Iranian Journal of Numerical Analysis and Optimization
Volume:13 Issue: 3, Summer 2023

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–Weir duality results under appropriate convexity hypotheses. The study concludes with examples designed to validate our findings.

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.

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–Nicolson method for time discretization. Also, the Newton–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.

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.

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− 12 }). Ample numerical results are addressed to con-firm the expected convergence order as well as the accuracy and efficiency of the proposed method.

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’s efficacy and precision, two examples are provided.

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.

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.

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.

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.