Papers by Farid Bozorgnia
Numerical Methods for Partial Differential Equations
A numerical scheme based on modified method of characteristics with adjusted advection (MMOCAA) i... more A numerical scheme based on modified method of characteristics with adjusted advection (MMOCAA) is proposed to approximate the solution of the system liquid chromatography with multi components case. For the case of one component, the method preserves the mass. Various examples and computational tests numerically verify the accuracy and efficiency of the approach.
arXiv (Cornell University), May 7, 2019
In this note, we study the asymptotic behavior, as t tends to infinity, of the solution u to the ... more In this note, we study the asymptotic behavior, as t tends to infinity, of the solution u to the evolutionary damped p-Laplace equation u tt + a u t = ∆ p u with Dirichlet boundary values. Let u * denote the stationary solution with same boundary values, then we prove the W 1,p-norm of u(t) − u * decays for large t like t − 1 (p−1)p , in the degenerate case p > 2.
Conference on Free Boundary Numerical investigation of long range segregation models June 24, 201... more Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 3 / 25 φ i • φ j = 0 on ∂Ω. Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 4 / 25 Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 5 / 25 ∇u j (y). Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 6 / 25 Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 9 / 25 u ε j → u in W 1,2 , v ε j → v in W 1,2. Conference on Free Boundary Numerical investigation of long range segregation models June 24, 2014 13 / 25 Thank you, Questions? Conference on Free Boundary Numerical investigation of long range segregation models
International Journal of Numerical Analysis and Modeling, 2014
In this work, numerical schemes to approximate the solution of one and multi phase quadrature dom... more In this work, numerical schemes to approximate the solution of one and multi phase quadrature domains are presented. We shall construct a monotone, stable and consistent finite difference method for both one and two phase cases, which converges to the viscosity solution of the partial differential equation arising from the corresponding quadrature domain theory. Moreover, we will discuss the numerical implementation of the resulting approach and present computational tests.
Applied Numerical Mathematics, 2019
In this paper, we study the first eigenvalue of a nonlinear elliptic system involving p-Laplacian... more In this paper, we study the first eigenvalue of a nonlinear elliptic system involving p-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, an upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems.
Applied Numerical Mathematics, 2019
In this paper, we study the first eigenvalue of a nonlinear elliptic system involving p-Laplacian... more In this paper, we study the first eigenvalue of a nonlinear elliptic system involving p-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, an upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems.
Journal of Scientific Computing, 2018
In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied.... more In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach.
Journal of Vibration and Control, 2018
We propose a direct numerical method for the solution of an optimal control problem governed by a... more We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre–Gauss–Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided...
Mathematical Methods in the Applied Sciences, 2017
In this paper, 2 extragradient methods for solving differential variational inequality (DVI) prob... more In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed-point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.
Computers & Mathematics with Applications, 2017
This paper is concerned with the two-phase obstacle problem, a type of a variational free boundar... more This paper is concerned with the two-phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of [22] and verify them numerically on two examples in two space dimensions. A solution algorithm is proposed for the construction of the finite element approximation to the two-phase obstacle problem. The algorithm is not based on the primal (convex and nondifferentiable) energy minimization problem but on a dual maximization problem formulated for Lagrange multipliers. The dual problem is equivalent to a quadratic programming problem with box constraints. The quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem. The majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound. A space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary.
Journal of Mathematical Analysis and Applications, 2018
In this work we investigate a coupled system of degenerate and nonlinear partial differential equ... more In this work we investigate a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that the system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions. We conclude, moreover, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case.
Numerical Methods for Partial Differential Equations, 2013
In this work, we present two numerical schemes for a free boundary problem called one phase quadr... more In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods. Contents 1. Introduction. 2 2. Notations and mathematical background of quadrature domains. 2 2.1. An estimate of quadrature domain. 5 3. An application (Hele Shaw flow). 6 4. First numerical method to approximate the solution of Problem (P). 7 4.1. Blow up techniques and the main idea. 7 4.2. A mixed boundary value problem and first algorithm. 9 4.3. Level set formulation. 10 5. Second numerical method to approach to the solution of Problem (P) based on shape optimization. 12 5.1. Shape optimization techniques for Problem (P) and second algorithm. 14 5.2. Alternative viewpoint. 16 6. Numerical examples. 17 References 23
Advances in Difference Equations, 2011
A perturbation formula for the two-phase membrane problem is considered. We perturb the data in t... more A perturbation formula for the two-phase membrane problem is considered. We perturb the data in the right-hand side of the two-phase equation. The stability of the solution and the free boundary with respect to perturbation in the coefficients and boundary value is shown. Furthermore, continuity and differentiability of the solution with respect to the coefficients are proved.
Acta Applicandae Mathematicae
We study a class of elliptic competition-diffusion systems of long range segregation models for t... more We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. We prove the uniqueness result for positive solution of those elliptic and related parabolic systems when the coupling in the right hand side involves a non-local term of integral form. Moreover, alternate proofs of some known results, such as existence of solutions in the elliptic case and the limiting configuration are given. The free boundary condition in a particular setting is given.
Discrete and Continuous Dynamical Systems, 2022
This work is devoted to study a class of singular perturbed elliptic systems and their singular l... more This work is devoted to study a class of singular perturbed elliptic systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behavior of limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero, which implies that all components can not coexist simultaneously. We present a novel method, which provides an explicit solution of the limiting problem for a special choice of parameters. Moreover, we present some numerical simulations of the asymptotic problem.
In this paper different numerical methods for a two-phase free boundary problem are discussed. In... more In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.
Two novel iterative methods for a class of population models of competitive type are considered. ... more Two novel iterative methods for a class of population models of competitive type are considered. This numerical solution is related to the positive solution as the competitive rate tends to infinity. Furthermore, the idea first is applied to an optimal partition problem.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright a b s t r a c t In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.
A perturbation formula for the two-phase membrane problem is considered. We perturb the data in t... more A perturbation formula for the two-phase membrane problem is considered. We perturb the data in the right-hand side of the two-phase equation. The stability of the solution and the free boundary with respect to perturbation in the coefficients and boundary value is shown. Furthermore, continuity and differentiability of the solution with respect to the coefficients are proved.
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Papers by Farid Bozorgnia