company's activities in a system of series network production; but it is shown that this model is not suitable for budgeting parallel systems. To fill this gap, a new model is presented in this paper to evaluate a company's optimal budgeting which has a parallel network system. The presented parallel network OSD via DEA models, allocates the optimal budget to each of the internal processes of the decision-making units (DMU) based on the efficiency of the parallel internal processes. The model, with a limited budget, also is able to identify the amount of budget deficit and congestion. In this regard, two real cases are studied using the suggested model, which is presented in the parallel network OSD via DEA and Forest production in Taiwan. The optimal budget, budget deficit and congestion, and also the advantages of the proposed model are discussed in these examples as well.A new public key cryptography using $M_{q}$ matrix
https://jmm.guilan.ac.ir/article_6980.html
We consider a new class of square Fibonacci $(q+1)\times(q+1)$-matrices in public key cryptography. This extends previous cryptography using generalized Fibonacci matrices. For a given integer $q$, a $(q+1)\times(q+1)$ binary matrix $M_{q}$ is a matrix which nonzero entries are located either on the super diagonal or on the last row of the matrix. In this article, we have proposed a modified public key cryptography using such matrices as key in Hill cipher and key agreement for encryption-decryption of terms of $M_{q}$-matrix. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers needed to be exchanged, which reduces the time complexity as well as the space complexity of the transmission and has a large key space.A robust optimization approach for multi-objective linear programming under uncertainty
https://jmm.guilan.ac.ir/article_6984.html
This paper proposes a new robust optimization approach for solving multi-objective linear programming problems under uncertainty. The uncertainty is assumed to be in the objective function coefficients and the constraint parameters. The proposed approach is based on an alternative model for obtaining robust efficient solutions to the original problem. A numerical example is given to test and illustrate the effectiveness of the proposed approach, and a comparison with a method given in the literature is discussed based on certain performance metrics.A covering-based algorithm for resolution of linear programming problems with max-product bipolar fuzzy relation equation constraints
https://jmm.guilan.ac.ir/article_7065.html
The linear programming problem provided to bipolar fuzzy relation equation constraints is considered in this paper. The structure of bipolar fuzzy relation equation system is studied with the max-product composition. Two new concepts, called covering and irredundant covering, are introduced in the bipolar fuzzy relation equation system. A covering-based sufficient condition is proposed to check its consistency. The relation between two concepts is discussed. Some sufficient conditions are presented to specify one of its optimal solutions or some its optimal components based on the concepts. Also, some covering-based sufficient conditions are given for uniqueness of its optimal solution. These conditions enable us to design some procedures for simplification and reduction of the problem. Moreover, a matrix-based branch-and-bound method is presented to solve the reduced problem. The sufficient conditions and algorithm are illustrated by some numerical examples. The algorithm is compared to existing methods.Estimate of the fractional advection-diffusion equation with a time-fractional term based on the shifted Legendre polynomials
https://jmm.guilan.ac.ir/article_7070.html
In this paper, we present a well-organized strategy to estimate the fractional advection-diffusion equations, which is an important class of equations that arises in many application fields. Thus, &nbsp;Lagrange square interpolation is applied in the discretization of the fractional temporal derivative, and the weighted and shifted Legendre polynomials as operators are exploited to discretize the spatial fractional derivatives of the space-fractional term in multi-termtime fractional advection-diffusion model. The privilege of the numerical method is the orthogonality of Legendre polynomials and its operational matrices which reduces time computation and increases speed. A second-order implicit technique is given, and its stability and convergence are investigated. Finally, we propose three numerical examples to check the validity and numerical results &nbsp; &nbsp;to illustrate the precision and efficiency of the new approach.&nbsp;