Article
Mathematics, Applied
Siping Tang, Xinlong Feng, Wei Wu, Hui Xu
Summary: In this paper, the authors propose a simplified neural network called polynomial interpolation physics-informed neural networks (PI-PINN) to solve nonlinear partial differential equations. By utilizing orthogonal polynomials to construct the neural network, the PI-PINN structure is shown to be effective in solving these equations. Numerical experiments and investigations on reverse problems demonstrate the accuracy and efficiency of the proposed approach.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Automation & Control Systems
Yan Zhao, Jiangbo Yu
Summary: This paper investigates the adaptive asymptotically stabilizing control problem for uncertain nonlinear systems using partial-state feedback, with a focus on input-to-state stability and the novel Nussbaum function to counteract unknown control directions. Damping terms with estimates of unknown disturbance bounds are added in control design to handle nonvanishing external disturbances, resulting in bounded signals and asymptotic convergence of system states to zero despite uncertainties. A simulation example is provided to demonstrate the effectiveness of the proposed control scheme.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2021)
Article
Mathematics
Qigui Yang, Xiaofang Lin, Caibin Zeng
Summary: This paper studies the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Holder rough paths with exponents in (1/3, 1/2]. The approach relies on rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, allowing the establishment of the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Muhammad Bhatti, Md Habibur Rahman, Nicholas Dimakis
Summary: A multivariable technique using B-polynomials is employed to estimate solutions of NPDE, with coefficients determined using the Galerkin method before conversion to an operational matrix equation. The method provides higher-order precision compared to finite difference in solving NPDE equations and has potential for solving complex partial differential equations in multivariable problems.
FRACTAL AND FRACTIONAL
(2021)
Article
Mathematics, Applied
Zhonghua Qiao, Xuguang Yang, Yuze Zhang
Summary: A novel lattice Boltzmann equation model is proposed to solve fourth order NPDE, featuring a source distribution function to eliminate unwanted terms. Through numerical experiments, it is shown that the performance of this model is superior to existing ones.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Mukesh Kumar Rawani, Lajja Verma, Amit Kumar Verma, Ravi P. Agarwal
Summary: In this paper, an efficient numerical scheme is developed for solving Burgers' equation and Fisher equation with different boundary conditions. By combining the Hermite approximation polynomial and Taylor series approximation, the numerical integration formula for the initial value problem is derived.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Engineering, Electrical & Electronic
Sandeep Malik, Mir Sajjad Hashemi, Sachin Kumar, Hadi Rezazadeh, W. Mahmoud, M. S. Osman
Summary: The purpose of this work is to find innovative exact solutions for nonlinear partial differential equations using the new Kudryashov approach. The technique provides novel exact solutions of soliton types. 3D and 2D plots of higher dimensional Klein-Gordon, Kadomtsev-Petviashvili, and Boussinesq equations are shown to better understand the nonlinear wave structures. The new Kudryashov technique is effective and simple, providing new generalized solitonic wave profiles that enhance the understanding of the development and dynamic nature of such models.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Materials Science, Multidisciplinary
Hijaz Ahmad, Tufail A. Khan, Predrag S. Stanimirovic, Wasfi Shatanawi, Thongchai Botmart
Summary: This study investigates the modified variational iteration algorithm-I, which is used for solving different types of nonlinear partial differential equations in modeling physical phenomena. The algorithm incorporates a supplementary parameter to ensure faster convergence. The results obtained from this algorithm are compared with exact and numerical solutions produced by various methods, demonstrating its efficiency, precision, and applicability. The proposed algorithm is highly valuable for solving practical problems in fields of applied physical sciences and engineering.
RESULTS IN PHYSICS
(2022)
Article
Mathematics, Applied
Elizabeth Qian, Ionut-Gabriel Farcas, Karen Willcox
Summary: We propose a new scientific machine learning method that learns from data to predict the evolution of a system governed by a time-dependent nonlinear partial differential equation. The method combines projection-based model reduction with supervised machine learning tools to reduce the computational cost of the model, achieving accurate predictions in test cases.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
A. Ghose-Choudhury, Sudip Garai
Summary: This article discusses the use of a comparison method to obtain exact solutions for nonlinear partial differential equations (PDEs) through their traveling wave reductions. The method, proposed by N. A. Kudryashov, is extended to include solutions expressed in terms of both the logistic function and the tanh$$ \tanh $$-class of functions. The article derives the standard set of second-order ordinary differential equations (ODEs) that have the logistic and tanh$$ \tanh $$ functions as solutions and also extends the analysis to third-order cases.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Interdisciplinary Applications
Erdogan Mehmet Ozkan
Summary: In this work, the F-expansion method is applied to obtain exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrodinger equation with beta derivative. More solutions defined by the Jacobi elliptic function are obtained with the assistance of Maple.
FRACTAL AND FRACTIONAL
(2022)
Article
Optics
Ya-Nan Zhao, Li-Feng Guo
Summary: This paper investigates a sixth-order nonlinear partial differential equation, which can describe the propagation pulse in optical fibers, using the methods of trial equation and complete discrimination system for polynomial. The study has yielded exact optical wave solutions, including singular solutions, solitary wave solutions, and elliptic function double periodic solutions, which can be completely classified in parameter space.
JOURNAL OF NONLINEAR OPTICAL PHYSICS & MATERIALS
(2023)
Article
Mathematics
Isaias Alonso-Mallo, Begona Cano
Summary: In order to avoid order reduction when applying Rosenbrock methods to solve nonlinear partial differential equations, a suitable choice of boundary values for the internal stages is used. The main challenge compared to the linear case lies in accurately calculating these boundary values in terms of data. Despite this difficulty, the implementation remains cheap and simple, requiring only the addition of some additional terms related to these boundary values at each stage.
Article
Engineering, Mechanical
Mohammad Hossein Abbasi, Laura Iapichino, Wil Schilders, Nathan van de Wouw
Summary: This paper proposes a data-based approach for model order reduction that preserves incremental stability properties. The method is applied to the discretized version of hyperbolic equations, successfully reducing complexity and maintaining the stability properties of the original system.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Hui -Min Zhu, Jia Zheng, Zhi-Yong Zhang
Summary: In this paper, an approximate symmetry method for (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed in the sense of Riemann-Liouville fractional derivative. By manipulating the fractional PDEs and performing symmetry reductions, the authors obtain the reduced equations and their solutions. The method is then illustrated with a time-fractional KdV equation and an anomalous diffusion equation.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Automation & Control Systems
Tong Ma, Chengyu Cao
EUROPEAN JOURNAL OF CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
EUROPEAN JOURNAL OF CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
Summary: This paper presents a constrained sampled-data output feedback controller for uncertain multivariable nonlinear systems, utilizing the filtered adaptive control framework. The control scheme prioritizes constraints violation avoidance over tracking performance and periodically solves a numerical constrained optimization problem to find the trade-off. System transformations are used to ensure stability of the closed-loop system with the sampled-data controller.
INTERNATIONAL JOURNAL OF CONTROL
(2021)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING
(2020)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2020)
Article
Automation & Control Systems
Tong Ma
Summary: In this paper, an extended filtered high-gain output feedback controller is developed for uncertain nonlinear systems subject to external disturbances. The controller integrates an extended state observer, high-gain, and low-pass filter to estimate the internal state and reject disturbances. A low-pass filtering mechanism is added to improve the system's robustness, and the filtered control law compensates for nonlinear uncertainties and achieves good tracking performance.
INTERNATIONAL JOURNAL OF CONTROL
(2022)
Article
Automation & Control Systems
Tong Ma
Summary: This paper presents a filtering adaptive tracking control method for uncertain switched multivariable nonlinear systems. The method utilizes a piecewise constant adaptive law to handle uncertainties and a filtering control law to achieve good tracking performance.
INTERNATIONAL JOURNAL OF CONTROL
(2022)
Article
Computer Science, Artificial Intelligence
Tong Ma
Summary: This article introduces a novel decentralized filtering adaptive neural network control framework for uncertain switched interconnected nonlinear systems. Each subsystem has its own decentralized controller to approximate nonlinear uncertainties and achieve the local objective tracking of the host system. By applying neural networks and adaptive laws, this framework effectively cancels uncertainties and achieves the desired performance.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
(2021)
Article
Automation & Control Systems
Tong Ma
Summary: This paper proposes the utilization of unscented Kalman filter (UKF) in the context of stochastic nonlinear model predictive control (SNMPC) for tracking a given trajectory in stochastic multivariable nonlinear systems. The UKF-SNMPC framework utilizes the conditional mean and covariance to reformulate the model cost and constraint functions, and introduces the concept of 'robust horizon' to ensure system stability.
INTERNATIONAL JOURNAL OF CONTROL
(2023)
Article
Automation & Control Systems
Tong Ma
Summary: This paper proposes the use of Gaussian processes (GPs) in the context of stochastic nonlinear model predictive control (SNMPC) to handle stochastic uncertainties. By utilizing GP regression to obtain predictions and uncertainty quantification, and leveraging the probability distribution of stochastic uncertainties, the proposed GP-SNMPC framework provides an effective solution for handling nonlinear constrained control problems with Gaussian parametric uncertainties.
INTERNATIONAL JOURNAL OF CONTROL
(2023)
Article
Automation & Control Systems
Tong Ma
Summary: Nonlinear model predictive control (NMPC) is a control method that can handle complex nonlinear systems. However, its performance highly depends on model accuracy and deterministic solutions may not work efficiently in uncertain stochastic cases. To address these issues, this paper proposes a model-and data-driven predictive control approach using Gaussian processes (GP-MDPC) for tracking control of stochastic nonlinear systems subject to model uncertainties and chance constraints.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2023)