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Mahir Rasulov - One of the best experts on this subject based on the ideXlab platform.

  • On Grid Method for Entropy Solution of the Problem of Simultaneous Motion of Two-Phase Fluid in a Natural Reservoir
    Asian Journal of Applied Sciences, 2016
    Co-Authors: Bahaddin Sinsoysal, Mahir Rasulov, Hasan Carfi
    Abstract:

    In this paper a method for obtaining an exact and numerical solution of the initial and initial-boundary value Problems for a first order partial differential equation with a non-convex state function is suggested, which models the macroscopic motion of the two- phase fluids in a porous medium. For this aim, an Auxiliary Problem is introduced such a way that it has some advantages over the main Problem, and it is equivalent to the main Problem in a definite sense. By use of this Auxiliary Problem it is proved that the exact and numerical solutions of the investigated Problems satisfy the entropy condition in the sense of Oleinik. To make use of this Auxiliary Problem, a method for fixing the location of shock which appears in the solution of the main Problem and its evolution in time is offered. The proposed Auxiliary Problem permits us also to prove convergence in the meaning of a numerical solution to the exact solution of the main Problem. Besides, the Auxiliary Problem permits us to write the higher sensitive and the higher order numerical scheme with respect to time variable whose solution expresses all the physical properties of the Problem accurately. Using the suggested algorithms, two laboratory experiments were carried out.

  • ON A METHOD FOR FINDING THE NUMERICAL SOLUTION OF CAUCHY Problem FOR 2D BURGERS EQUATION
    European Scientific Journal ESJ, 2015
    Co-Authors: Mahir Rasulov, Emine Boran
    Abstract:

    In this paper a new method is proposed for finding the numerical solution of the Cauchy Problem for 2D Burgers equation with initial function consisting of four piecewise constants in a class of discontinuous functions. For this goal, a special Auxiliary Problem which has some advantages over the main Problem is introduced. Using these advantages of the Auxiliary Problem, the numerical solution of the main Problem is obtained. Some computer experiments are carried out.

  • NAA - Grid Method for Solving the Flow of Traffic Problem on the Highway in a Class of Discontinuous Functions
    Lecture Notes in Computer Science, 2009
    Co-Authors: Mahir Rasulov, Kenan Göçer
    Abstract:

    In this paper a new method for obtaining an exact and numerical solution of initial value Problem for a first order nonlinear partial equation which describes the traffic flow on highway in a class of discontinuous functions is developed. For this goal, at first, the special Auxiliary Problem having some advantages over the main Problem has been suggested and using the solution of the Auxiliary Problem an original method for finding the weak solution of the main Problem has been suggested. Using the advantages of the Auxiliary Problem the new numerical method for obtaining a solution which expresses the all physical properties accurately is purposed, too.

  • NAA - Efficient Numerical Method of the 1D Motion of Two-Phase Fluid through Porous Medium in a Class of Discontinuous Functions
    Lecture Notes in Computer Science, 2009
    Co-Authors: Bahaddin Sinsoysal, Mahir Rasulov
    Abstract:

    In this paper a new numerical method for nonlinear system of partial differential equations which describes motion of a two-phase compressible fluid in microscopic porous medium in a class of discontinuous functions is suggested. For this aim, at first, the system of equations is split in two equations according to physical parameters. First one with respect to water saturation, and the other one with respect to pressure. Then, the special Auxiliary Problem having some advantages over the main Problem is introduced. This Auxiliary Problem permits to develop the efficient numerical algorithm for obtaining the solution. Using the solution of the Auxiliary Problem, the solution of the main Problem, which expresses all the physical properties is found.

  • A new algorithm for the numerical solution of Maxwell's equations in a class of generalized functions
    2006
    Co-Authors: Turhan Karaguler, Mahir Rasulov
    Abstract:

    In this paper a new method for the numerical solution of the Cauchy Problem for the Maxwell's equations in a class of generalized functions is suggested. For this purpose, the special Auxiliary Problem having some advantages over the main Problem is introduced. On the basis of the Auxiliary Problem, an effective and economical algorithm for the numerical solution has been developed.

Daoli Zhu - One of the best experts on this subject based on the ideXlab platform.

  • Auxiliary Problem Principle of augmented Lagrangian with Varying Core Functions for Large-Scale Structured Convex Problems
    arXiv: Optimization and Control, 2015
    Co-Authors: Lei Zhao, Daoli Zhu, Bo Jiang
    Abstract:

    The Auxiliary Problem principle of augmented Lagrangian (APP-AL), proposed by Cohen and Zhu (1984), aims to find the solution of a constrained optimization Problem through a sequence of Auxiliary Problems involving augmented Lagrangian. The merits of this approach are two folds. First, the core function is usually separable, which makes the subProblems at each step decomposable and particularly attractive for parallel computing. Second, the choice of the core function is quite flexible. Consequently, by carefully specifying this function, APP-AL may reduce to some standard optimization algorithms. In this paper, we pursue enhancing such flexibility by allowing the core function to be non-identical at each step of the algorithm, and name it varying Auxiliary Problem principle (VAPP-AL). Depending on the Problem structure, the varying core functions in VAPP-AL can be adapted to design new flexible and suitable algorithm for parallel and distributed computing. The convergence and O(1/t) convergence rate of VAPP-AL for convex Problem with coupling objective and constraints is proved. Moreover, if this function is specialized to be quadratic, an o(1/t) convergence rate can be established. Interestingly, the new VAPP framework can cover several variants of Jacobian type augmented Lagrangian decomposition methods as special cases. Furthermore, our technique works for the convex Problem with nonseparable objective and multi-blocks coupled linear constraints, which usually can not be handled by ADMM.

  • Auxiliary Problem Methods with Varying Core Functions for Large-Scale Structured Convex Problems
    arXiv: Optimization and Control, 2015
    Co-Authors: Lei Zhao, Daoli Zhu, Bo Jiang
    Abstract:

    The Auxiliary Problem principle (APP), proposed by Cohen (1978, 1980), Cohen and Zhu (1984), aims to find the solution of an optimization Problem through a sequence of Auxiliary Problems. The merits of this approach are two folds. First, the core function is usually separable, which makes the subProblems at each step decomposable and particularly attractive for parallel computing. Second, the choice of the core function is quite flexible. Consequently, by carefully specifying this function, APP may reduce to many basic optimization algorithms. In this paper, we pursue enhancing such flexibility by allowing the core function to be non-identical at each step of the algorithm, and name it varying Auxiliary Problem principle (VAPP). The convergence of VAPP for convex Problem with coupling constraints is proved if the "difference" of the core functions associated with the two consecutive steps is bounded. Moreover, if this function is specialized to be quadratic, an o(1/k) convergence rate can be established. Interestingly, the new VAPP framework can cover several variants of Jacobian type ADMM as special cases. In the absence of the proximal terms, the condition used in our proof is weaker than that required by other literature for ADMM. Furthermore, our technique works for the convex Problem with nonseparable objective and coupled linear constraints, which usually can not be handled by ADMM. Numerical results are presented to illustrate the efficiency of the proposed new framework.

  • Coupling the Auxiliary Problem principle with descent methods of pseudoconvex programming
    European Journal of Operational Research, 1995
    Co-Authors: Daoli Zhu, Patrice Marcotte
    Abstract:

    Abstract The descent Auxiliary Problem method allows one to find the solution of minimization Problems by solving a sequence of Auxiliary Problems which incorporate a linesearch strategy. We derive the basic algorithm and study its convergence properties within the framework of infinite dimensional pseudoconvex minimization. We also introduce a partial descent type Auxiliary Problem method which partially linearizes the objective functional and includes the Auxiliary term only for of subset of variables. Numerous examples of this very general scheme are provided.

Patrice Marcotte - One of the best experts on this subject based on the ideXlab platform.

  • Coupling the Auxiliary Problem principle with descent methods of pseudoconvex programming
    European Journal of Operational Research, 1995
    Co-Authors: Daoli Zhu, Patrice Marcotte
    Abstract:

    Abstract The descent Auxiliary Problem method allows one to find the solution of minimization Problems by solving a sequence of Auxiliary Problems which incorporate a linesearch strategy. We derive the basic algorithm and study its convergence properties within the framework of infinite dimensional pseudoconvex minimization. We also introduce a partial descent type Auxiliary Problem method which partially linearizes the objective functional and includes the Auxiliary term only for of subset of variables. Numerous examples of this very general scheme are provided.

Bahaddin Sinsoysal - One of the best experts on this subject based on the ideXlab platform.

  • A Higher-Order Sensitive Finite Differences Scheme Of The Cauchy Problem For 2D Linear Hyperbolic Equations With Constant Coefficients In A Class Of Discontinuous Functions
    Beykent Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 2020
    Co-Authors: Öykü Yener, Bahaddin Sinsoysal, Mahir Resulov
    Abstract:

    In this study we develop a finite difference schema for practical calculation of the Cauchy Problem for the 2D scalar advection equation with a higher accuracy oder constant coefficient, encountered in different areas of hydrodynamics. For this aim to develop an Auxiliary Problem having some advantages over the main Problem is introduced. The proposed Auxiliary Problem permits us construct a higher order sensitive finite differences schema.

  • On Grid Method for Entropy Solution of the Problem of Simultaneous Motion of Two-Phase Fluid in a Natural Reservoir
    Asian Journal of Applied Sciences, 2016
    Co-Authors: Bahaddin Sinsoysal, Mahir Rasulov, Hasan Carfi
    Abstract:

    In this paper a method for obtaining an exact and numerical solution of the initial and initial-boundary value Problems for a first order partial differential equation with a non-convex state function is suggested, which models the macroscopic motion of the two- phase fluids in a porous medium. For this aim, an Auxiliary Problem is introduced such a way that it has some advantages over the main Problem, and it is equivalent to the main Problem in a definite sense. By use of this Auxiliary Problem it is proved that the exact and numerical solutions of the investigated Problems satisfy the entropy condition in the sense of Oleinik. To make use of this Auxiliary Problem, a method for fixing the location of shock which appears in the solution of the main Problem and its evolution in time is offered. The proposed Auxiliary Problem permits us also to prove convergence in the meaning of a numerical solution to the exact solution of the main Problem. Besides, the Auxiliary Problem permits us to write the higher sensitive and the higher order numerical scheme with respect to time variable whose solution expresses all the physical properties of the Problem accurately. Using the suggested algorithms, two laboratory experiments were carried out.

  • NAA - Finite Differences Method for One Dimensional Nonlinear Parabolic Equation with Double Nonlinearity
    Lecture Notes in Computer Science, 2013
    Co-Authors: Bahaddin Sinsoysal, Turgay Coruhlu
    Abstract:

    The finite differences scheme for finding a numerical solution of the parabolic equation with double nonlinearity is suggested. For this purpose a special Auxiliary Problem having some advantages over the main Problem is introduced. Some properties of the numerical solution are studied and using the advantages of the proposed Auxiliary Problem, the convergence of the numerical solution to the exact solution in the sense of mean is proven.

  • The Analytical and a Higher-Accuracy Numerical Solution of a Free Boundary Problem in a Class of Discontinuous Functions
    Mathematical Problems in Engineering, 2012
    Co-Authors: Bahaddin Sinsoysal
    Abstract:

    A new method is suggested for obtaining the exact and numerical solutions of the initial-boundary value Problem for a nonlinear parabolic type equation in the domain with the free boundary. With this aim, a special Auxiliary Problem having some advantages over the main Problem and being equivalent to the main Problem in a definite sense is introduced. The Auxiliary Problem allows us to obtain the weak solution in a class of discontinuous functions. Moreover, on the basis of the Auxiliary Problem a higher-resolution numerical method is developed so that the solution accurately describes all physical properties of the Problem. In order to extract the significance of the numerical solutions obtained by using the suggested Auxiliary Problem, some computer experiments are carried out.

  • A New Numerical Method for Stefan-Type Problems in a Class of Unsmooth Functions
    2010
    Co-Authors: Bahaddin Sinsoysal
    Abstract:

    In this paper a new numerical method for obtaining the weak solution of the initial-boundary value Problem for a nonlinear parabolic type equation in the domain with the free boundary is suggested. For this aim, a special Auxiliary Problem which has some advantages over the main Problem and is equivalent to the main Problem in a definite sense is introduced. Using the Auxiliary Problem the efficient and economical numerical algorithms from a computer point of view for obtaining the weak solution of the main Problem is suggested. In addition to these, the Auxiliary Problem also permits us to determine the order of differentiability properties of the solution of the main Problem.

Ali Ebrahimnejad - One of the best experts on this subject based on the ideXlab platform.

  • Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients
    International Journal of Operations Research and Information Systems, 2011
    Co-Authors: Ali Ebrahimnejad, Seyed Hadi Nasseri, Sayyed Mehdi Mansourzadeh
    Abstract:

    In most practical Problems of linear programming Problems with fuzzy cost coefficients, some or all variables are restricted to lie within lower and upper bounds. In this paper, the authors propose a new method for solving such Problems called the bounded fuzzy primal simplex algorithm. Some researchers used the linear programming Problem with fuzzy cost coefficients as an Auxiliary Problem for solving linear programming with fuzzy variables, but their method is not efficient when the decision variables are bounded variables in the Auxiliary Problem. In this paper the authors introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus.

  • a fuzzy primal simplex algorithm and its application for solving flexible linear programming Problems
    European Journal of Industrial Engineering, 2010
    Co-Authors: Seyed Hadi Nasseri, Ali Ebrahimnejad
    Abstract:

    One of the most interesting models of the linear programming is the flexible linear programming Problem. It is shown that by using a suitable membership function for their constraints, we can obtain an equivalent Fuzzy Variable Linear Programming (FVLP) Problem. Some methods have been developed for solving these Problems by introducing and solving a certain Auxiliary Problem. Here, we first propose a fuzzy primal simplex algorithm for solving the flexible linear programming Problem and then suggest the fuzzy primal simplex method to solve the flexible linear programming Problems directly without solving any Auxiliary Problem. This method will also be especially useful for sensitivity analysis using the primal simplex tableaus. Thereafter, we shall illustrate our method with some examples. [Received 05 April 2008; Revised 05 October 2008; Revised 14 September 2009; Accepted 15 September 2009]