arXiv subjects

Kareem T. Elgindy

Publications and source records attributed to Kareem T. Elgindy.

At least 19 recordsLinked to original sources

arXiv2025

Periodic Fractional Control in Bioprocesses for Clean Water and Ecosystem Health

This paper introduces a novel fractional-order chemostat model (FOCM) incorporating Caputo fractional derivative with sliding memory (CFDS) to capture microbial memory effects in biological water treatment, addressing limitations of integer-order models that overlook time-dependent behaviors and fail to capture microbial memory such as delayed growth responses to past nutrient availability, history-dependent adaptation to inflow fluctuations, and persistent historical effects over hours to days, which are biologically critical in wastewater treatment. By optimizing periodic dilution rate control, we minimize the average pollutant output, constrained by treatment capacity and periodic boundaries. Key contributions include: (1) a rigorous fractional framework linking microbial kinetics to memory-driven control; (2) reduction to a 1D fractional-order differential equation (FDE) for computational efficiency; (3) proofs of optimal periodic solution (OPS) existence/uniqueness via Schauder's theorem and convexity; (4) bang-bang control derivation using fractional Pontryagin maximum principle (PMP) and Fourier-Gegenbauer pseudospectral (FG-PS) method with a specialized edge-detection technique to handle control discontinuities, ensuring efficient numerical resolution of switching points and discontinuities inherent in bang-bang strategies; and (5) a comprehensive sensitivity analysis revealing how the fractional order α, scaling parameter {\vartheta}, and memory length L critically influence system performance, with simulations showing up to 40% reduction in substrate concentrations versus steady-state and demonstrating computational tractability through efficient discretization that scales favorably for practical implementation. Scientifically, this advances fractional calculus in bioprocesses, revealing memory's role in improving responsiveness and efficiency.

math.OC↗

arXiv2025

Hybrid Shifted Gegenbauer Integral-Pseudospectral Method for Solving Time-Fractional Benjamin-Bona-Mahony-Burgers Equation

This paper presents a high-order hybrid shifted Gegenbauer integral-pseudospectral (HSG-IPS) method for solving the time-fractional Benjamin-Bona-Mahony-Burgers (FBBMB) equation. A key innovation of our approach is the transformation of the original equation into a fractional partial-integro differential form that contains only a first-order derivative, which can be accurately approximated using a first-order shifted Gegenbauer differentiation matrix (SGDM), while all other terms in the transformed equation are resolved using highly accurate quadrature rules. The method combines several advanced numerical techniques including the shifted Gegenbauer pseudospectral (SGPS) method, Gegenbauer-based fractional approximation (GBFA), shifted Gegenbauer integration matrix (SGIM), shifted Gegenbauer integration row vector (SGIRV), and SGDM to achieve spectral accuracy. Numerical experiments demonstrate that the HSG-IPS method outperforms existing numerical approaches, achieving significantly lower average absolute errors (AAEs) with computational times as low as 0.04-0.05 seconds. The method's robustness is validated across various fractional orders, showing excellent agreement with analytical solutions. The transformation strategy effectively circumvents the numerical instability associated with direct approximation of high-order derivatives in the original equation, while the use of shifted Gegenbauer (SG) polynomials and barycentric representations ensures numerical stability and efficiency. This work provides a powerful computational framework for modeling wave propagation, dispersion, and nonlinearity in fractional calculus applications.

math.NA↗

arXiv2025

Sustainable Water Treatment through Fractional-Order Chemostat Modeling with Sliding Memory and Periodic Boundary Conditions: A Mathematical Framework for Clean Water and Sanitation

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Caratheodory framework and fixed-point theorems, ensuring the system's well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation).

math.OC↗

arXiv2025

Super-Exponential Approximation of the Riemann-Liouville Fractional Integral via Gegenbauer-Based Fractional Approximation Methods

This paper introduces a Gegenbauer-based fractional approximation (GBFA) method for high-precision approximation of the left Riemann-Liouville fractional integral (RLFI). By using precomputable fractional-order shifted Gegenbauer integration matrices (FSGIMs), the method achieves super-exponential convergence for smooth functions, delivering near machine-precision accuracy with minimal computational cost. Tunable shifted Gegenbauer (SG) parameters enable flexible optimization across diverse problems, while rigorous error analysis verifies a fast reduction in approximation error when appropriate parameter choices are applied. Numerical experiments demonstrate that the GBFA method outperforms MATLAB's integral, MATHEMATICA's NIntegrate, and existing techniques by up to two orders of magnitude in accuracy, with superior efficiency for varying fractional orders 0 < α < 1. Its adaptability and precision make the GBFA method a transformative tool for fractional calculus, ideal for modeling complex systems with memory and non-local behavior, where understanding underlying structures often benefits from recognizing inherent symmetries or patterns.

math.NA↗

arXiv2025

The Numerical Approximation of Caputo Fractional Derivative of Higher Orders Using A Shifted Gegenbauer Pseudospectral Method: Two-Point Boundary Value Problems of the Bagley Torvik Type Case Study

This work presents a new framework for approximating Caputo fractional derivatives (FDs) of any positive order using a shifted Gegenbauer pseudospectral (SGPS) method. By transforming the Caputo FD into a scaled integral of the $m$th-derivative of the Lagrange interpolating polynomial (with $m$ being the ceiling of the fractional order $α$), we mitigate the singularity near zero, improving stability and accuracy. The method links $m$th-derivatives of shifted Gegenbauer (SG) polynomials with SG polynomials of lower degrees, allowing for precise integration using SG quadratures. We employ orthogonal collocation and SG quadratures in barycentric form to obtain an accurate and efficient approach for solving fractional differential equations. We provide error analysis showing that the SGPS method is convergent in a semi-analytic framework and conditionally convergent with exponential rate for smooth functions in finite-precision arithmetic. This exponential convergence improves accuracy compared to wavelet-based, operational matrix, and finite difference methods. The SGPS method is flexible, with adjustable SG parameters for optimal performance. A key contribution is the fractional SG integration matrix (FSGIM), which enables efficient computation of Caputo FDs via matrix-vector multiplications and accelerates the SGPS method through pre-computation and storage. The method remains within double-precision limits, making it computationally efficient. It handles any positive fractional order $α$ and outperforms existing schemes in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley-Torvik type.

math.NA↗

arXiv2025

Fourier-Gegenbauer Integral-Galerkin Method for Solving the Advection-Diffusion Equation With Periodic Boundary Conditions

This study presents the Fourier-Gegenbauer Integral-Galerkin (FGIG) method, a novel and efficient numerical framework for solving the one-dimensional advection-diffusion equation with periodic boundary conditions. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Distinctively, this approach eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of shifted-Gegenbauer-Gauss quadrature to ensure high accuracy and stability for relatively low Péclet numbers. Numerical experiments validate the method's superior performance over traditional techniques, demonstrating its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering.

math.NA↗

arXiv2023

Direct Integral Pseudospectral and Integral Spectral Methods for Solving a Class of Infinite Horizon Optimal Output Feedback Control Problems Using Rational and Exponential Gegenbauer Polynomials

This study is concerned with the numerical solution of a class of infinite-horizon linear regulation problems with state equality constraints and output feedback control. We propose two numerical methods to convert the optimal control problem into nonlinear programming problems (NLPs) using collocations in a semi-infinite domain based on rational Gegenbauer (RG) and exponential Gegenbauer (EG) basis functions. We introduce new properties of these basis functions and derive their quadratures and associated truncation errors. A rigorous stability analysis of the RG and EG interpolations is also presented. The effects of various parameters on the accuracy and efficiency of the proposed methods are investigated. The performance of the developed integral spectral method is demonstrated using two benchmark test problems related to a simple model of a divert control system and the lateral dynamics of an F-16 aircraft. Comparisons of the results of the current study with available numerical solutions show that the developed numerical scheme is efficient and exhibits faster convergence rates and higher accuracy.

math.OC↗

arXiv2023

Fourier-Gegenbauer Pseudospectral Method for Solving Time-Dependent One-Dimensional Fractional Partial Differential Equations with Variable Coefficients and Periodic Solutions

In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafa et al. [1] or their reduced forms obtained by Elgindy [2] as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures. The reduced linear system has a simple special structure and can be solved accurately and rapidly by using standard linear system solvers. A rigorous study of the computational storage requirements as well as the error and convergence of the proposed method is presented. The idea and results presented in this paper are expected to be useful in the future to address more general problems involving FPDEs with periodic solutions.

math.NA↗

arXiv2023

Numerical Solution of Nonlinear Periodic Optimal Control Problems Using a Fourier Integral Pseudospectral Method

This paper presents a Fourier integral pseudospectral (FIPS) method for a general class of nonlinear, periodic optimal control (OC) problems with equality and/or inequality constraints and sufficiently smooth solutions. In this scheme, the integral form of the problem is collocated at an equispaced set of nodes, and all necessary integrals are approximated using highly accurate Fourier integration matrices (FIMs). The proposed method leads to a nonlinear programming problem (NLP) with algebraic constraints, which is solved using a direct numerical optimization method. Sharp convergence and error estimates are derived for Fourier series, interpolants, and quadratures used for smooth and continuous periodic functions. Two nonlinear examples are considered to show the accuracy and efficiency of the numerical scheme.

math.OC↗

arXiv2023

Fourier-Gegenbauer Pseudospectral Method for Solving Periodic Fractional Optimal Control Problems

This paper introduces a new accurate model for periodic fractional optimal control problems (PFOCPs) using Riemann-Liouville (RL) and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. The paper also provides a novel numerical method for solving PFOCPs using Fourier and Gegenbauer pseudospectral methods. By employing Fourier collocation at equally spaced nodes and Fourier and Gegenbauer quadratures, the method transforms the PFOCP into a simple constrained nonlinear programming problem (NLP) that can be treated easily using standard NLP solvers. We propose a new transformation that largely simplifies the problem of calculating the periodic FDs of periodic functions to the problem of evaluating the integral of the first derivatives of their trigonometric Lagrange interpolating polynomials, which can be treated accurately and efficiently using Gegenbauer quadratures. We introduce the notion of the αth-order fractional integration matrix with index L based on Fourier and Gegenbauer pseudospectral approximations, which proves to be very effective in computing periodic FDs. We also provide a rigorous priori error analysis to predict the quality of the Fourier-Gegenbauer-based approximations to FDs. The numerical results of the benchmark PFOCP demonstrate the performance of the proposed pseudospectral method.

math.OC↗

arXiv2023

Fourier-Gegenbauer Pseudospectral Method for Solving Periodic Higher-Order Fractional Optimal Control Problems

In [1], we inaugurated a new area of optimal control (OC) theory that we called "periodic fractional OC theory," which was developed to find optimal ways to periodically control a fractional dynamic system. The typical mathematical formulation in this area includes the class of periodic fractional OC problems (PFOCPs), which can be accurately solved numerically for a fractional order α in the range 0 < α < 1 using Fourier collocation at equally spaced nodes and Fourier and Gegenbauer quadratures. In this study, we extend this earlier work to cover periodic higher-order fractional OC problems (PHFOCPs) of any positive non-integer fractional order α.

math.OC↗

arXiv2023

New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method

In its simplest form, a chemostat consists of microorganisms or cells that grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions of the cell growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "overyielding" state in which the performance becomes higher than that at steady-state operation. In this paper, we show that an optimal periodic control policy for maximizing chemostat performance can be accurately and efficiently derived numerically using a novel class of integral pseudospectral (IPS) methods and adaptive h-IPS methods composed through a predictor-corrector algorithm. New formulas for the construction of Fourier pseudospectral (PS) integration matrices and barycentric-shifted Gegenbauer (SG) quadratures are derived. A rigorous study of the errors and convergence rates of SG quadratures, as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions, is presented. We also introduce a novel adaptive scheme for detecting jump discontinuities and reconstructing a piecewise analytic function from PS data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.

math.NA↗

arXiv2023

Optimal Periodic Control of Unmanned Aerial Vehicles Based on Fourier Integral Pseudospectral and Edge-Detection Methods

This study describes the development of a novel numerical optimization framework to maximize the endurance of unmanned aerial vehicles (UAVs). We address the problem of numerically determining the optimal thrust and cruise angle of attack in a two-dimensional space for a UAV under certain initial, periodic, and bound constraints. The time horizon of the free final time optimal control problem (OCP) is first normalized, and the normalized OCP in integral form is discretized in physical space into a nonlinear programming problem (NLP) using Fourier collocation and quadrature based on equispaced points. Great attention in this work is placed on the accurate detection of jump discontinuities and resolving the thrust history effectively directly from the Fourier pseudospectral (FPS) data through a novel edge-detection method without any smoothing techniques. The numerical results demonstrate that the proposed method is simple, stable, and easy to implement.

math.OC↗

arXiv2022

A Direct Integral Pseudospectral Method for Solving a Class of Infinite-Horizon Optimal Control Problems Using Gegenbauer Polynomials and Certain Parametric Maps

We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems (FHOCs) in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points, and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Two illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS- and IPS-methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large, if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.

math.NA↗

arXiv2017

High-Order Adaptive Gegenbauer Integral Spectral Element Method for Solving Nonlinear Optimal Control Problems

In this work, we propose an adaptive spectral element algorithm for solving nonlinear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer-Gauss points combined with very accurate and stable numerical quadratures to fully discretize the multiple-phase integral form of the optimal control problem. The proposed algorithm relies on exploiting the underlying smoothness properties of the solutions for computing approximate solutions efficiently. In particular, the method brackets discontinuities and "points of nonsmoothness" through a novel local adaptive algorithm, which achieves a desired accuracy on the discrete dynamical system equations by adjusting both the mesh size and the degree of the approximating polynomials. A rigorous error analysis of the developed numerical quadratures is presented. Finally, the efficiency of the proposed method is demonstrated on three test examples from the open literature.

math.OC↗

arXiv2017

High-Order Shifted Gegenbauer Integral Pseudospectral Method for Solving Differential Equations of Lane-Emden Type

We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudospectral method (SGIPSM) to solve numerically Lane-Emden equations provided with some mixed Neumann and Robin boundary conditions. The framework of the proposed method includes: (i) recasting the problem into its integral formulation, (ii) collocating the latter at the shifted flipped-Gegenbauer-Gauss-Radau (SFGGR) points, and (iii) replacing the integrals with accurate and well-conditioned numerical quadratures constructed via SFGGR-based shifted Gegenbauer integration matrices. The integral formulation is eventually discretized into linear/nonlinear system of equations that can be solved easily using standard direct system solvers. The implementation of the proposed method is further illustrated through four efficient computational algorithms. Moreover, we furnish rigorous error and convergence analyses of the SGIPSM. Five numerical test examples are presented to verify the effectiveness, accuracy, exponential convergence, and numerical stability of the proposed method. The numerical simulations are associated with extensive numerical comparisons with other rival methods in the literature to demonstrate further the power of the proposed method. The SGIPSM is broadly applicable and represents a strong addition to common numerical methods for solving linear/nonlinear differential equations when high-order approximations are required using a relatively small number of collocation points.

math.NA↗

arXiv2017

Optimization via Chebyshev Polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.

math.OC↗

arXiv2016

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

math.NA↗