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On Generalized High Order Derivatives of Nonsmooth Functions

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American Journal of Computational Mathematics 2014年第4期4卷 317-328页

作者： Samaneh Soradi Zeid ali vahidian kamyad Department of Mathematics Ferdowsi University of Mashhad Mashhad Iran

In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.

关键词： Generalized Derivative Smooth and Nonsmooth Functions Nonsmooth Optimization Problem Linear Programming

Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem

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American Journal of Computational Mathematics 2016年第2期6卷 141-152页

作者： Samaneh Soradi Zeid Mostafa Yousefi ali vahidian kamyad Department of Mathematics Faculty of Mathematics University of Sistan and Baluchestan Zahedan Iran National Iranian Oil Products Distribution Company (NIOPDC) Zahedan Region Zahedan Iran Department of Mathematics Ferdowsi University of Mashhad Mashhad Iran

In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4+T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs.

关键词： Riemann-Liouville Derivative Fractional HIV Model Optimization Linear Programming Discritezation

A Consecutive Quasilinearization Method for the Optimal Boundary Control of Semilinear Parabolic Equations

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Applied Mathematics 2014年第4期5卷 691-706页

作者： Mohammad Dehghan Nayyeri ali vahidian kamyad Applied Mathematics Department Ferdowsi University of Mashhad Mashhad Iran

Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.

关键词： Quasilinearization Optimal Boundary Control SQP Methods Semilinear Parabolic PDE’s

Optimal Immunotherapy Control of Aggressive Tumors Growth

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Intelligent Control and Automation 2012年第2期3卷 168-175页

作者： Elnaz Keikhosoro Kiani ali vahidian kamyad Hedayatollah Shirzad Department of Electrical Engineering Islamic Azad University Esfahan Iran Department of Mathematical Sciences Ferdowsi University Mashhad Iran Department of Microbiology & Immunology Shahrekord University of Medical Sciences Shahrekord Iran

Tumor cells can evade immune surveillance by secreting immuno-suppressive factors such as transforming growth factor-beta (TGF-β) and also, Interlukin-10 (IL-10). In this paper the optimal control of mathematical model for aggressive tumor growth via a new and proper approach known as AVK method has been considered. Moreover, we have implemented a special treatment so-called small interfering RNA (siRNA) to reduce presence and effect of TGF-β in tumor cells and also we have added Interlukin-2 (IL-2) into our treatment model to minimize the population of tumor cells. Further research and experimentation with these combination therapies may provide an effective solution in addressing the immuno-suppressive effects of TGF-β. Finally, we analyze the optimal control and system optimality of these equations using numerical techniques.

关键词： Aggressive Tumor Immunotherapy Optimal Control AVK Method

The Best Piecewise Linearization of Nonlinear Functions

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Applied Mathematics 2014年第20期5卷 3270-3276页

作者： Mohammad Mehdi Mazarei ali Asghar Behroozpoor ali vahidian kamyad Department of Applied Mathematics School of Mathematical Sciences Ferdowsi University of Mashhad International Campus Mashhad Iran

In this paper, we propose a method for finding the best piecewise linearization of nonlinear functions. For this aim, we try to obtain the best approximation of a nonlinear function as a piecewise linear function. Our method is based on an optimization problem. The optimal solution of this optimization problem is the best piecewise linear approximation of nonlinear function. Finally, we examine our method to some examples.

关键词： Nonlinear Systems Piecewise Linearization Optimization Problem Linear Programming

Maximizing of Asymptomatic Stage of Fast Progressive HIV Infected Patient Using Embedding Method

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Intelligent Control and Automation 2010年第1期1卷 48-58页

作者： Hassan Zarei ali vahidian kamyad Sohrab Effati 不详

A system of ordinary differential equations, which describe various aspects of the interaction of HIV with healthy cells in fast progressive patient, is utilized, and an optimal control problem is constructed to prolong survival and delay the progression to AIDS as far as possible, subject to drug costs. Optimal control problem is approximated by linear programming model using measure theoretical approach and suboptimal combinations of reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) drug efficacies are proposed. The Comparison of healthy CD4+ Tcells counts, virus particles and immune response, before and after the treatment is introduced.

关键词： HIV Model Optimal Control Measure Theory Linear Programming

A New Definition for Generalized First Derivative of Nonsmooth Functions

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Applied Mathematics 2011年第10期2卷 1252-1257页

作者： ali vahidian kamyad Mohammad Hadi Noori Skandari Hamid Reza Erfanian 不详

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

关键词： Generalized Derivative Smooth and Nonsmooth Functions Fourier analysis Linear Programming Functional Optimization

A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems

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Intelligent Control and Automation 2011年第2期2卷 144-151页

作者： Amin Jajarmi Hamidreza Ramezanpour Mohammad Dehghan Nayyeri ali vahidian kamyad 不详

This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.

关键词： Path Planning Optimal Control Measure Theory Linear Programming

A New Approach for a Class of Optimal Control Problems of Volterra Integral Equations

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Intelligent Control and Automation 2011年第2期2卷 121-125页

作者： Mohammad Hadi Noori Skandari Hamid Reza Erfanian ali vahidian kamyad 不详

In this paper, we propose a new approach for a class of optimal control problems governed by Volterra integral equations which is based on linear combination property of intervals. We convert the nonlinear terms in constraints of problem to the corresponding linear terms. Discretization method is also applied to convert the new problems to the discrete-time problem. In addition, some numerical examples are presented to illustrate the effectiveness of the proposed approach.

关键词： Volterra Integral Equations Optimal Control Linear Programming