Iranian Journal of Numerical Analysis and Optimization
https://ijnao.um.ac.ir/
Iranian Journal of Numerical Analysis and Optimizationendaily1Fri, 01 Sep 2023 00:00:00 +0330Fri, 01 Sep 2023 00:00:00 +0330On optimality and duality for multiobjective interval-valued programming problems with vanishing constraints
https://ijnao.um.ac.ir/article_43541.html
In this study, we explore the theoretical features of a multiobjective interval-valued programming problem with vanishing constraints. In view of this, we have defined a multiobjective interval-valued programming prob-lem with vanishing constraints in which the objective functions are consid-ered to be interval-valued functions, and we define an LU-efficient solution by employing partial ordering relations. Under the assumption of general-ized convexity, we investigate the optimality conditions for a (weakly) LU-efficient solution to a multiobjective interval-valued programming problem with vanishing constraints. Furthermore, we establish Wolfe and Mond&ndash;Weir duality results under appropriate convexity hypotheses. The study concludes with examples designed to validate our findings.Error estimates for approximating fixed points and best proximity points for noncyclic and cyclic contraction mappings
https://ijnao.um.ac.ir/article_43574.html
In this article, we find a priori and a posteriori error estimates of the fixed point for the Picard iteration associated with a noncyclic contraction map, which is defined on a uniformly convex Banach space with a modulus of convexity of power type. As a result, we obtain priori and posteriori error estimates of Zlatanov for approximating the best proximity points ofcyclic contraction maps on this type of space.Solving two-dimensional coupled Burgers equations via a stable hybridized discontinuous Galerkin method
https://ijnao.um.ac.ir/article_43622.html
The purpose of this paper is to design a fully discrete hybridized discon-tinuous Galerkin (HDG) method for solving a system of two-dimensional (2D) coupled Burgers equations over a specified spatial domain. The semi-discrete HDG method is designed for a nonlinear variational formulation on the spatial domain. By exploiting broken Sobolev approximation spaces in the HDG scheme, numerical fluxes are defined properly. It is shown that the proposed method is stable under specific mild conditions on the stabi-lization parameters to solve a well-posed (in the sense of energy method) 2D coupled Burgers equations, which is imposed by Dirichlet boundary conditions. The fully discrete HDG scheme is designed by exploiting the Crank&ndash;Nicolson method for time discretization. Also, the Newton&ndash;Raphson method that has the order of at least two is nominated for solving the obtained nonlinear system of coupled Burgers equations over the rect-angular domain. To reduce the complexity of the proposed method and the size of the linear system, we exploit the Schur complement idea. Numerical results declare that the best possible rates of convergence are achieved for approximate solutions of the 2D coupled Burgers equations and their first-order derivatives. Moreover, the proposed HDG method is examined for two other types of systems, that is, a system with high Reynolds numbers and a system with an unavailable exact solution. The acceptable results of examples show the flexibility of the proposed method in solving various problems.Evaluation of iterative methods for solving nonlinear scalar equations
https://ijnao.um.ac.ir/article_43267.html
This study is aimed at performing a comprehensive numerical evalua-tion of the iterative solution techniques without memory for solving non-linear scalar equations with simple real roots, in order to specify the most efficient and applicable methods for practical purposes. In this regard, the capabilities of the methods for applicable purposes are be evaluated, in which the ability of the methods to solve different types of nonlinear equations is be studied. First, 26 different iterative methods with the best performance are reviewed. These methods are selected based on performing more than 46000 analyses on 166 different available nonlinear solvers. For the easier application of the techniques, consistent mathematical notation is employed to present reviewed approaches. After presenting the diverse methodologies suggested for solving nonlinear equations, the performances of the reviewed methods are evaluated by solving 28 different nonlinear equations. The utilized test functions, which are selected from the re-viewed research works, are solved by all schemes and by assuming different initial guesses. To select the initial guesses, endpoints of five neighboring intervals with different sizes around the root of test functions are used. Therefore, each problem is solved by ten different starting points. In order to calculate novel computational efficiency indices and rank them accu-rately, the results of the obtained solutions are used. These data include the number of iterations, number of function evaluations, and convergence times. In addition, the successful runs for each process are used to rank the evaluated schemes. Although, in general, the choice of the method de-pends on the problem in practice, but in practical applications, especially in engineering, changing the solution method for different problems is not feasible all the time, and accordingly, the findings of the present study can be used as a guide to specify the fastest and most appropriate solution technique for solving nonlinear problems.Effective numerical methods for nonlinear singular two-point boundary value Fredholm integro-differential equations
https://ijnao.um.ac.ir/article_43734.html
We deal with some effective numerical methods for solving a class of nonlinear singular two-point boundary value Fredholm integro-differential equations. Using an appropriate interpolation and a q-order quadrature rule of integration, the original problem will be approximated by the non-linear finite difference equations and so reduced to a nonlinear algebraic system that can be simply implemented. The convergence properties of the proposed method are discussed, and it is proved that its convergence order will be of O(hmin{ 72 ,q&minus; 12 }). Ample numerical results are addressed to con-firm the expected convergence order as well as the accuracy and efficiency of the proposed method.An optimal control approach for solving an inverse heat source problem applying shifted Legendre polynomials
https://ijnao.um.ac.ir/article_43685.html
This study addresses the inverse issue of identifying the space-dependent heat source of the heat equation, which is stated using the optimal con-trol framework. For the numerical solution of this class of problems, an approach based on shifted Legendre polynomials and the associated oper-ational matrix is presented. The approach turns the primary problem into the solution of a system of nonlinear algebraic equations. To do this, the temperature and heat source variables are enlarged in terms of the shifted Legendre polynomials with unknown coefficients employed in the objectivefunction, inverse problem, and initial and Neumann boundary conditions. When paired with their operational matrix, these basis functions provide a quadratic optimization problem with linear constraints, which is then solved using the Lagrange multipliers approach. To assess the method&rsquo;s efficacy and precision, two examples are provided.Numerical non-linear model solutions for the hepatitis C transmission between people and medical equipment using Jacobi wavelets method
https://ijnao.um.ac.ir/article_43410.html
In this work, we present a new mathematical model for the spread of hepatitis C disease in two populations: the human population and the medical equipment population. Then, we apply the Jacobi wavelets method combined with the decoupling and quasilinearization technique to solve this set of nonlinear differential equations for numerical simulation.Analysis and optimal control of a fractional MSD model
https://ijnao.um.ac.ir/article_43688.html
In this research, we aim to analyze a mathematical model of Maize streak virus disease as a problem of fractional optimal control. For dynamical analysis, the boundedness and uniqueness of solutions have been investi-gated and proven. Also, the basic reproduction number is obtained, and local stability conditions are given for the equilibrium points of the model. Then, an optimal control strategy is proposed for the purpose of examining the best strategy to fight the maize streak disease. We solve the fractional optimal control problem by a forward-backward sweep iterative algorithm. In this algorithm, the state variable is obtained in a forward and co-state variable by a backward method where an explicit Runge-Kutta method is used to solve differential equations arising from fractional optimal control problems. Some comparative results are presented in order to verify the model and show the efficacy of the fractional optimal control treatments.A generalized form of the parametric spline methods of degree (2k + 1) for solving a variety of two-point boundary value problems
https://ijnao.um.ac.ir/article_43684.html
In this paper, a high-order accuracy method is developed for finding the approximate solution of two-point boundary value problems (BVPs). The present approach is based on a special algorithm, taken from Pascal's triangle, for obtaining a generalized form of the parametric splines of degree (2k+1), k=1,2,..., which has a lower computational cost and gives a better approximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm.The approximate solution converges to the exact solution of order O(h^{4k}), where k is a quantity related to the degree of parametric splines and the number of matrix bands that are applied in this paper. Some examples are given to illustrate the applicability of the method and we compared the computed results with other existing known methods. It is observed that our approach produced better results.Cubic hat-functions approximation for linear and nonlinear fractional integral-differential equations with weakly singular kernels
https://ijnao.um.ac.ir/article_43722.html
In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures.Nearest fuzzy number of type L-R to an arbitrary fuzzy number with applications to fuzzy linear system
https://ijnao.um.ac.ir/article_43823.html
The fuzzy operations on fuzzy numbers of type L-R are much easier than general fuzzy numbers. It would be interesting to approximate a fuzzy number by a fuzzy number of type L-R. In this paper, we state and prove two significant application inequalities in the monotonic functions set. These inequalities show that under a condition, the nearest fuzzy number of type L-R to an arbitrary fuzzy number exists and is unique. After that, the nearest fuzzy number of type L-R can be obtained by solving a linear system. Note that the trapezoidal fuzzy numbers are a particular case of the fuzzy numbers of type L-R. The proposed method can represent the nearest trapezoidal fuzzy number to a given fuzzy number. Finally, to approximate fuzzy solutions of a fuzzy linear system, we apply our idea to construct a framework to find solutions of crisp linear systems instead of the fuzzy linear system. The crisp linear systems give the nearest fuzzy numbers of type L-R to fuzzy solutions of a fuzzy linear system. The proposed method is illustrated with some examples.A novel integral transform operator and its applications
https://ijnao.um.ac.ir/article_43780.html
The proposed study is focused to introduce a novel integral transform op-erator, called Generalized Bivariate (GB) transform. The proposed trans-form includes the features of the recently introduced Shehu transform, ARA transform, and Formable transform. It expands the repertoire of existing Laplace-type bivariate transforms. The primary focus of the present work is to elaborate fashionable properties and convolution theorems for the proposed transform operator. The existence, inversion, and duality of the proposed transform have been established with other existing transforms. Implementation of the proposed transform has been demonstrated by ap-plying it to different types of differential and integral equations. It validates the potential and trustworthiness of the GB transform as a mathematical tool. Furthermore, weighted norm inequalities for integral convolutions have been constructed for the proposed transform operator.A shifted fractional-order Hahn functions tau method for time-fractional PDE with non-smooth solution
https://ijnao.um.ac.ir/article_43790.html
In this paper, a new orthogonal system of non-polynomial basis functions is introduced and used to solve a class of time-fractional partial differential equations that have non-smooth solutions. In fact, unlike polynomial bases, such basis functions have a singularity and are constructed with a fractional variable change on Hahn polynomials. This feature leads to obtaining more accurate spectral approximations than polynomial bases. The introduced method is a spectral method that uses the operational matrix of fractional order integral of fractional-order shifted Hahn functions and finally converts the equation into a matrix equation system. In the introduced method, no collocation method has been used and initial and boundary conditions are applied during the execution of the method. Error and convergence analysis of the numerical method has been investigated in a Sobolev space. Finally, some numerical experiments are considered in the form of tables and figures for demonstrating the accuracy and capability of the proposed method.Collocation-based numerical method for multi-order fractional integro-differential equations
https://ijnao.um.ac.ir/article_43824.html
In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equationsusing Caputo sense. We obtain the integral form of the problem and transform it into a system of linear algebraic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method's uniqueness as well as the convergence of the method. Numerical examples show that the developed method is efficient in problem-solving and competes favorably with the existing method.Optimal control analysis for modeling HIV transmission
https://ijnao.um.ac.ir/article_43825.html
In this study, a modi ed model of HIV with therapeutic and preventive controls is developed. moreover, a simple evaluation of the optimal control problem is investigated. We construct the Hamiltonian function by way of integrating the Pontryagin's maximal principle to achieve the point-wise optimal solution. The effects obtained from the version analysis strengthen public health education to an conscious population, PrEP for early activation of HIV infection prevention, and early treatment with artwork for safe life after HIV infection. moreover, numerical simulations had been done using the MATLAB platform to illustrate the qualitative conduct of the HIV infection. In the end, we received that adhering to art, protective prone people the usage of PrEP along with different prevention control are safer controlmeasures.A robust uniformly convergent scheme for two parameters singularly perturbed parabolic problems with time delay
https://ijnao.um.ac.ir/article_43907.html
A singularly perturbed time delay parabolic problem with two small parameters is considered. The paper develops a finite difference scheme that is exponentially fitted on a uniform mesh in the spatial direction and uses the implicit-Euler method to discretize the time derivative in the temporal direction in order to obtain a better numerical approximation to the solutions of this class of problems. We establish the parameter-uniform error estimate and discuss the stability of the suggested approach. In order to demonstrate the improvement in terms of accuracy, numerical results are also shown to validate the theoretical conclusions and are contrasted with the current hybrid scheme.Numerical Study of sine-Gordon Equations using Bessel Collocation Method
https://ijnao.um.ac.ir/article_43967.html
The non-linear space-time dynamics have been discussed in terms of a hyperbolic equation known as the sine Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel polynomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously differentiable function. The system of equations involves one linear and the other non-linear diffusion equation. The convergence of the present technique has been discussed through absolute error, $L_2$-norm, and $L_{\infty}$-norm. The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form.Improving the Performance of the FCM Algorithm in Clustering using the DBSCAN Algorithm
https://ijnao.um.ac.ir/article_44117.html
The $FCM$ algorithm is one of the most famous fuzzy clustering algorithms, but it gets stuck in local optima. In addition, this algorithm requires a number of clusters. Also, the $ DBSCAN $ algorithm, which is a density-based clustering algorithm, unlike the $ FCM $ algorithm should not be pre-numbered. If the clusters are specific and depend on the number of clusters, it can determine the number of clusters. Another advantage of the $ DBSCAN $ clustering algorithm over $FCM$ is its ability to cluster data of different shapes. In this paper, in order to overcome these limitations, a hybrid approach for clustering is proposed which uses $ FCM $ and $ DBSCAN $ algorithms. In this method, the optimal number of clusters and the optimal location for the centers of the clusters are determined based on the changes that take place according to the data set in three phases by predicting the possibility of the problems stated in the $FCM$ algorithm. With this improvement, the values of none of the initial parameters of $FCM$ algorithm are random, and in the first phase, it has been tried to replace these random values to the optimal in the $FCM$ algorithm, which has a significant effect on the convergence of the algorithm because it helps to reduce iterations. The proposed method has been examined on the Iris flower and compared the results with the basic $FCM$ algorithm and another algorithm. Results show the better performance of the proposed method.Numerical solution of fractional Bagley-Torvik equation using Lucas polynomials
https://ijnao.um.ac.ir/article_44136.html
The aim of this article is to present a new method based on Lucas polynomials and residual error function for numerical solution of the fractional Bagley-Torvik equation. Here, the approximate solution is expanded as a linear combination of Lucas polynomials, and by using collocation method the original problem reduces to a system of linear equations. So, the approximate solution to the problem could be found by solving this system. Then, by using the residual error function and approximating the error function by utilizing the same approach, we achieve more accurate results. In addition, convergence analysis of the method is investigated. Numerical examples demonstrate the validity and applicability of the method.Singularly perturbed two-point boundary value problem by applying exponential fitted finite difference method
https://ijnao.um.ac.ir/article_44171.html
The present study addresses an exponentially fitted finite difference method to obtain the solution of singularly perturbed two point boundary value problems(BVPs) having boundary layer at one end (left or right) point on uniform mesh . A fitting factor is introduced in the derived scheme using the theory of singular perturbations. Thomas algorithm is employed to solve the resulting tri-diagonal system of equations. Convergence of the presented method are investigated. Several model example problems are solved using the proposed method. The results are presented in terms of maximum absolute errors which demonstrate the accuracy and efficiency of the method. It is observed that the proposed method is capable of producing highly accurate results with minimal computational effort for a fixed value of step size h, when perturbation parameter tends to zero. From the graphs, we also observed that numerical solution approximate exact solution very well in the boundary layers for smaller value of 𝞮.Efficient numerical methods on modified graded mesh for singularly perturbed parabolic problem with time delay
https://ijnao.um.ac.ir/article_44180.html
In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. The proposed numerical approach comprises of an upwind difference scheme with modified graded mesh in the spatial direction and a backward Euler scheme on uniform mesh in the temporal direction. In order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. The proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and first-order in the temporal variable. Finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings.Chebyshev wavelet-based method for solving various stochastic optimal control problems and its application in finance
https://ijnao.um.ac.ir/article_44251.html
In this paper, a computational method based on parameterizing state and control variables is presented for solvingStochastic Optimal Control(SOC) problems. By using Chebyshev wavelets with unknown coefficients, state, and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting SO problem is approximated by the Sample Average Approximation (SAA), thereby the problem can be solved by optimization methods, more easily. For facilitating and guaranteeing convergence of the presented method a new theorem is proved. Finally, the proposed method is implemented,&nbsp; based on a newly designed algorithm for solving one of the well-known problems in mathematical finance, the Merton portfolio allocation problem in finite horizon. The simulation results illustrate the improvement of the constructed portfolio return. A new numerical approach to the solution of the non-linear Kawahara equation by using combined Taylor-Dickson approximation
https://ijnao.um.ac.ir/article_44278.html
This article presents a novel numerical approach to the solution of the nonlinear Kawahara equation. The desired approximations are obtained from the combination of Dickson polynomials and Taylor&rsquo;s expansion. The combined approach is based on Taylor's expansion for discretizing the time derivative and Dickson polynomials for space derivatives. The problem will be converted into a system of linear algebraic equations for each time step via some suitable collocation points. Error estimation is presented after obtaining the approximate solution. The newly proposed technique is compared with some existing numerical methods in order to show the applicability, accuracy, and efficacy of the method. Two problems are solved to demonstrate the method's power and effect, and the results are presented in the form of a table and graphics.