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Omid Soleymani-Fard

Associate Professor of Applied Mathematics

Selected Publications

Nosratipour, H., Fard, O.S., Borzabadi, A.H. Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai–Borwein gradient method (2018) International Journal of Computer Mathematics, 95 (9), pp. 1858-1873.

DOI: 10.1080/00207160.2017.1343472

An adaptive nonmonotone spectral gradient method for the solution of distributed optimal control problem (OCP) for the viscous Burgers equation is presented in a black-box framework. Regarding the implicit function theorem, the OCP is transformed into an unconstrained nonlinear optimization problem (UNOP). For solving UNOP, an adaptive nonmonotone Barzilai–Borwein gradient method is proposed in which to make a globalization strategy, first an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity is presented and then is incorporated into an inexact line search approach to construct a more relaxed line search procedure. Also an adjoint technique is used to effectively evaluate the gradient. The low memory requirement and the guaranteed convergence property make the proposed method quite useful for large-scale OCPs. The efficiency of the presented method is supported by numerical experiments. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.

AUTHOR KEYWORDS: 65K10; 90C30; Distributed optimal control; finite elements method; implicit function theorem; nonmonotone Barzilai–Borwein method; viscous Burgers equation
INDEX KEYWORDS: Gradient methods; Nonlinear programming; Numerical methods; Optimal control systems; Partial differential equations, Distributed optimal control; Distributed optimal control problems; Guaranteed convergence; Implicit function theorem; Non-linear optimization problems; Nonmonotone; Spectral gradient method; Viscous Burgers equation, Finite element method
PUBLISHER: Taylor and Francis Ltd.

Abchouyeh, M.A., Mohebbi, R., Fard, O.S. Lattice Boltzmann simulation of nanofluid natural convection heat transfer in a channel with a sinusoidal obstacle (2018) International Journal of Modern Physics C, 29 (9), art. no. 1850079, .

DOI: 10.1142/S0129183118500791

The aim of this work is to conduct numerical study of fluid flow and natural convection heat transfer by utilizing the nanofluid in a two-dimensional horizontal channel consisting of a sinusoidal obstacle by lattice Boltzmann method (LBM). The fluid in the channel is a water-based nanofluid containing Cuo nanoparticles. Thermal conductivity and nanofluid's viscosity are calculated by Patel and Brinkman models, respectively. A wide range of parameters such as the Reynolds number (Re=100-400) and the solid volume fraction ranging (φ=0-0.05) at different non-dimensional amplitude of the wavy wall of the sinusoidal obstacle (A=0-20) on the streamlines and temperature contours are investigated in the present study. In addition, the local and average Nusselt numbers are illustrated on lower wall of the channel. The sensitivity to the resolution and representation of the sinusoidal obstacle's shape on flow field and heat transfer by LBM simulations are the main interest and innovation of this study. The results showed that increasing the solid volume fraction φ and Reynolds number Re leads to increase the average Nusselt numbers. The maximum average Nusselt number occurs when the Reynolds number and solid volume fraction are maximum and amplitude of the wavy wall is minimum. Also, by decreasing the A, the vortex shedding forms up at higher Reynolds number in the wake region downstream of the obstacle. © 2018 World Scientific Publishing Company.

AUTHOR KEYWORDS: heat transfer; laminar channel flow; lattice Boltzmann method; nanofluid; Sinusoidal obstacle
PUBLISHER: World Scientific Publishing Co. Pte Ltd

Soolaki, J., Fard, O.S., Borzabadi, A.H. A necessary condition of Pontryagin type for fuzzy control systems (2018) Computational and Applied Mathematics, 37 (2), pp. 1263-1278.

DOI: 10.1007/s40314-016-0399-4

In the present paper, we prove necessary optimality conditions of Pontryagin type for a class of fuzzy optimal control problems. The new results are illustrated by computing the extremals of two fuzzy optimal control systems, which improve recent results of Najariyan and Farahi. © 2016, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

AUTHOR KEYWORDS: Fuzzy Hamiltonian function; Fuzzy Pontryagin maximum principle; Fuzzy variational problems
PUBLISHER: Springer Science and Business Media, LLC

Fard, O.S., Soolaki, J., Almeida, R. A formulation of Noether’s theorem for fuzzy problems of the calculus of variations (2018) Afrika Matematika, 29 (1-2), pp. 33-46.

DOI: 10.1007/s13370-017-0524-6

The theory of the calculus of variations for fuzzy systems was recently initiated in Farhadinia (Inf Sci 181:1348–1357, 2011), with the proof of the fuzzy Euler–Lagrange equation. Using fuzzy Euler–Lagrange equation, we obtain here a Noether–like theorem for fuzzy variational problems. © 2017, African Mathematical Union and Springer-Verlag GmbH Deutschland.

AUTHOR KEYWORDS: Fuzzy conservation law; Fuzzy Euler–Lagrange conditions; Fuzzy Noether’s theorem
PUBLISHER: Springer Verlag

Fard, O.S., Soolaki, J., Torres, D.F.M. A necessary condition of pontryagin type for fuzzy fractional optimal control problems (2018) Discrete and Continuous Dynamical Systems - Series S, 11 (1), pp. 59-70.

DOI: 10.3934/dcdss.2018004

We prove necessary optimality conditions of Pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. The new results are illustrated by computing the extremals of three fuzzy optimal control systems, which improve recent results of Najariyan and Farahi. © 2018, American Institute of Mathematical Sciences. All rights reserved.

AUTHOR KEYWORDS: Fractional calculus; Fuzzy control systems; Fuzzy fractional Hamiltonian function; Fuzzy optimal control; Pontryagin maximum principle
PUBLISHER: American Institute of Mathematical Sciences

Fard, O.S., Ramezanzadeh, M. Optimality conditions for optimal control under parametric representation of interval uncertainty (2018) International Journal of Systems, Control and Communications, 9 (3), pp. 210-229.

DOI: 10.1504/IJSCC.2018.093397

The purpose of this paper is to offer a new approach that enables the decision maker to investigate optimal control problems under uncertainty (undetermined) processes. To this end, using the novel parametric representations of interval quantities as in Ramezanzadeh et al. (2015) the approach is developed to find a candidate for the solution of interval optimal control problem. Furthermore, necessary and sufficient optimality conditions are provided which guarantee the candidate solution to be optimal. Finally, some examples are given to show the main results, more specifically, a discussion on the interval optimal control governed by half-model of a car is also presented. Copyright © 2018 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Interval number; Interval optimal control problem; Interval-valued function; Necessary condition; Sufficient condition
PUBLISHER: Inderscience Publishers

Fard, O.S., Heidari, M., Borzabadi, A.H. Fuzzy Taylor formula: An approach via fuzzification of the derivative and integral operators (2018) Fuzzy Sets and Systems, . Article in Press.

DOI: 10.1016/j.fss.2018.04.007

Using the concepts of derivative and integral of fuzzy functions in the sense of fuzzification, this paper is devoted to studying a new version fuzzy fundamental theorem of calculus as well as a new variant of fuzzy Taylor formula with an integral remainder in the univariate and multivariate cases. Here, the fuzzification of derivative and integral means using Zadeh's extension principle on the corresponding classical operators. Indeed, by presenting appropriating symbols, it is shown in this work, contrary to what was supposed to be, Zadeh's extension principle is capable of making the ability to compute and introduce many quantities and concepts in univariate and multivariate calculus such as integral, derivative, Taylor expansion and etc. © 2018 Elsevier B.V.

AUTHOR KEYWORDS: Fuzzy derivative and integral; Fuzzy fundamental theorem of calculus; Fuzzy Taylor formula; Zadeh's extension
INDEX KEYWORDS: Fuzzy set theory, Fundamental theorem of calculus; Fuzzy derivatives; Fuzzy function; Integral operators; Taylor expansions; Taylor formula; Zadeh's extension; Zadeh's extension principles, Calculations
PUBLISHER: Elsevier B.V.

Nosratipour, H., Hashemi Borzabadi, A., Solaymani Fard, O. On the nonmonotonicity degree of nonmonotone line searches (2017) Calcolo, 54 (4), pp. 1217-1242.

DOI: 10.1007/s10092-017-0226-3

The nonmonotone globalization technique is useful in difficult nonlinear problems, because of the fact that it may help escaping from steep sided valleys and may improve both the possibility of finding the global optimum and the rate of convergence. This paper discusses the nonmonotonicity degree of nonmonotone line searches for the unconstrained optimization. Specifically, we analyze some popular nonmonotone line search methods and explore, from a computational point of view, the relations between the efficiency of a nonmonotone line search and its nonmonotonicity degree. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm. Hence in an attempt to control the nonmonotonicity degree, two adaptive nonmonotone rules based on the morphology of the objective function are proposed. The global convergence and the convergence rate of the proposed methods are analysed under mild assumptions. Numerical experiments are made on a set of unconstrained optimization test problems of the CUTEr (Gould et al. in ACM Trans Math Softw 29:373–394, 2003) collection. The performance data are first analysed through the performance profile of Dolan and Moré (Math Program 91:201–213, 2002). In the second kind of analyse, the performance data are analysed in terms of increasing dimension of the test problems. © 2017, Springer-Verlag Italia.

AUTHOR KEYWORDS: Global convergence; Nonmonotone line search; Nonmonotonicity degree; Unconstrained optimization
PUBLISHER: Springer-Verlag Italia s.r.l.

Fard, O.S., Bidgoli, T. Existence and uniqueness of solutions to the second order fuzzy dynamic equations on time scales (2017) Advances in Difference Equations, 2017 (1), art. no. 231, .

DOI: 10.1186/s13662-017-1296-x

In this paper, we introduce a new metric space to study the existence and uniqueness of solutions to second order fuzzy dynamic equations on time scales. In this regard, we use Banach’s fixed point theorem to prove this result. Also, we see that this metric guarantees an elegant and easier proof for the existence of solutions to second order fuzzy dynamic equations on time scales. © 2017, The Author(s).

AUTHOR KEYWORDS: Banach’s fixed point theorem; metric space; second order fuzzy dynamic equations; time scales
PUBLISHER: Springer International Publishing

Nosratipour, H., Fard, O.S., Borzabadi, A.H., Sarani, F. Stable equilibrium configuration of two bar truss by an efficient nonmonotone global Barzilai–Borwein gradient method in a fuzzy environment (2017) Afrika Matematika, 28 (3-4), pp. 333-356.

DOI: 10.1007/s13370-016-0451-y

The stable equilibrium configuration of structures is a main goal in structural optimization. This goal may be achieved through minimizing the potential energy function. In the real world, sometimes, the input data and parameters of structural engineering design problems may be considered as fuzzy numbers which lead us to develop structural optimization methods in a fuzzy environment. In this regard, the present paper is intended to propose a fuzzy optimization scheme according to the nonmonotone globalization technique, the Barzilai–Borwein (BB) gradient method and the generalized Hukuhara differentiability (gH-differentiability). In fact, using the best benefits of BB-like methods i.e., simplicity, efficiency and low memory requirements, a modified global Barzilai–Borwein (GBB) gradient method is proposed for obtaining a non-dominated solution of the unconstrained fuzzy optimization related to the two bar asymmetric shallow truss in a fuzzy environment. The global convergence to first-order stationary points is also proved and the R-linear convergence rate is established under suitable assumptions. Furthermore, some numerical examples are given to illustrate the main results. © 2016, African Mathematical Union and Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Barzilai–Borwein gradient method; Generalized Hukuhara differentiability; Nonmonotone line search; Truss; Unconstrained fuzzy optimization
PUBLISHER: Springer Verlag

Nosratipour, H., Fard, O.S., Borzabadi, A.H. An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization (2017) Optimization, 66 (4), pp. 641-655.

DOI: 10.1080/02331934.2017.1287702

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

AUTHOR KEYWORDS: Barzilai–Borwein gradient method; global convergence; nonmonotone line search; Unconstrained optimization
PUBLISHER: Taylor and Francis Ltd.

Fard, O.S., Ramezanzadeh, M. On fuzzy portfolio selection problems: A parametric representation approach (2017) Complexity, 2017, art. no. 9317924, .

DOI: 10.1155/2017/9317924

Fuzzy portfolio selection problem is a major issue in the financial field and a special case of constrained fuzzy-valued optimization problems (CFOPs). In this respect, the present paper aims to investigate the CFOP with regard to the features of the parametric representation of fuzzy numbers named as convex constraint function (CCF) which is proposed by Chalco-Cano et al. in 2014. Furthermore, relying on this parametric representation, some proper conditions are provided for the existence of solutions to a CFOP. To this end, by the increasing representation of CCF, the main problem is converted to a parametric multiobjective programming problem and some solution concepts from a similar framework in the multiobjective programming are proposed for the CFOP. Eventually to illustrate the proposed results, the fuzzy portfolio selection problem is discussed. © 2017 Omid Solaymani Fard and Mohadeseh Ramezanzadeh.

PUBLISHER: Hindawi Limited

Bejandi, P.R., Fard, O.S., Haghighatkhah, H., Zadeh, H.S. Segmentation of pelvic bone in STIR MRI with active contour (2016) 2016 24th Iranian Conference on Electrical Engineering, ICEE 2016, art. no. 7585727, pp. 1330-1335.

DOI: 10.1109/IranianCEE.2016.7585727

Intensity inhomogeneities often occur in STIR Sequence MR images. For this reason, segmentation of these images is challenging. In STIR images from pelvis, signal of bone marrow is similar to subcutaneous fat and adjacent muscular structures so bone marrow pathologic lesions could be hidden. In this paper, we propose a method for image segmentation of pelvic bones in STIR MR images. This method is an extension of active contour methods. In this method, radius of localization is adaptively determined for each point of the contour. We combine this local structure with RSF model and we use the edge information to reach an appropriate segmentation. The proposed model outperforms RSF model in segmenting the images that have intensity inhomogeneities. We apply our algorithm on more than 120 MRI slices of 6 subjects to show its accuracy. Finally, we compare the result of the proposed model with those of two other active contours models. © 2016 IEEE.

AUTHOR KEYWORDS: Active contour; Segmentation; STIR MRI
INDEX KEYWORDS: Bone; Magnetic resonance imaging, Active contour method; Active contours; Active contours model; Edge information; Inhomogeneities; Local structure; STIR-sequence; Subcutaneous fat, Image segmentation
PUBLISHER: Institute of Electrical and Electronics Engineers Inc.

Heidari, M., Zadeh, M.R., Fard, O.S., Borzabadi, A.H. On Unconstrained Fuzzy-Valued Optimization Problems (2016) International Journal of Fuzzy Systems, 18 (2), pp. 270-283.

DOI: 10.1007/s40815-016-0165-1

In this paper, based on the parametric representation of fuzzy-valued function, an unconstrained fuzzy-valued optimization problem is converted to a general unconstrained optimization problem. Two solutions for the unconstrained fuzzy-valued optimization problem are proposed which are parallel to the concept of efficient solution in the case of multi-objective programming problem. Also it is proven that the optimal solution of its corresponding general unconstrained optimization problem is the optimal solution of original problem. Finally, some numerical examples are given to illustrate the discussed suitability scheme. In the first example, the various solutions are discussed in details and a classic motivating example in the mathematical study of variational inequalities, namely the Elliptic Obstacle Problem, is expressed in the second one. Non-convex Fuzzy Rosenbrock Function have been solved in third example. © 2016 Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Fuzzy matrix; Fuzzy optimization; Fuzzy-valued convex function
INDEX KEYWORDS: Functions; Optimal systems; Optimization; Variational techniques, Convex functions; Elliptic obstacle problem; Fuzzy matrix; Fuzzy optimization; Multi-objective programming problem; Parametric representations; Unconstrained optimization problems; Variational inequalities, Multiobjective optimization
PUBLISHER: Springer Berlin Heidelberg

Fard, O.S., Torres, D.F.M., Zadeh, M.R. A hukuhara approach to the study of hybrid fuzzy systems on time scales (2016) Applicable Analysis and Discrete Mathematics, 10 (1), pp. 152-167.

DOI: 10.2298/AADM160311004F

We introduce a new approach to study the practical stability of hybrid fuzzy systems on time scales in the Lyapunov sense. Our method is based on the delta-Hukuhara derivative for fuzzy valued functions and allow us to obtain new interesting stability criteria. We also show the validity of the results of M. Sambandham: Hybrid fuzzy systems on time scales, Dynam. Systems Appl., 12 (1-2) (2003), 217-227, by embedding the space of all fuzzy subsets into a suitable Banach space.

AUTHOR KEYWORDS: Fuzzy dynamical systems; Fuzzy hybrid systems; Practical stability; Time scales
PUBLISHER: University of Belgrade

Soolaki, J., Fard, O.S., Borzabadi, A.H. Fuzzy fractional variational problems under Jumarie's Riemann-Liouville H-differentiability (2016) 4th Iranian Joint Congress on Fuzzy and Intelligent Systems, CFIS 2015, art. no. 7391703, .

DOI: 10.1109/CFIS.2015.7391703

We prove necessary optimality conditions of Euler-Lagrange type for both fuzzy unconstrained and constrained fractional variational problems where the fuzzy fractional derivative is described in the Jumarie's Riemann-Liouville sense. The new results are illustrated by computing the extremals of two fuzzy variational problems. © 2015 IEEE.

INDEX KEYWORDS: Intelligent systems, Euler-Lagrange; Fractional derivatives; Necessary optimality condition; New results; Riemann-liouville h-differentiability; Riemann-Liouville sense; Variational problems, Variational techniques
PUBLISHER: Institute of Electrical and Electronics Engineers Inc.

Soolaki, J., Fard, O.S., Borzabadi, A.H. Generalized Euler–Lagrange equations for fuzzy fractional variational calculus (2016) Mathematical Communications, 21 (2), pp. 199-218.

This paper presents the necessary optimality conditions of Euler–Lagrange type for variational problems with natural boundary conditions and problems with holonomic constraints where the fuzzy fractional derivative is described in the combined Caputo sense. The new results are illustrated by computing the extremals of two fuzzy variational problems. © 2016 Department of Mathematics, University of Osijek.

AUTHOR KEYWORDS: Fuzzy fractional euler–Lagrange equations; Fuzzy fractional holonomic problem; Fuzzy fractional variational problem
PUBLISHER: Udruga Matematicara Osijek

Ghasemian, J., Farnoosh, R., Fard, O.S. The Tikhonov-type regularization approach in multivariate fuzzy nonparametric regression (2015) Journal of Advanced Research in Dynamical and Control Systems, 7 (3), pp. 105-118.

This paper deals with Tikhonov-type regularization of fuzzy nonparametric regression models using quasi-Gaussian and quadratic fuzzy numbers. Implementing Tikhonov regularization in the Lagragian dual space, this estimation method is obtained. The distance measure for fuzzy numbers that suggested by Xu [27] is used and the local linear smoothing technique with the k–fold cross-validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method. © 2015 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Fuzzy nonparametric regression; Fuzzy regression; Local linear smoothing; Tikhonov regularization
PUBLISHER: Institute of Advanced Scientific Research, Inc.

Fard, O.S., Salehi, M. A survey on fuzzy fractional variational problems (2014) Journal of Computational and Applied Mathematics, 271, pp. 71-82.

DOI: 10.1016/j.cam.2014.03.019

This paper focuses on introducing and studying generalized fuzzy Euler-Lagrange equation and fuzzy isoperimetric problem. According to the concept of Caputo-type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability, we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional Euler-Lagrange equation for both constrained and unconstrained fuzzy fractional variational problems. The fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy Euler-Lagrange. Finally, two examples are given to describe the proposed approach. © 2014 Elsevier B.V. All rights reserved.

AUTHOR KEYWORDS: Caputo-type fuzzy fractional derivative; Euler-Lagrange equations; Isoperimetric problem
INDEX KEYWORDS: Equations of motion, Differentiability; Euler-Lagrange; Euler-Lagrange equations; Fractional calculus; Fractional derivatives; Fractional euler-lagrange equations; Isoperimetric problem; Variational problems, Variational techniques
PUBLISHER: Elsevier

Fard, O.S., Salahshour, S., Karimi Dizicheh, A. Erratum: Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients (Information Sciences (2011) 181 (4765-4779)) (2014) Information Sciences, 255, p. 221.

DOI: 10.1016/j.ins.2013.07.028

Fard, O.S., Bidgoli, T.A. Calculus of fuzzy functions on time scales (I) (2014) Soft Computing, 19 (2), pp. 293-305.

DOI: 10.1007/s00500-014-1252-6

This paper deals with fuzzy-number-valued functions on time scales, and more particularly focuses on a class of new derivative and Henstock–Kurzweil integral of such fuzzy functions. Furthermore, the corresponding fundamental properties of the introduced derivative and integral are studied and discussed. © 2014, Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Fuzzy Henstock–Kurzweil delta integral; Fuzzy-number-valued functions; Hukuhara–Hilger derivative; Time scale
INDEX KEYWORDS: Calculations; Fuzzy rules; Fuzzy set theory; Fuzzy sets; Time measurement, Delta integrals; Fundamental properties; Fuzzy function; Fuzzy-number-valued functions; Time-scales, Integral equations
PUBLISHER: Springer Verlag

Farnoosh, R., Ghasemian, J., Fard, O.S. Integrating ridge-type regularization in fuzzy nonlinear regression (2012) Computational and Applied Mathematics, 31 (2), pp. 323-338.

DOI: 10.1590/S1807-03022012000200006

In this paper, we deal with the ridge-type estimator for fuzzy nonlinear regression modelsusingfuzzynumbersandGaussianbasisfunctions. Shrinkageregularizationmethodsare used in linear and nonlinear regression models to yield consistent estimators. Here, we propose a weighted ridge penalty on a fuzzy nonlinear regression model, then select the number of basis functions and smoothing parameter. In order to select tuning parameters in the regularization method, we use the Hausdorff distance for fuzzy numbers which was first suggested by Dubois and Prade [8]. The cross-validation procedure for selecting the optimal value of the smoothing parameterandthenumberofbasisfunctionsarefuzzifiedtofitthepresentedmodel. Thesimulation results show that our fuzzy nonlinear modelling performs well in various situations. © 2012 SBMAC.

AUTHOR KEYWORDS: Basis expansion; Fuzzynonlinear regression; Gaussian; MonteCarlo method; Regularization method

Farnoosh, R., Ghasemian, J., Solaymani Fard, O. A modification on ridge estimation for fuzzy nonparametric regression (2012) Iranian Journal of Fuzzy Systems, 9 (2), pp. 75-88.

This paper deals with ridge estimation of fuzzy nonparametric regression models using triangular fuzzy numbers. This estimation method is obtained by implementing ridge regression learning algorithm in the Lagrangian dual space. The distance measure for fuzzy numbers that suggested by Diamond is used and the local linear smoothing technique with the cross validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method.

AUTHOR KEYWORDS: Fuzzy nonparametric regression; Fuzzy regression; Local linear smoothing; Ridge estimation

Borzabadi, A.H., Fard, O.S., Mehne, H.H. A hybrid algorithm for approximate optimal control of nonlinear Fredholm integral equations (2012) International Journal of Computer Mathematics, 89 (16), pp. 2259-2273.

DOI: 10.1080/00207160.2012.705279

In this paper, a novel hybrid method based on two approaches, evolutionary algorithms and an iterative scheme, for obtaining the approximate solution of optimal control governed by nonlinear Fredholm integral equations is presented. By converting the problem to a discretized form, it is considered as a quasi-assignment problem and then an iterative method is applied to find an approximate solution for the discretized form of the integral equation. An analysis for convergence of the proposed iterative method and its implementation for numerical examples are also given. © 2012 Copyright Taylor and Francis Group, LLC.

AUTHOR KEYWORDS: Approximation; Discretization; Evolutionary algorithm; Fredholm integral equation; Iterative method; Optimal control
INDEX KEYWORDS: Approximate solution; Approximation; Discretizations; Fredholm integral equations; Iterative schemes; Nonlinear Fredholm integral equation; Numerical example; Optimal controls, Control; Evolutionary algorithms; Integral equations; Numerical methods, Iterative methods

Fard, O.S., Zadeh, M.S. Note on "Necessary optimality conditions for fuzzy variational problems" (2012) Journal of Advanced Research in Dynamical and Control Systems, 4 (3), pp. 1-9.

Farhadinia in [6] studied the fuzzy Euler-Lagrange conditions for fuzzy constrained and unconstrained variational problems based on Buckley and Feuring's derivative [3]. The main goal of this paper is to show that the concept of α-differentiability [11] allows to obtain an extended fuzzy Euler-Lagrange condition. © 2011 Institute of Advanced Scientic Research.

AUTHOR KEYWORDS: a-Differentiability; Fuzzy Euler-Lagrange condition; Fuzzy variational problem

Solaymani Fard, O., Esfahani, A., Vahidian Kamyad, A. On solution of a class of fuzzy BVPs (2012) Iranian Journal of Fuzzy Systems, 9 (1), pp. 49-60.

This paper investigates the existence and uniqueness of solutions to first-order nonlinear boundary value problems (BVPs) involving fuzzy differential equations and two-point boundary conditions. Some sufficient conditions are presented that guarantee the existence and uniqueness of solutions under the approach of Hukuhara differentiability.

AUTHOR KEYWORDS: Boundary value problems; Fuzzy differential equations; Fuzzy numbers

Sanchooli, M., Fard, O.S. Numerical scheme for Fredholm integral equations optimal control problems (2012) Applied Mechanics and Materials, 110-116, pp. 4284-4291.

DOI: 10.4028/www.scientific.net/AMM.110-116.4284

This paper deals with a class of optimal control problems governed by linear Fredholm integral equations. A direct scheme based on the Taylor expansion and parametrization to calculate an approximate-analytical solution of the problem is proposed. This method produces an approximation with a controlled level of accuracy. Moreover, a hybrid algorithm to show the procedure of the scheme is given. The convergence of the proposed scheme is also discussed in detail. Some numerical examples illustrate the potential, efficiency and accuracy of the algorithm. © (2012) Trans Tech Publications, Switzerland.

AUTHOR KEYWORDS: Approximate- Analytical solution; Fredholm integral equations; Iterative methods; Numerical scheme; Optimal control problem
INDEX KEYWORDS: Analytical solutions; Fredholm integral equations; Hybrid algorithms; Numerical example; Numerical scheme; Optimal control problem; Parametrizations; Taylor expansions, Aerospace engineering; Algorithms; Control; Iterative methods; Optimal control systems; Optimization, Integral equations

Borzabadi, A.H., Ghasemi, S.G., Fard, O.S. A hybrid iterative scheme for optimal control problems based on L-M and penalty techniques (2011) Journal of Advanced Research in Dynamical and Control Systems, 3 (2), pp. 53-65.

This paper presents an investigation on applying the methods of solving nonlinear least square problems (NLSP)'s for detecting approximate solutions of optimal control problems (OCP)'s under an iterative hybrid process. Levenberg- Marquardt method as a successful classic approach is applied for solving the created NLSP's. Results of implementing the proposed algorithm for solving some linear and nonlinear OCP's are given. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Approximation; Nonlinear least square problem; Optimal control; Penalty method

Fard, O.S., Ghal-Eh, N. Numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients (2011) Information Sciences, 181 (20), pp. 4765-4779.

DOI: 10.1016/j.ins.2011.06.007

In this paper, a novel iterative method is proposed to obtain approximate-analytical solutions for the linear systems of first-order fuzzy differential equations (FDEs) with fuzzy constant coefficients (FCCs) while avoiding the complexities of eigen-value computations. A theorem for the convergence and the validity of the approach is also presented in detail. Numerical experiments and comparisons with exact solutions reveal that the proposed method is capable of generating accurate results. © 2011 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Approximate-analytical solution; System of first-order linear FDEs; Variational iteration method
INDEX KEYWORDS: Approximate-analytical solution; Constant coefficients; Eigen-value; Exact solution; First-order; Fuzzy differential equations; Numerical experiments; Numerical solution; Variational iteration method, Differential equations; Iterative methods; Linear systems; Numerical methods, Differentiation (calculus)

Fard, O.S., Bidgoli, T.A. The Nyström method for hybrid fuzzy differential equation IVPs (2011) Journal of King Saud University - Science, 23 (4), pp. 371-379.

DOI: 10.1016/j.jksus.2010.07.020

In this paper, the Nyström method is developed to approximate the solutions for hybrid fuzzy differential equation initial value problems (IVPs) using the Seikkala derivative. A proof of convergence of this method is also discussed in detail. The accuracy and efficiency of the proposed method are demonstrated by applying it to two different numerical experiments. © 2010.

AUTHOR KEYWORDS: Fuzzy differential equations; Fuzzy interpolation; Fuzzy polynomials; Hybrid systems; Nyström method; Seikkala derivative

Fard, O.S., Kamyad, A.V. Modified k-step method for solving fuzzy initial value problems (2011) Iranian Journal of Fuzzy Systems, 8 (1), pp. 49-63.

We are concerned with the development of a α-step method for the numerical solution of fuzzy initial value problems. Convergence and stability of the method are also proved in detail. Moreover, a specific method of order 4 is found. The numerical results show that the proposed fourth order method is efficient for solving fuzzy differential equations.

AUTHOR KEYWORDS: Fuzzy differential equations; Fuzzy numbers; Modified k-step method

Fard, O.S., BeheshtAien, R. A note on fuzzy best approximation using chebyshev's polynomials (2011) Journal of King Saud University - Science, 23 (2), pp. 217-221.

DOI: 10.1016/j.jksus.2010.07.012

This paper deals with the best approximation for fuzzy valued functions using Chebyshev nodes. We prove a result on the best near-minimax approximation in the fuzzy sense. As an application, Runge's phenomenon is fuzzified in two different cases, i.e. the best approximation and the best near-minimax approximation. © 2010.

AUTHOR KEYWORDS: Fuzzy Approximation; Fuzzy Linear Programming; Fuzzy numbers; Fuzzy polynomials; Runge's Phenomenon

Fard, O.S., Bidgoli, T.A., Borzabadi, A.H. Approximate-analytical approach to nonlinear FDEs under generalized differentiability (2010) Journal of Advanced Research in Dynamical and Control Systems, 2 (1), pp. 56-74.

The current research attempts to offer an approximate-analytical scheme to solve nonlinear fuzzy differential equations under generalized differentiability. In comparison with existing numerical methods, one may find the better capability and e±ciency of the given scheme. The proposed method is illustrated by some numerical examples. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Generalized differentiability; Local variational iteration method; Nonlinear fuzzy differential equation

Borzabadi, A.H., Azizsefat, M., Fard, O.S. An iterative scheme for optimal control of linear Volterra integral equations (2010) Journal of Advanced Research in Dynamical and Control Systems, 2 (2), pp. 13-25.

In this study, a numerical scheme for solving a class of optimal control problems governed by linear Volterra integral equations is presented. The approach is based on the homotopy perturbation theory. The numerical results reveal that the given scheme is extremely effective. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Approx- imation; Homotopy perturbation; Numerical method; Optimal control prblem; Volterra integral equation

Sanchooli, M., Fard, O.S. Numerical scheme for Fredholm integral equations optimal control problems via Bernstein polynomials (2010) Australian Journal of Basic and Applied Sciences, 4 (11), pp. 5675-5682.

In this paper, we present a novel iterative method to approximate the solution to a class of optimal control problems governed by Fredholm integral equations. We are willing to construct a direct scheme based on the Bernstein polynomials and parameterization. The convergence of the method is also discussed in details and at the end, some numerical examples illustrate the efficiency and accuracy of the method. © 2010, INSInet Publication.

AUTHOR KEYWORDS: Approximate-analytical solution; Fredholm integral equations; Iterative methods; Numerical scheme; Optimal control problem

Fard, O.S., Sanchooli, M. Two successive schemes for numerical solution of linear fuzzy Fredholm integral equations of the second kind (2010) Australian Journal of Basic and Applied Sciences, 4 (5), pp. 817-825.

In this paper, two successive schemes for solving linear fuzzy Fredholm integral equations are presented. Using the parametric form of fuzzy numbers, we convert linear fuzzy Fredholm integral equation of the second kind to a linear system of integral equations of the second kind in the crisp case. We use two schemes, successive approximation and Taylor-successive approximation methods, to find the approximate solutions of the converted system, which are the approximate solutions for the fuzzy Fredholm integral equation of the second kind. The proposed methods are illustrated by two numerical examples. © 2010, INSInet Publication.

AUTHOR KEYWORDS: Fuzzy integral equation; Successive approximation method; System of linear Fredholm integral equations of the second kind; Taylor-successive approximation method

Borzabadi, A.H., Fard, O.S. A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind (2009) Journal of Computational and Applied Mathematics, 232 (2), pp. 449-454.

DOI: 10.1016/j.cam.2009.06.038

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach. © 2009 Elsevier B.V. All rights reserved.

AUTHOR KEYWORDS: Approximation; Discretization; Iterative methods; Nonlinear Fredholm integral equations
INDEX KEYWORDS: Approximate solution; Approximation; Discretization; Exact solution; Iterative approach; Nonlinear Fredholm integral equation; Nonlinear Fredholm integral equations; Numerical scheme, Integral equations; Iterative methods, Nonlinear equations

Tahmasbi, A., Fard, O.S. Numerical solution of linear Volterra integral equations system of the second kind (2008) Applied Mathematics and Computation, 201 (1-2), pp. 547-552.

DOI: 10.1016/j.amc.2007.12.041

There are several numerical approaches for solving linear Volterra integral equations system of the second kind. In this paper, we have presented a method for numerical solution of linear Volterra integral equations system based on the power series method, the major advantage of which is being derivative-free. Also, this method reproduces the analytical solution when the exact solutions are polynomial. The numerical results prove that the presented method is very effective and simple. The software used for the numerical calculations in this study was MATLAB®7.4. © 2008 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Linear Volterra integral equations system; Numerical method; Power series method
INDEX KEYWORDS: Integral equations; Linear equations; Numerical methods; Polynomials, Linear Volterra integral equations system; Power series method, Linear systems

Fard, O.S., Kamyad, A.V. A numerical scheme to solve nonlinear BSDES with lipschitz and non-lipschitz coefficients (2005) Journal of Applied Mathematics and Computing, 18 (1-2), pp. 73-93.

DOI: 10.1007/BF02936557

In this paper, we attempt to present a new numerical approach to solve non-linear backward stochastic differential equations. First, we present some definitions and theorems to obtain the conditions, from which we can approximate the non-linear term of the backward stochastic differential equation (BSDE) and we get a continuous piecewise linear BSDE correspond with the original BSDE. We use the relationship between backward stochastic differential equations and stochastic controls by interpreting BSDEs as some stochastic optimal control problems, to solve the approximated BSDE and we prove that the approximated solution converges to the exact solution of the original non-linear BSDE in two different cases. © 2005 Korean Society for Computational & Applied mathematics and Korean SIGCAM.

AUTHOR KEYWORDS: Approximation theory; BSDE; LQ control; Nonlinearity; Stochastic control; Stochastic Riccati equation
PUBLISHER: Journal of Applied Mathemathics

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