Article
Automation & Control Systems
Xisheng Dai, Jing Wu, Jianxiang Zhang, Guopeng Zhou
Summary: This paper studies the iterative learning control (ILC) for a class of second-order nonlinear hyperbolic impulsive partial differential systems. Firstly, a P-type learning law is adopted to track the discontinuous desired output, and sufficient conditions for the convergence of the tracking error are established under identified initial state value. The rigorous analysis is also given using the impulsive Gronwall inequality. Secondly, the tracking error of output trajectory is considered in systems with state initial values shifting based on an initial learning algorithm. The results show that the tracking error on the finite time interval can uniformly converge to 0 as the iteration index goes to infinity if the impulse number of the systems is only a finite number. Finally, numerical simulation examples are provided to verify the effectiveness of the theoretical results.
IET CONTROL THEORY AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Maseim Kenmoe, Ronald Kriemann, Matteo Smerlak, Anton S. Zadorin
Summary: A novel eigenvalue algorithm called Iterative Perturbative Theory (IPT), inspired by Rayleigh-Schrödinger perturbation theory, is introduced. IPT computes any number of eigenpairs efficiently using a parallel iterative procedure, making it more efficient than classical methods. Sufficient conditions for linear convergence are provided, and performance is demonstrated on various types of matrices, including one from quantum chemistry.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Mawia Osman, Yonghui Xia, Omer Abdalrhman Omer, Ahmed Hamoud
Summary: In this article, the fuzzy Adomian decomposition method and fuzzy modified Laplace decomposition method are presented to solve the fuzzy fractional Navier-Stokes equations. The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are also investigated for solving fuzzy linear-nonlinear Schrodinger differential equations. Comparisons are made with other methods, and the proposed methods are found to be simpler and consistent with analytical and numerical results.
Article
Mathematics, Applied
Hirotada Okawa, Kotaro Fujisawa, Yu Yamamoto, Ryosuke Hirai, Nobutoshi Yasutake, Hiroki Nagakura, Shoichi Yamada
Summary: This article proposes a new method for solving nonlinear systems of equations. Inspired by the mathematical property of damped oscillators, the method can be seen as a simple extension to the Newton-Raphson method. It exhibits the same local convergence as the NR method but has a significantly wider convergence region or global convergence.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Quantum Science & Technology
Caio B. D. Goes, Thiago O. O. Maciel, Giovani G. G. Pollachini, Juan P. L. C. Salazar, Rafael G. G. Cuenca, Eduardo I. I. Duzzioni
Summary: This paper proposes a hybrid algorithm based on machine learning and quantum ensemble learning to find an approximate solution to a partial differential equation with good precision and favorable scaling in the required number of qubits. The classical component trains multiple regressors capable of approximately solving the equation using machine learning, while the quantum component adapts the QBoost algorithm to build an ensemble of classical learners. The algorithm is successfully applied to solve the 1D Burgers' equation with viscosity, demonstrating the improved solutions compared to classical weak-learners, and implemented on D-Wave Systems with good performance compared to other methods.
QUANTUM INFORMATION PROCESSING
(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
Sebastian Buedo-Fernandez, Daniel Cao Labora, Rosana Rodriguez-Lopez
Summary: This paper studies a class of nonlinear second-order functional differential equations with piecewise constant arguments, which are related to a thermostat controlled by the introduction of functional terms in temperature and temperature change rate. The paper first proves some comparison results for boundary value problems associated with linear delay differential equations, allowing control of the derivatives of the solutions. Then, the method of upper and lower solutions and the monotone iterative technique are applied to deduce the existence and approximations of solutions for a class of boundary value problems, including the periodic case. The paper also provides estimates for these derivatives due to the non-monotonic sequences of derivatives for the approximate solutions. Some examples and conclusions are presented.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Yuesheng Xu, Taishan Zeng
Summary: As the amount of available data in applications increases, more competent learning models are demanded for data processing. Data often have embedded sparsity structures, where their energies can concentrate on a small number of basis functions if represented appropriately. This paper focuses on the numerical study of adaptive approximation of solutions of nonlinear partial differential equations using deep neural networks (DNNs) with a sparse regularization with multiple parameters. By employing a penalty with multiple parameters, the proposed DNNs with multi-scale sparse regularization (SDNN) effectively represent functions with singularities. Numerical examples confirm the sparsity and accuracy of solutions generated by the proposed SDNN.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Sandip Maji, Srinivasan Natesan
Summary: This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial-boundary-value problems. The non-linearity is tackled using the Newton linearization process. The non-symmetric interior penalty Galerkin method is applied for the spatial variable, and a semi-discrete problem is obtained in the time variable. By using L1-scheme and L2-scheme for the time-fractional derivative, and trapezoidal rule for the integral term, a fully-discrete scheme is derived.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Alicia Cordero, Neus Garrido, Juan Ramon Torregrosa, Paula Triguero-Navarro
Summary: This paper studies different approaches to introduce memory into a parametric family of optimal two-step iterative methods. The convergence and stability of the resulting methods are examined through real dynamics for comparison. Several numerical experiments are also conducted to observe the behavior of the methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
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
T. S. Jang
Summary: The presented iterative semi-analytical method converts differential equations to integral equations with the introduction of a pseudo-parameter, allowing for solutions without relying on small perturbation parameters. This method offers a wide application in handling strongly nonlinear physical problems effectively.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Computer Science, Interdisciplinary Applications
Liyao Lyu, Zhen Zhang, Minxin Chen, Jingrun Chen
Summary: In recent years, there has been a significant focus on solving partial differential equations (PDEs) using deep learning techniques. This study introduces a deep mixed residual method (MIM) for solving PDEs with high-order derivatives. The MIM approach provides better approximations than traditional numerical methods in many cases and exhibits interesting connections with them.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics
Ying Li, Longxiang Xu, Shihui Ying
Summary: In this paper, a deep wavelet neural network model is proposed to approximate natural phenomena described by classical partial differential equations (PDEs). The effectiveness of the proposed method is demonstrated through various numerical experiments, showing that it outperforms the state-of-the-art approach.
Article
Mathematics
Hassan Kamil Jassim, Mohammed Abdulshareef Hussein
Summary: Recently, researchers have developed a new method called the Hussein-Jassim (HJ) method for solving nonlinear fractional ordinary differential equations, which have wide-ranging applications in various scientific fields. The HJ method is based on a power series of fractional order and provides approximate solutions for these equations. The results of this study demonstrate a high degree of consistency between the approximate solutions obtained using the HJ method and the exact solutions.
Article
Mathematics, Applied
Peijun Chen, Jianguo Huang, Xiaoqun Zhang
Article
Mathematics, Applied
Xuehai Huang, Jianguo Huang
JOURNAL OF SCIENTIFIC COMPUTING
(2013)
Article
Mathematics, Applied
Huang JianGuo, Lai JunJiang, Tang Tao
SCIENCE CHINA-MATHEMATICS
(2013)
Article
Mathematics, Applied
Fang Feng, Weimin Han, Jianguo Huang
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Jianguo Huang, Lili Ju, Bo Wu
APPLIED MATHEMATICS LETTERS
(2019)
Article
Mathematics, Applied
Yuling Guo, Jianguo Huang, Junjiang Lai
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Computer Science, Interdisciplinary Applications
Jianguo Huang, Lili Ju, Bo Wu
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Mathematics, Applied
Yuling Guo, Jianguo Huang
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Fang Feng, Weimin Han, Jianguo Huang
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Binbin Du, Jianguo Huang, Haibiao Zheng
Summary: Three two-grid Arrow-Hurwicz (A-H) methods are proposed for solving the steady incompressible Navier-Stokes equations, which combine the existing A-H method with different one-step schemes on fine mesh to enhance efficiency. Error analyses and numerical tests demonstrate the theoretical results and efficiency of these methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jianguo Huang, Sen Lin
Summary: This paper introduces and analyzes residual-type a posteriori error estimates for a non-consistent virtual element method for reaction-diffusion equations. The reliability and efficiency of the error bound are derived under a weak mesh assumption, allowing for the development of an adaptive VEM. Numerical experiments confirm the agreement with theoretical results.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Fang Feng, Weimin Han, Jianguo Huang
Summary: This paper focuses on the numerical solution of a fourth-order elliptic variational inequality using the virtual element method (VEM) for the obstacle problem of the Kirchhoff-Love plate. Both conforming and fully nonconforming VEMs are studied, with optimal order error estimates derived in the discrete energy norm. The primal-dual active algorithm is employed to solve the discrete problems, and numerical examples demonstrate the performance and convergence orders of the numerical solutions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Fang Feng, Weimin Han, Jianguo Huang
Summary: This paper investigates a fourth-order hemivariational inequality for a Kirchhoff plate problem and proves the existence and uniqueness of the solution through analyzing a corresponding minimization problem. It develops a nonconforming virtual element method to solve the hemivariational inequality and derives an optimal error estimate for the virtual element solutions under appropriate solution regularity assumptions. The computer simulation results on a numerical example validate the numerical convergence orders.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Jianguo Huang, Yue Yu
Summary: This paper develops estimates on virtual element methods (VEMs), including inverse inequalities, norm equivalence, and interpolation error estimates, for star-shaped polyhedral meshes with faces that allow virtual regular and quasi-uniform triangulations. The regularity of these meshes covers the usual ones used for theoretical analysis of VEMs, and the proofs are carried out straightforwardly.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)