Showing papers in "Numerical Algebra, Control and Optimization in 2021"
TL;DR: The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.
Abstract: One of the challenges facing supply chain designers is designing a sustainable and resilient supply chain network. The present study considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion for the first time. The study suggests a two-stage mixed-integer linear programming model for the problem. Further, the robust counterpart model is used to handle uncertainties. Furthermore, conditional value at risk criterion in the model is considered in order to create real-life conditions. The sustainability goals addressed in the present study include minimizing the costs, \begin{document}$ \text{CO}_2 $\end{document} emission, and energy, along with maximizing employment. In addition, effective environmental and social life-cycle evaluations are provided to assess the associated effects of the model on society, environment, and energy consumption. The model aims to answer the questions regarding the establishment of facilities and amount of transported goods between facilities. The model is implemented in a car assembler company in Iran. Based on the results, several managerial insights are offered to the decision-makers. Due to the complexity of the problem, a constraint relaxation is applied to produce quality upper and lower bounds in medium and large-scale models. Moreover, the LP-Metric method is used to merge the objectives to attain an optimal solution. The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.
142 citations
TL;DR: A feature selection approach based on the binary whale optimization algorithm with different kinds of updating techniques for the time-varying transfer functions is proposed and it proved that BWOA-TV2 has consistency in feature selection and it gives rise to the high accuracy of the classification with more congruent in the convergence.
Abstract: Feature selection is a valuable tool in supervised machine learning research fields, such as pattern recognition or classification problems. Feature selection used to eliminate irrelevant and noise features that adversely affect results. Swarm algorithms are usually used in feature selection problem; these algorithms need transfer functions that change search space from continuous to the discrete. However, transfer functions are the backbone of all binary swarm algorithms. Transfer functions in the current formula cannot provide binary swarm algorithms with a fit balance between exploration and exploitation stages. In this work, a feature selection approach based on the binary whale optimization algorithm with different kinds of updating techniques for the time-varying transfer functions is proposed. To evaluate the performance of the proposed method, three of each chemical and biological binary datasets are used. The results proved that BWOA-TV2 has consistency in feature selection and it gives rise to the high accuracy of the classification with more congruent in the convergence. It worth mentioning that the proposed method is proved advance in performance over competitor optimization algorithms, such as particle swarm optimization (PSO) and firefly optimization (FO) that commonly used in this field.
29 citations
TL;DR: In this article, conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map, and the relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in higher gaps.
Abstract: In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.
25 citations
TL;DR: Li et al. as discussed by the authors developed a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system, which is obtained by combining its search direction with the projection method.
Abstract: By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.
14 citations
TL;DR: A methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM) and results correspond that the primal aim of chaos control in the given system have been attained computationally.
Abstract: In this manuscript, we design a methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM). Initially, an ACM has been proposed and analysed systematically for controlling the microscopic chaos found in the discussed system which is essentially described by employing Lyapunov stability theory (LST). The required asymptotic stability criterion of the state variables of the discussed system having unknown parameters is derived by designing appropriate control functions and parameter updating laws. In addition, numerical simulation results in MATLAB software are performed to illustrate the effective presentation of the considered strategy. Simulations outcomes correspond that the primal aim of chaos control in the given system have been attained computationally.
10 citations
TL;DR: It is shown that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach and that the method can solve problems of this type much faster in comparison with (very few) available options.
Abstract: We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.
9 citations
TL;DR: An observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible and guarantees that theclosed-loop system is regular, impulse-free and stable in the event of actuator failures.
Abstract: A method of designing observer-based feedback controller against actuator failures for uncertain singular fractional order systems (SFOS) is presented in this paper. By establishing actuator fault model and state observer, an observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible. The controller designed by the proposed method guarantees that the closed-loop system is regular, impulse-free and stable in the event of actuator failures. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.
9 citations
TL;DR: A two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractions-order plant model and the presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain.
Abstract: The approximation of the fractional-order controller (FOC) has already been recognized as a distinguished field of research in the literature of system and control. In this paper, a two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractional-order plant model. The design goals are based on some frequency domain specifications. The first stage of the work is focused on developing the pure continuous-time FOC, while the second stage actually realizes the FOPI controller in discrete-time representation. The presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain. In the process of realizing the FOC, the delta operator has been involved as a generating function due to its exclusive competency to unify the discrete-time system and its continuous-time counterpart at low sampling time limit. The well-known continued fraction expansion (CFE) method has been employed to approximate the FOPI controller in delta-domain. Simulation outcomes exhibit that the discrete-time FOPI controller merges to its continuous-time counterpart at the low sampling time limit. The robustness of the overall system is also investigated in delta-domain.
9 citations
TL;DR: Yang et al. as discussed by the authors presented an iterative method for solving the convex constraint nonlinear equation problem, which incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method.
Abstract: We present an iterative method for solving the convex constraint nonlinear equation problem. The method incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method by Yang et al. for solving the unconstrained optimization problem. The proposed method does not require the Jacobian information, nor does it require to store any matrix at each iteration. Thus, it has the potential to solve large-scale non-smooth problems. Under some standard assumptions, the convergence analysis of the method is established. Finally, to show the applicability of the proposed method, the proposed method is used to solve the \begin{document}$ \ell_1 $\end{document} -norm regularized problems to restore blurred and noisy images. The numerical experiment indicates that our result is a significant improvement compared with the related methods for solving the convex constraint nonlinear equation problem.
8 citations
TL;DR: A rigorous analysis in damping of oscillations in a power network utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics.
Abstract: The assimilation of flexible AC transmission (FACTS) controllers to the existing power network outweigh the numerous alternatives in enhancing the damping behavior for the inter-area /intra-area system oscillations of a power network. This paper provides a rigorous analysis in damping of oscillations in a power network. It utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics. A comprehensive system mathematical modelling has been developed for demonstrating the system behavior under different loading conditions. A novel hybrid augmented grey wolf optimization-particle swarm optimization (AGWO-PSO) is proposed for the coordinated design of controllers static synchronous compensator (STATCOM) and power system stabilizers (PSSs). A multi-objective function, comprising damping ratio improvement and drifting the real part to the left-hand side of S-plane of the system poles, has been developed to achieve the objective and the effectiveness of the proposed algorithms have been analyzed by monitoring the system performance under different loading conditions. Eigenvalue analysis and damping nature of the system states under perturbation have been presented for the proposed algorithms under different loading conditions, and the performance evaluation of the proposed algorithms have been done by means of time of execution and the convergence characteristics.
7 citations
TL;DR: In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered, and stability and convergence of the proposed scheme are discussed using mathematical induction.
Abstract: In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.
TL;DR: The so-called vector E $\ end{document} -dual problem in the sense of Mond-Weir is defined for the considered E $\end{ document} -differentiable multiobjective programming problem and several theorems are derived also under appropriate assumptions.
Abstract: In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems with \begin{document}$ E $\end{document} -differentiable functions. Namely, for an \begin{document}$ E $\end{document} -differentiable vector-valued function, the concept of \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity is defined as a generalization of the \begin{document}$ E $\end{document} -differentiable \begin{document}$ E $\end{document} -invexity notion and the concept of \begin{document}$ V $\end{document} -invexity. Further, the sufficiency of the so-called \begin{document}$ E $\end{document} -Karush-Kuhn-Tucker optimality conditions are established for the considered \begin{document}$ E $\end{document} -differentiable vector optimization problems with both inequality and equality constraints under \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity hypotheses. Furthermore, the so-called vector \begin{document}$ E $\end{document} -dual problem in the sense of Mond-Weir is defined for the considered \begin{document}$ E $\end{document} -differentiable multiobjective programming problem and several \begin{document}$ E $\end{document} -duality theorems are derived also under appropriate \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity assumptions.
TL;DR: In this article, the authors discuss conditions for exact (approximate) controllability and exact observability of stochastic implicit systems in Banach spaces, in terms of the GE-evolution operator and the dual principle.
Abstract: This paper discusses exact (approximate) controllability and exact (approximate) observability of stochastic implicit systems in Banach spaces. Firstly, we introduce the stochastic GE-evolution operator in Banach space and discuss existence and uniqueness of the mild solution to stochastic implicit systems by stochastic GE-evolution operator in Banach space. Secondly, we discuss conditions for exact (approximate) controllability and exact (approximate) observability of the systems considered in terms of stochastic GE-evolution operator and the dual principle. Finally, an illustrative example is given.
TL;DR: In this article, the authors analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space.
Abstract: This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.
TL;DR: This approach proposes to formulate any piecewise smooth function as the expectation of a random variable and proposes to use the Boltzmann distribution as a smoothing approximation for this probability distribution.
Abstract: In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.
TL;DR: A new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex, and it was shown that the method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.
Abstract: In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.
TL;DR: It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal controlProblem when use as a delayed boundary value solver of the necessary conditions.
Abstract: There are a limited number of user-friendly, publicly available optimal control software packages that are designed to accommodate problems with delays. GPOPS-Ⅱ is a well developed MATLAB based optimal control code that was not originally designed to accommodate problems with delays. The use of GPOPS-Ⅱ on optimal control problems with delays is examined for the first time. The use of various formulations of delayed optimal control problems is also discussed. It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal control problem when used as a delayed boundary value solver of the necessary conditions.
TL;DR: The class separability term improved the method RMDS significantly and also outperforms other discriminant analysis methods such as Linear discriminantAnalysis (LDA) which is documented through numerical experiments.
Abstract: Regularized Multidimensional Scaling with Radial basis function (RMDS) is a nonlinear variant of classical Multi-Dimensional Scaling (cMDS). A key issue that has been addressed in RMDS is the effective selection of centers of the radial basis functions that plays a very important role in reducing the dimension preserving the structure of the data in higher dimensional space. RMDS uses data in unsupervised settings that means RMDS does not use any prior information of the dataset. This article is concerned on the supervised setting. Here we have incorporated the class information of some members of data to the RMDS model. The class separability term improved the method RMDS significantly and also outperforms other discriminant analysis methods such as Linear discriminant analysis (LDA) which is documented through numerical experiments.
TL;DR: A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point.
Abstract: A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems. The main idea of this method is to formulate an unconstrained optimization problem as an optimal control problem to obtain an optimal trajectory. However, the optimal trajectory can only be generated by impulsive control variables and it is a straight line joining a guessed initial point to a minimum point. Thus, a priori bounds are imposed on the control variables in order to obtain a feasible solution. As a result, the optimal trajectory is made up of bang-bang control sub-arcs, which form an iterative model based on the Lyapunov function's theorem. This is to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point. However, a chattering behavior may occur near the solution. To avoid this behavior, the Newton iterations are then applied to the proposed method via a two-phase approach to achieve fast convergence. Numerical experiments show that this new approach is efficient and cost-effective to solve the unconstrained optimization problems.
TL;DR: In this paper, the authors define and introduce some new concepts of higher order strongly general biconvex functions involving the arbitrary bifunction and a function, and prove that the optimality conditions of these functions are characterized by a class of variational inequalities.
Abstract: In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving the arbitrary bifunction and a function. Some new relationships among various concepts of higher order strongly general biconvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general biconvex functions, which is itself an novel application. It is proved that the optimality conditions of the higher order strongly general biconvex functions are characterized by a class of variational inequalities, which is called the higher order strongly general bivariational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general bivariational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.
TL;DR: In this article, the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor are discussed.
Abstract: This article focuses on the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor. Conservation equations are subjected to linear perturbations and further analyzed by using the Normal Mode technique. A dispersion relation incorporating the effects of surface tension, Atwood number, rotation factor and volume fraction of nanoparticles is obtained. Using Routh-Hurtwitz criterion the stable and unstable modes of Rayleigh-Taylor instability are discussed in the presence/absence of nanoparticles and presented through graphs. It is observed that in the absence/presence of nanoparticles, surface tension helps to stabilize the system and Atwood number has a destabilizing impact without the consideration of rotation factor. But if rotation parameter is considered (in the absence/presence of nanoparticles) then surface tension destabilizes the system while Atwood number has a stabilization effect (for a particular range of wave number). The volume fraction of nanoparticles destabilizes the system in the absence of rotation but in the presence of rotation the stability of the system is significantly stimulated by the nanoparticles.
TL;DR: A PID control method which combined optimal control strategy is proposed in this paper, and through the numerical simulation, the effectiveness of the proposed method is vertified.
Abstract: A PID control method which combined optimal control strategy is proposed in this paper. The posterior unmodeled dynamics measurement data information are made full use to compensate the unknown nonlinearity of the system, and the unknown increment of the unmodeled dynamics is estimated. Then, a nonlinear PID controller with compensation of the posterior unmodeled dynamics measurement data and the estimation of the increment of the unmodeled dynamics is designed. Finally, through the numerical simulation, the effectiveness of the proposed method is vertified.
TL;DR: In this paper, the authors investigated the controllability and observability of second-order linear systems with discrete/continuous time varying and linear time-invariant continuous systems in matrix form.
Abstract: The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.
TL;DR: The purpose of this article is to introduce a methodology to select the list of stocks for investment purpose, and to employ a stochastic fractional programming model to assign money into selected stocks.
Abstract: This article proposes an efficient approach for solving portfolio type problems. It is highly suitable to help fund allocators and decision makers to set up appropriate portfolios for investors. Stock selection is based upon the risk benefits analysis using MADM approach in fuzzy environment. This sort of analysis allows decision makers to identify the list of acceptable portfolios where they can assign some portions of their asset to them. The purpose of this article is two folds; first, to introduce a methodology to select the list of stocks for investment purpose, and second, to employ a stochastic fractional programming model to assign money into selected stocks. This article proposes a hybrid methodology for finding an optimal or new optimal solution of the problem. This hybrid approach considers risks and benefits at the time of stocks prioritization. This is followed by solving a fractional programming to determine the percentages of the budget to be allocated to stocks while dealing with two sets of suitable and non-suitable stocks. For clarification purposes, a sample example problem is solved.
TL;DR: In this paper, interior point modified Nelder Mead method for nonlinearly constrained optimization is described, which neither uses nor estimates objective function or constraint gradients, and a modified logarithmic barrier function is used.
Abstract: An interior point modified Nelder Mead method for nonlinearly constrained optimization is described. This method neither uses nor estimates objective function or constraint gradients. A modified logarithmic barrier function is used. The method generates a sequence of points which converges to KKT point(s) under mild conditions including existence of a Slater point. Numerical results are presented that show the algorithm performs well in practice.
TL;DR: A method for controlling a fixed-wing UAV to get time-optimal using the re-scaling and parameterization techniques, which are useful and effective in maximizing the performance of the gradient-based methods as a sequential quadratic programming method.
Abstract: The route prediction of unmanned aerial vehicles (UAVs) according to their missions is a strategic issue in the aviation field. In some particular missions, the UAV tasks are to start a movement from a defined point to a target reign in the shortest time. This paper proposes a practical method to find the guidance law of the fixed-wing UAV to achieve time-optimal considering the ambient wind. The unique features of this paper are that the environment includes the moving and fixed obstacles as the route constraints, and the fixed-wing UAVs have to keep a given distance from these obstacles. Also, we consider the specific kinematic equation of the fixed-wing UAV and limitations on the flight-path angle and bank-angles as other constraints. We suggest a method for controlling a fixed-wing UAV to get time-optimal using the re-scaling and parameterization techniques. These techniques are useful and effective in maximizing the performance of the gradient-based methods as a sequential quadratic programming method ( \begin{document}$ SQP $\end{document} ) for numerical solutions. Then, all constraints of the time-optimal control problem are converted to a constraint using an exact penalty function. Due to being exact, finding the control variables and switching times is more accurate and faster. Finally, some numerical examples are simulated to explore the effectiveness of our proposed study in reality.
TL;DR: In this article, a novel version of Differential Evolution (NDE) is used to solve the economic power dispatch problem (EPP), which is an influential optimization problem which is a highly nonconvex and non-linear optimization problem.
Abstract: In power systems, Economic Power dispatch Problem (EPP) is an influential optimization problem which is a highly non-convex and non-linear optimization problem. In the current study, a novel version of Differential Evolution (NDE) is used to solve this particular problem. NDE algorithm enhances local and global search capability along with efficient utilization of time and space by making use of two elite features: selfadaptive control parameter and single population structure. The combined effect of these concepts improves the performance of Differential Evolution (DE) without compromising on quality of the solution and balances the exploitation and exploration capabilities of DE. The efficiency of NDE is validated by evaluating on three benchmark cases of the power system problem having constraints such as power balance and power generation along with nonsmooth cost function and is compared with other optimization algorithms. The Numerical outcomes uncovered that NDE performed well for all the benchmark cases and maintained a trade-off between convergence rate and efficiency.
TL;DR: In this article, the authors used the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method to solve the nonlinear fractional type Volterra integro-differential equation.
Abstract: The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation \begin{document}$ \begin{equation*} \sum\limits_{k = 1}^{m}F_{k}(x)D^{(k\alpha )}y(x)+\lambda \int_{0}^{x}K(x, t)D^{(\alpha )}y(t)dt = g(x)y^{2}(x)+h(x)y(x)+P(x). \end{equation*} $\end{document} In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.
TL;DR: The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain using the Reverse Hilbert Uniqueness Method (RHUM approach) and resolve the problem that relays on computing a control with minimum cost.
Abstract: The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain \begin{document}$ \Omega $\end{document} using the Reverse Hilbert Uniqueness Method (RHUM approach) introduced by Lions in 1988. It consists in finding the control \begin{document}$ u $\end{document} that steers the system from an initial state \begin{document}$ y_{_{0}} $\end{document} to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of \begin{document}$ \omega = \Omega $\end{document} and in the regional case where \begin{document}$ \omega $\end{document} is a part of \begin{document}$ \Omega $\end{document} .
TL;DR: The aim of this work is to reconstruct the depletion region in pn junction by establishing the simplified equation for the considered semiconductor and establishing several numerical examples to prove the validity of theoretical results using the proposed algorithm.
Abstract: The aim of this work is to reconstruct the depletion region in pn junction. Starting with famous drift diffusion model, we establish the simplified equation for the considered semiconductor. There we call the shape optimization technique to formulate a minimization problem from the inverse problem at hand. The existence of an optimal solution of the optimization problem is proved. The proposed numerical algorithm is a combined Domain Decomposition method with an efficient hybrid conjugate gradient guided by differential evolution heuristic algorithm, the finite element method is used to discretize the state equation. At the end we establish several numerical examples, to prove the validity of theoretical results using the proposed algorithm, in addition we show some simulation of the depletion region approximation under two different functioning modes.