East Asian Journal on Applied Mathematics
type: channel
description: The East Asian Journal on Applied Mathematics (EAJAM) aims at promoting study and research in Applied Mathematics in East Asia. It is the editorial policy of EAJAM to accept refereed papers in all active areas of Applied Mathematics and related Mathematical Sciences. Novel applications of Mathematics in real situations are especially welcome. Substantial survey papers on topics of exceptional interest will also be published occasionally.
publisher: Global Science Press
language: en
publication name: East Asian Journal on Applied Mathematics
issn: 2079-7362
items: Integrable mKdV Models as Reductions of AKNS Integrable Systems via Dual Similarity Transformations
items: Solving Schrödinger Bridge Problem via Continuous Normalizing Flow
items: Piecewise Linear Maximum Entropy Method for Fredholm Integral Equations with Weakly Singular Kernel
items: A Reduced Order Modeling Method with Variable Separation-Based Domain Decomposition for Parametric Dynamical Systems
items: A New Hybrid Trigonometric WENO Scheme for Hamilton-Jacobi Equations
items: A Neural Network Modeling for MHD–Radiative Natural Convection Williamson Fluid Between Concentric Cylinders
items: Heterogeneous Optimized Schwarz Methods for Heat Conduction in Composites with Thermal Contact Resistance
items: Dynamics of the Center of Mass in Arbitrary-Angle Rotating Bose-Einstein Condensates
items: Energy-Based Adaptive Deep Unfitted Nitsche Method for Elliptic Interface Problems
items: Painlevé Analysis and Analytic Solutions of a Variable-Coefficient Sawada-Kotera System in Shallow Water, Ion-Acoustic Waves and Fluid Flow Dynamics
items: Energy-Stable Full Discretization of the Modified Elastic Flow of Closed Curves
items: Quasi-Periodic Breathers of the Hirota-Maccari System
items: High-Order BDFk Parametric Finite Element Methods for Anisotropic Surface Diffusion Flows and Applications in Solid-State Dewetting
items: A Greedy Randomized Average Block Projection Method for Linear Feasibility Problems
items: An Energy Stable TFPM-Based Petrov-Galerkin Scheme for Solving the Allen-Cahn Equation
items: Piecewise Smooth $N$-Dimensional Nonlinear Singular Singularly Perturbed Boundary Value Problems
items: Schwartz Duality for Singularly Perturbed Differential Equations with Chebyshev Spectral Methods
items: A New Multilevel Homotopic Adaptive Finite Element Algorithm for Convection-Dominated Diffusion-Reaction Problems
items: A Fully Decoupled Linear Second-Order Energy Stable Numerical Scheme for the Moving Contact Line Problem
items: The Quasi-Reversibility Regularization for Source Inversion of a Linear Kuramoto-Sivashinsky Equation
items: A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations
items: Geometric Approach to Symmetric Positive Definite Linear Systems
items: Identifying the Order and a Space Source Term in a Time Fractional Diffusion-Wave Equation
items: Asymptotic Behavior of Solutions for a Free Boundary Problem with a Monostable Nonlinearity
items: High-Order Operator-Compensation Schemes with Mass and Energy Conservation for the Multi-Dimensional Zakharov System
items: Exact Solutions for a Higher-Order Boussinesq Equation and a New Higher-Order Boussinesq-Like Equation
items: Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation
items: A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model
items: Error Analysis of $hp$ Spectral Element Approximation for Optimal Control Problems with Control Constraint
items: Multi-View Data Clustering via Dynamical Optimization of Consensus Laplacian Matrix

Integrable mKdV Models as Reductions of AKNS Integrable Systems via Dual Similarity Transformations
type: item
description: This paper explores integrable modified Korteweg-de Vries (mKdV) models using dual similarity transformations. Two representative examples involving distinct similarity transformations are presented, along with their corresponding reduced Ablowitz-Kaup-Newell–Segur matrix spectral problems. These examples illustrate how reduced matrix mKdV integrable models can be systematically generated through such transformations.
subject: Articles
  topic: section
subject: Lax pair
  topic: keywords
subject: AKNS matrix spectral problem
  topic: keywords
subject: zero-curvature equation
  topic: keywords
subject: integrable hierarchy
  topic: keywords
subject: similarity transformation
  topic: keywords
creator: Wen-Xiu Ma
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-24
publication date: 2026-02-24
doi: 10.4208/eajam.2025-103.181125

Solving Schrödinger Bridge Problem via Continuous Normalizing Flow
type: item
description: The Schrödinger bridge problem (SBP), which can be understood as an entropy-regularized optimal transport, seeks to compute stochastic dynamic mappings connecting two given distributions. SBP has shown significant theoretical importance and broad practical potential, with applications spanning a wide range of interdisciplinary fields. While theoretical aspects of the SBP are well-understood, practical computational solutions for general cases have remained challenging. This work introduces a computational framework that leverages continuous normalizing flows and score matching methods to approximate the drift in the dynamic formulation of the SBP. The learned drift term can be used for building generative models, opening new possibilities for applications in probability flow-based methods. We also provide a rigorous $Γ$−convergence analysis for our algorithm, demonstrating that the neuron network solutions converge to the theoretical ones as the regularization parameter tends to infinity. Lastly, we validate our algorithm through numerical experiments on fundamental cases.
subject: Articles
  topic: section
subject: Schrödinger bridge
  topic: keywords
subject: continuous normalizing flow
  topic: keywords
subject: $Γ$-convergence
  topic: keywords
creator: Yang Jing
creator: Lei Li
creator: Jingtong Zhang
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-24
publication date: 2026-02-24
doi: 10.4208/eajam.2025-085.181125

Piecewise Linear Maximum Entropy Method for Fredholm Integral Equations with Weakly Singular Kernel
type: item
description: Approximate solution of Fredholm integral equations with certain types of weakly singular algebraic kernels is obtained by the piecewise linear maximum entropy method. During implementation of the method, two-dimension numerical integration is reduced to one-dimension numerical integration, which allows to lower the computational cost. The results of numerical experiments are consistent with theoretical findings and show the effectiveness of the method.
subject: Articles
  topic: section
subject: Fredholm integral equation
  topic: keywords
subject: piecewise linear maximum entropy method
  topic: keywords
subject: weakly singular kernel
  topic: keywords
creator: Yucheng Song
creator: Tingting Li
creator: Jiu Ding
creator: Congming Jin
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-24
publication date: 2026-02-24
doi: 10.4208/eajam.2025-140.241225

A Reduced Order Modeling Method with Variable Separation-Based Domain Decomposition for Parametric Dynamical Systems
type: item
description: This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing frequency variables and parameters inherited from the original model. To reduce the computational cost of the frequency-variable elliptic equations, we extend the variable-separation-based domain decomposition method to the complex-valued context, resulting in an offline-online procedure for solving the parametric dynamical systems. At the offline stage, separate representations of the solutions for the interface problem and the subproblems are constructed. At the online stage, the solutions of the parametric dynamical systems for new parameter values can be directly derived by utilizing the separate representations and implementing the inverse Fourier transform. The proposed approach is capable of being highly efficient because the online stage is independent of the spatial discretization. Finally, we present three specific instances of parametric dynamical systems to demonstrate the effectiveness of the proposed method.
subject: Articles
  topic: section
subject: Parametric dynamical system
  topic: keywords
subject: model order reduction
  topic: keywords
subject: domain decomposition
  topic: keywords
subject: variable-separation method
  topic: keywords
subject: Fourier transformation
  topic: keywords
creator: Yuming Ba
creator: Liang Chen
creator: Yaru Chen
creator: Qiuqi Li
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-23
publication date: 2026-02-23
doi: 10.4208/eajam.2025-099.281025

A New Hybrid Trigonometric WENO Scheme for Hamilton-Jacobi Equations
type: item
description: In this paper, we introduce a new hybrid weighted essentially non-oscillatory (WENO) scheme grounded in trigonometric polynomials for the solution of Hamilton-Jacobi equations. This innovative approach utilizes trigonometric polynomial reconstruction as an alternative to the conventional algebraic polynomial reconstruction. Notably, the proposed scheme demonstrates reduced truncation errors in smooth regions and enhanced resolution in nonsmooth regions, outperforming the classical trigonometric WENO scheme — cf. [J. Zhu and J.X. Qiu, Commun. Comput. Phys. 8 (2010)]. In particular, we have designed a novel streamlined hybrid strategy aimed at improving the computational efficiency of the scheme, which surpasses the classical WENO scheme in this regard. A comprehensive suite of numerical experiments has been undertaken to demonstrate the merits of the proposed scheme.
subject: Articles
  topic: section
subject: WENO scheme
  topic: keywords
subject: Hamilton-Jacobi equation
  topic: keywords
subject: finite difference scheme
  topic: keywords
subject: hybrid scheme
  topic: keywords
creator: Liang Li
creator: Zhengwei Hou
creator: Liuyong Pang
creator: Jun Zhu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-23
publication date: 2026-02-23
doi: 10.4208/eajam.2025-065.211025

A Neural Network Modeling for MHD–Radiative Natural Convection Williamson Fluid Between Concentric Cylinders
type: item
description: This study investigates the natural convection flow of Williamson fluid between two concentric cylinders while affected by the radiation effect and magnetic field. The inner cylinder remains fixed while the outer cylinder rotates. Additionally, magnetic field is oriented radially, which influences the flow of the fluid. Applying a proper transformation, one transform the non-linear partial differential equations of the Williamson fluid model into ordinary differential equations. Artificial neural networks (ANN) facilitate the computation of solutions to these nonlinear ordinary differential equations. Trial functions employ a multilayer perceptron neural network with tunable parameters, including weights and biases. The governing equations are satisfied by determining the trial solution’s changeable parameters by applying the Adam (adaptive moment estimation algorithm) optimization technique. Compared to the analytical solutions, the ANN’s result demonstrates good accuracy. Moreover, graphs show how pertinent parameters affect the velocity and temperature profiles. The temperature and velocity profiles get smaller as the magnetic parameter value increases. Furthermore, the temperature and velocity profiles increase as the Hall parameter value rises.
subject: Articles
  topic: section
subject: Artificial neural network
  topic: keywords
subject: concentric cylinders
  topic: keywords
subject: magnetic field
  topic: keywords
subject: natural convection
  topic: keywords
subject: radiation effect
  topic: keywords
creator: Subham Jangid
creator: Kaladhar Kolla
rights: Copyright (c) 2026 East Asian Journal on Applied Mathematics
date: 2026-02-15
publication date: 2026-02-15
starting page: 662
ending page: 678
doi: 10.4208/eajam.2024-219.110225

Heterogeneous Optimized Schwarz Methods for Heat Conduction in Composites with Thermal Contact Resistance
type: item
description: Heat transfer in composites is crucial in engineering, where imperfect layer contact induces thermal contact resistance (TCR), causing interfacial temperature jumps. We propose to solve this numerically using the optimized Schwarz method (OSM), which decouples the heterogeneous problem into homogeneous subproblems. This approach avoids ill-conditioned systems typical of monolithic methods under high contrast and interface discontinuities. Convergence of the algorithm with a standard Robin transmission condition is established via energy estimates and Fourier analysis. For accelerated convergence, a scaled Robin condition is introduced, with rigorous optimization of its free parameter. Key findings emerge due to TCR: first, larger TCR values speed up OSM convergence, achieving asymptotic mesh-independence. This contrasts with the mesh-dependent behavior observed without TCR. Second, greater heterogeneity contrast enhances convergence; third, unlike TCR-free cases, higher thermal conductivity also promotes convergence, similar to heterogeneity; finally, the scaled Robin condition outperforms the standard one in both theory and practice. Numerical tests validate the results and demonstrate the method's potential for nonlinear problems on irregular domains.
subject: Articles
  topic: section
subject: Optimized Schwarz method
  topic: keywords
subject: heat conduction in composites
  topic: keywords
subject: thermal contact resistance
  topic: keywords
subject: domain decomposition
  topic: keywords
subject: optimized transmission condition
  topic: keywords
creator: Huan Zhang
creator: Hui Zhang
creator: Yan Wang
creator: Yingxiang Xu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-02-03
publication date: 2026-02-03
starting page: 632
ending page: 661
doi: 10.4208/eajam.2025-094.071025

Dynamics of the Center of Mass in Arbitrary-Angle Rotating Bose-Einstein Condensates
type: item
description: This paper investigates the dynamics of the center of mass in three-dimensional arbitrary-angle rotating Bose-Einstein condensates (ARotBECs), which are governed by the Gross-Pitaevskii equation (GPE) with an arbitrary-angle angular momentum rotation term. The second-order ordinary differential equations (ODEs) which govern the motion of the center of mass of ARotBECs are analytically solved. Subsequently, a novel numerical scheme, which is mass conservative — i.e. the Lagrangian multiplier-based Crank-Nicolson leap-frog (LagM-CNLF) method, integrated with Fourier pseudo-spectral spatial discretization, is proposed to efficiently and accurately simulate the GPE with arbitrary-angle rotation term. Finally, distinct motion patterns of the center of mass are systematically categorized based on analytical solutions and rigorously validated through direct numerical simulations of the GPE, demonstrating robust consistency between theoretical predictions and numerical computational results. The dynamics of quantized vortices is also studied to show the effectiveness of the LagM-CNLF method.
subject: Articles
  topic: section
subject: Bose-Einstein condensate
  topic: keywords
subject: arbitrary-angle rotation
  topic: keywords
subject: Gross-Pitaevskii equation
  topic: keywords
subject: the center of mass
  topic: keywords
subject: Lagrangian multiplier method
  topic: keywords
creator: Qingzhou Shu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-01-28
publication date: 2026-01-28
starting page: 600
ending page: 631
doi: 10.4208/eajam.2025-053.100925

Energy-Based Adaptive Deep Unfitted Nitsche Method for Elliptic Interface Problems
type: item
description: The paper proposes an energy-based adaptive sampling strategy to enhance the performance of the deep unfitted Nitsche method (DUNM) for elliptic interface problems. Instead of relying on fixed or random training points, the proposed refinement indicator dynamically concentrates samples in high-energy regions, including sharp coefficient jumps and interface singularities. This targeted allocation improves both accuracy and efficiency compared to random sampling. Numerical experiments in both two- and three-dimensional settings demonstrate that the method achieves robust accuracy and efficiency across diverse scenarios, from standard geometries to highly irregular interfaces, while effectively handling high-contrast coefficients and multi-subdomain configurations. These results confirm that the proposed adaptive strategy not only reduces training cost but also ensures reliable performance in complex interface problems.
subject: Articles
  topic: section
subject: Interface problem
  topic: keywords
subject: deep learning
  topic: keywords
subject: unfitted Nitsche method
  topic: keywords
subject: adaptive method
  topic: keywords
creator: Qimu Luo
creator: Ren Zhao
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-01-26
publication date: 2026-01-26
starting page: 573
ending page: 599
doi: 10.4208/eajam.2025-069.100925

Painlevé Analysis and Analytic Solutions of a Variable-Coefficient Sawada-Kotera System in Shallow Water, Ion-Acoustic Waves and Fluid Flow Dynamics
type: item
description: A variable-coefficient Sawada-Kotera system is investigated that models the nonlinear behaviors of waves in shallow water, ion-acoustic waves in plasma environments and fluid flow dynamics. The Painlevé integrability is tested by the WTC method with the simplified form of Krustal. The Hirota bilinear method is employed to derive the bilinear form. Consequently, we obtain a variety of analytic solutions, including soliton, lump, and breather solutions. In addition, the interactions between the lump soliton and one stripe soliton, among with the breather soliton and one stripe soliton are discussed.
subject: Articles
  topic: section
subject: Sawada-Kotera system
  topic: keywords
subject: Painlevé integrability
  topic: keywords
subject: soliton solution
  topic: keywords
subject: lump solution
  topic: keywords
subject: breather solution
  topic: keywords
creator: Hao-Qing Chen
creator: Guang-Mei Wei
creator: Yu-Xin Song
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2026-01-13
publication date: 2026-01-13
starting page: 552
ending page: 572
doi: 10.4208/eajam.2025-051.060425

Energy-Stable Full Discretization of the Modified Elastic Flow of Closed Curves
type: item
description: An energy-stable full discretization for the modified elastic flow of closed curves is proposed. This is a gradient flow of a modified elastic energy combining bending and Dirichlet energies. The minimization of Dirichlet energy can lead to improved mesh quality. Gradient flows for both isotropic and anisotropic cases are considered. We derive new evolution equations for the parameterization and curvature vector of curves in arbitrary codimension. The proposed formulation is discretized by a parametric finite element method in space and a first-order implicit scheme in time. We establish the unconditional energy stability for the fully discretized scheme. Additionally, the second-order accuracy of the BDF2 scheme is demonstrated. Numerical examples in two and three dimensions illustrate the efficiency, energy stability, and asymptotic mesh distribution of the method for simulating the modified elastic flow.
subject: Articles
  topic: section
subject: Curve evolution
  topic: keywords
subject: elastic flow
  topic: keywords
subject: parametric finite element method
  topic: keywords
subject: unconditional stability
  topic: keywords
subject: Dirichlet energy
  topic: keywords
creator: Cuiling Ma
creator: Xufeng Xiao
creator: Xinlong Feng
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-12-08
publication date: 2025-12-08
starting page: 527
ending page: 551
doi: 10.4208/eajam.2025-149.031125

Quasi-Periodic Breathers of the Hirota-Maccari System
type: item
description: The Hirota-Maccari (HM) system generalizing the Hirota equation, serves as a generalized (2+1)-dimensional model in fluid dynamics, plasma physics, and optical fiber communication. In this paper, we obtain the quasi-periodic breathers to HM system by using the Hirota’s bilinear method and the theta function. Asymptotic analysis demonstrates that the quasi-periodic breathers can be reduced to regular breathers under small amplitude limits. Moreover, we also classified solutions based on their asymptotic behavior. Numerical examples are given to confirm the theoretical analysis.
subject: Articles
  topic: section
subject: Quasi-periodic breather
  topic: keywords
subject: Hirota-Maccari system
  topic: keywords
subject: Hirota’s bilinear method
  topic: keywords
subject: theta function
  topic: keywords
creator: Mao-Wei Yin
creator: Jian-Qing Sun
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-25
publication date: 2025-11-25
starting page: 505
ending page: 526
doi: 10.4208/eajam.2025-092.100825

High-Order BDFk Parametric Finite Element Methods for Anisotropic Surface Diffusion Flows and Applications in Solid-State Dewetting
type: item
description: In this paper, we extend the BGN formulation [J.W. Barrett, H. Garcke and R. Nürnberg, J. Comput. Phys. 222 (2007)] by incorporating the $k$-order backward differentiation formulae (BDFk) for time discretization. This allows us to develop high-order temporal parametric finite element methods for simulating anisotropic surface diffusion flows and solid-state dewetting problems, achieving accuracy levels from second-order to fourth-order. We prove the well-posedness of the constructed high-order schemes. The proposed schemes maintain good mesh quality characteristic of the classical first-order BGN scheme. Finally, we present several numerical simulations to demonstrate the high-order temporal accuracy and verify the preservation of good mesh quality and energy stability throughout the evolution.
subject: Articles
  topic: section
subject: Parametric finite element method
  topic: keywords
subject: BGN scheme
  topic: keywords
subject: BDFk
  topic: keywords
subject: high-order accuracy
  topic: keywords
subject: energy stability
  topic: keywords
creator: Lechuan Gu
creator: Yihang Guo
creator: Meng Li
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-20
publication date: 2025-11-20
starting page: 477
ending page: 504
doi: 10.4208/eajam.2025-007.190525

A Greedy Randomized Average Block Projection Method for Linear Feasibility Problems
type: item
description: The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the standard RP method by using two ingredients: the greedy probability criterion and the average block approach, and obtains a greedy randomized average block projection (GRABP) method for solving large-scale systems of linear inequalities. We demonstrate that the GRABP method achieves deterministic linear convergence with various extrapolated step sizes. Numerical experiments on both randomly generated and real-world data show the advantage of GRABP over several state-of-the-art solvers, such as the RP method, the sampling Kaczmarz Motzkin (SKM) method, the generalized SKM method, and the Nesterov acceleration of SKM method.
subject: Articles
  topic: section
subject: Linear feasibility
  topic: keywords
subject: randomized projection
  topic: keywords
subject: average block
  topic: keywords
subject: greedy probability criterion
  topic: keywords
subject: convergence property
  topic: keywords
creator: Lin Zhu
creator: Yuan Lei
creator: Jiaxin Xie
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-20
publication date: 2025-11-20
starting page: 457
ending page: 476
doi: 10.4208/eajam.2024-176.150325

An Energy Stable TFPM-Based Petrov-Galerkin Scheme for Solving the Allen-Cahn Equation
type: item
description: An energy stable tailored finite point method based Petrov-Galerkin scheme to solve Allen-Cahn equation is proposed. In time discretization, we present both first-order and second-order semi-discrete schemes based on stabilized and convex-splitting techniques, which satisfy unconditional energy stability. We prove the maximum bound preserving principle for first-order schemes. Due to nonlinearity, the well-posedness of weak formulations based on semi-discrete schemes are demonstrated. As the nature of singularly perturbation in semi-discrete level remains when $ε$ is extremely small, we establish a specified Petrov-Galerkin scheme which leads to a unified way for space discretization. To this end, we set up nonlinear solvers which are proved to be stable and convergent. Then we construct our Petrov-Galerkin scheme, which is built upon problem-dependent test function space. The stability and second-order convergence of this scheme are rigorously proved in one dimension. In order to compute test functions, specialized TFPM schemes are incorporated into the scheme. Numerical experiments show the accuracy, efficiency, and the good performance of the method on uniform meshes even when mesh size $h$ is much larger than $ε$.
subject: Articles
  topic: section
subject: Allen-Cahn equation
  topic: keywords
subject: energy stability
  topic: keywords
subject: nonlinear solvers
  topic: keywords
subject: Petrov-Galerkin scheme
  topic: keywords
subject: tailored finite point method
  topic: keywords
creator: Yueran Wang
creator: Wenli Yang
creator: Zhongyi Huang
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-14
publication date: 2025-11-14
starting page: 427
ending page: 456
doi: 10.4208/eajam.2025-022.190725

Piecewise Smooth $N$-Dimensional Nonlinear Singular Singularly Perturbed Boundary Value Problems
type: item
description: Internal layer phenomena are often appear in various piecewise smooth nonlinear singular singularly perturbed problems of natural sciences. To characterize these special structures, we study piecewise smooth $n$-dimensional nonlinear singular singularly perturbed boundary value problems. In particular, we show the existence of solutions with an internal layer and construct their asymptotic expansions. The remainder estimations of the approximate solutions are also given. Finally, an example aimed to verify the correctness of the developed theory is presented.
subject: Articles
  topic: section
subject: Asymptotic method
  topic: keywords
subject: singularly perturbed equation
  topic: keywords
subject: discontinuous equation
  topic: keywords
creator: Shitao Liu
creator: Mingkang Ni
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-14
publication date: 2025-11-14
starting page: 349
ending page: 364
doi: 10.4208/eajam.2024-162.260525

Schwartz Duality for Singularly Perturbed Differential Equations with Chebyshev Spectral Methods
type: item
description: Singularly perturbed differential equations with the Dirac delta function usually yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been previously proposed, yielding superior results without oscillations. However, this approach has primarily been applied to linear problems and still exhibits the Gibbs phenomenon when extended to nonlinear or higher-dimensional problems. In this paper, we propose a consistent yet simple approach based on Schwartz duality that can handle such problems. Our proposed approach utilizes a modified direct projection method with a consistent discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. As numerical examples, we consider several problems, including the Burgers’ equation and the two-dimensional time-dependent advection equation. The proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction for the problems considered.
subject: Articles
  topic: section
subject: Differential equations with singular sources
  topic: keywords
subject: Dirac delta function
  topic: keywords
subject: Gibbs phenomenon
  topic: keywords
subject: Schwartz duality
  topic: keywords
subject: spectral collocation method
  topic: keywords
creator: Eunwoo Heo
creator: Kwanghyuk Park
creator: Jae-Hun Jung
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-14
publication date: 2025-11-14
starting page: 388
ending page: 412
doi: 10.4208/eajam.2025-089.130625

A New Multilevel Homotopic Adaptive Finite Element Algorithm for Convection-Dominated Diffusion-Reaction Problems
type: item
description: The multilevel homotopic adaptive finite element method (MHAFEM) was developed for solving convection-dominated diffusion-reaction problems on triangular meshes, where the mesh adaptation (refinement and mesh smoothing/moving) is based on an optimal interpolation error estimate in $L^p$ norm. In this paper, we follow the framework of the MHAFEM to give a new multilevel homotopic adaptive finite element algorithm. This new MHAFEM executes mesh refinement based on an a posteriori error estimator instead of the interpolation error estimate. We apply this algorithm to 1D and 2D example problems using 1D linear elements and 2D quadrilateral transition elements, respectively. Numerical results demonstrate the remarkable efficiency of the algorithm.
subject: Articles
  topic: section
subject: Convection-dominated diffusion-reaction problem
  topic: keywords
subject: multilevel homotopic adaptive finite element method
  topic: keywords
subject: quadrilateral transition element
  topic: keywords
subject: layer
  topic: keywords
creator: Jie Zhang
creator: Guozhu Yu
creator: Xiaoping Xie
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-14
publication date: 2025-11-14
starting page: 326
ending page: 348
doi: 10.4208/eajam.2024-207.280425

A Fully Decoupled Linear Second-Order Energy Stable Numerical Scheme for the Moving Contact Line Problem
type: item
description: A decoupled linear second-order accurate unconditionally energy stable scheme for a phase-field model of the moving contact line (MCL) problem is proposed. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations with the generalized Navier boundary condition. The scheme proposed introduces three scalar auxiliary variables to decouple nonlinear coupling terms while retaining stability and accuracy. Two of these variables represent the bulk and boundary energy, and one captures the “zero-energy-contribution” property between convection and surface tension terms. All related terms are decoupled by applying semi-explicit treatments while preserving stability and accuracy. Further coupling between velocity and pressure is removed by adopting a projection method. The overall scheme is second-order accurate, and the unconditional stability of energy is rigorously proved. Numerical results demonstrate the accuracy, efficiency, and stability of the method for the MCL problem. Besides, a consistent implementation of the contact angle hysteresis (CAH) is conducted to model important CAH phenomena.
subject: Articles
  topic: section
subject: Moving contact line
  topic: keywords
subject: phase-field model
  topic: keywords
subject: fully decoupled
  topic: keywords
subject: second-order
  topic: keywords
subject: unconditional energy stability
  topic: keywords
creator: Min Gao
creator: Yi Shi
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-14
publication date: 2025-11-14
starting page: 365
ending page: 387
doi: 10.4208/eajam.2024-187.020625

The Quasi-Reversibility Regularization for Source Inversion of a Linear Kuramoto-Sivashinsky Equation
type: item
description: The source inversion problem for the linear Kuramoto-Sivashinsky equation is studied. To address the ill-posedness of the source term inversion, a quasi-reversibility regularization method is used for the stable reconstruction of the source term. In the case of noisy measurement data, the convergence (error estimate) of regularized solutions with respect to noise level is analyzed under a priori choice of the regularization parameter. Subsequently, a modified Morozov discrepancy principle is proposed to select regularization parameters for the a posteriori principle. The convergence (error estimate) of regularized solutions is established under this a posteriori principle. Numerical examples are provided to verify the effectiveness of the proposed quasi-reversibility regularization method.
subject: Articles
  topic: section
subject: Kuramoto-Sivashinsky equation
  topic: keywords
subject: source term inversion
  topic: keywords
subject: quasi-reversibility regularization
  topic: keywords
subject: ill-posed problem
  topic: keywords
subject: convergence
  topic: keywords
creator: Bin Hu
creator: Xingyu Wang
creator: Zewen Wang
creator: Wen Zhang
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-10
publication date: 2025-11-10
starting page: 413
ending page: 426
doi: 10.4208/eajam.2025-029.190725

A Fast Cascadic Multigrid Method for Exponential Compact FD Discretization of Singularly Perturbed Convection-Diffusion Equations
type: item
description: Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.
subject: Articles
  topic: section
subject: Singular perturbation
  topic: keywords
subject: convection-diffusion problem
  topic: keywords
subject: boundary layer
  topic: keywords
subject: high efficiency
  topic: keywords
subject: exponential finite difference method
  topic: keywords
subject: multigrid
  topic: keywords
subject: high-order compact scheme
  topic: keywords
creator: Kejia Pan
creator: Huaqing Wang
creator: Pinxia Wu
creator: Jinxuan Wang
creator: Jiajia Xie
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-10
publication date: 2025-11-10
starting page: 301
ending page: 325
doi: 10.4208/eajam.2024-212.120525

Geometric Approach to Symmetric Positive Definite Linear Systems
type: item
description: This paper compares the performance of the conjugate gradient method and geometric approach in the case of symmetric positive definite (SPD) linear systems. This approach is based on the geometric theory of ODEs which was effectively initiated by Poncaré and Liapunov. The simplest and most obvious advantage of the geometric approach over the conjugate gradient method (the MATLAB code pcg) is that this approach can find the inverse of the underlying positive definite matrix and the solution. We present various numerical examples, which demonstrate the superiority of the geometric approach. For SPD linear systems, this approach provides much higher accuracy than the conjugate gradient method. In particular, since it is a one-stop procedure, it can avoid the growth of accumulated round-off errors to some extent.
subject: Articles
  topic: section
subject: Geometric approach
  topic: keywords
subject: linear algebraic system
  topic: keywords
subject: direct method
  topic: keywords
subject: conjugate gradient method
  topic: keywords
subject: matrix inversion
  topic: keywords
creator: Xinyuan Wu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-11-10
publication date: 2025-11-10
starting page: 227
ending page: 257
doi: 10.4208/eajam.2024-074.161024

Identifying the Order and a Space Source Term in a Time Fractional Diffusion-Wave Equation
type: item
description: This paper is devoted to identifying the order of time fractional derivative and a space-dependent source term in a time fractional diffusion-wave equation from some additional measured data in a subdomain or on a subboundary with a small time period. The Lipschitz continuity of forward operators mapping the unknown order and source term into the given data are established based on the stability estimates of solution for the direct problem. We prove the uniqueness of the considered inverse problems by using the asymptotic behavior of the solution at $t$ = 0, the Titchmarsh convolution theorem and the Duhamel principle. Moreover, a Tikhonov-type regularization method is proposed with $H^1$-norm as a penalty term. The existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice are obtained. Then we employ a linearized iteration algorithm combined with the piecewise linear finite element approximation to find simultaneously the approximate order and space source term. Three numerical examples for one- and two-dimensional cases are tested and the numerical results demonstrate the effectiveness of the proposed method.
subject: Articles
  topic: section
subject: Time fractional diffusion-wave equation
  topic: keywords
subject: order and source
  topic: keywords
subject: uniqueness
  topic: keywords
subject: numerical method
  topic: keywords
creator: Ting Wei
creator: Jianming Xu
creator: Xi Yue
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-10-23
publication date: 2025-10-23
starting page: 168
ending page: 203
doi: 10.4208/eajam.2024-198.060425

Asymptotic Behavior of Solutions for a Free Boundary Problem with a Monostable Nonlinearity
type: item
description: A free boundary problem with Dirichlet boundary conditions is studied. Such problems can be used for describing the spread of chemical substances or biological species, which live in a moving region $[0,h(t)].$ In this case, the free boundary $h(t)$ represents the spreading front. If the density of the substance or the population at the boundary exceeds a threshold value, they will be going to spread outwards. On the other hand, the outside environment may be not very beneficial for spreading and this generates a decay rate. We mainly analyze how the decay rate and threshold value affect the solutions. There is a trichotomy result: the solution is either shrinking, or the transition case, or spreading. Besides, if the decay rate or threshold value is large, only the shrinking happens.
subject: Articles
  topic: section
subject: Free boundary problem
  topic: keywords
subject: asymptotic behavior
  topic: keywords
subject: monostable
  topic: keywords
subject: eaction-diffusion equation
  topic: keywords
creator: Jingjing Cai
creator: Siqi Liu
creator: Li Xu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-10-17
publication date: 2025-10-17
starting page: 152
ending page: 167
doi: 10.4208/eajam.2024-193.170325

High-Order Operator-Compensation Schemes with Mass and Energy Conservation for the Multi-Dimensional Zakharov System
type: item
description: This paper presents high-order operator-compensation (OC) methods and the Crank-Nicolson (CN) technology to solve the Zakharov system (ZS). Firstly, the OC method is used for spatial discretization, and semi-discrete high-order OC schemes are obtained, which can maintain mass and energy conservation. Subsequently, the CN technology is adopted for temporal discretization, and full-discrete high-order OC schemes are obtained, which can inherit mass and energy conservation and achieve arbitrary even-order accuracy in space and second-order accuracy in time. In the process of solving ZS, we employ the linearization method to handle the nonlinear ZS, which significantly improves the computational efficiency. Considering the large stencil of numerical schemes, we propose a fast numerical iterative method to deal with the problem, which saves the computational cost. Numerical experiments are given to test accuracy order, verify conservation property, and simulate dynamic evolution.
subject: Articles
  topic: section
subject: Zakharov system
  topic: keywords
subject: high-order operator-compensation method
  topic: keywords
subject: conservation
  topic: keywords
subject: Crank-Nicolson technology
  topic: keywords
creator: Yanjing Li
creator: Xianggui Li
creator: Dongying Hua
creator: Yongyong Cai
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-10-14
publication date: 2025-10-14
starting page: 105
ending page: 128
doi: 10.4208/eajam.2024-195.010325

Exact Solutions for a Higher-Order Boussinesq Equation and a New Higher-Order Boussinesq-Like Equation
type: item
description: A kind of higher-order Boussinesq equation is studied in this work. Based on the Bell polynomials theories, bilinear representations of the equation are derived, and diverse interaction solutions are constructed through Hirota’s bilinear method. These include interaction solutions characterized by hyperbolic cosine and cosine functions, as well as interactions between lump and soliton solutions. In addition, generalized bilinear operators are used in order to construct a new higher-order Boussinesq-like equation, while lump and breather solutions are also developed utilizing Hirota’s bilinear technique. For the various explicit solutions obtained in this work, several of them are considered to selected particular values for the relevant parameters in order to plot different kind of three-dimensional surfaces with associated two-dimensional density profiles to give a comprehensive understanding of the evolution for various solutions.
subject: Articles
  topic: section
subject: Higher-order Boussinesq equation
  topic: keywords
subject: higher-order Boussinesq-like equation
  topic: keywords
subject: binary Bell polynomial
  topic: keywords
subject: Hirota’s bilinear method
  topic: keywords
subject: generalized bilinear method
  topic: keywords
creator: Wenxia Chen
creator: Weixu Ni
creator: Yi Wang
creator: Lixin Tian
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-05-28
publication date: 2025-05-28
starting page: 72
ending page: 89
doi: 10.4208/eajam.2024-046.220824

Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation
type: item
description: The reconstruction of continuous solutions using all available mechanisms and data is essential for high-precision simulations and forecasts. This paper presents an improved random feature method (IRFM) that combines observational data with models for solution reconstruction. For the multi-region neuron approximation, we employ a global activation function with robust local approximation capabilities instead of the traditional piecewise method. This enhances smoothness across regions and accelerates convergence. We also derived the equivalence condition for the optimal solution of multi-constrained optimization problems, established criteria for determining the weights in the cost function and the distribution of randomly generated collocation points, reducing biases from subjective choices. Additionally, we introduce a weighted scheme for computing the cost function related to sparse observations, reducing interpolation errors and improving stability against noise. Numerical examples demonstrate that the IRFM is more accurate and converges faster than the original RFM. Its efficiency and accuracy are validated through comparisons with physics-informed neural networks, and its flexibility is shown by successful continuous solution reconstruction in complex domains.
subject: Articles
  topic: section
subject: Improved random feature method
  topic: keywords
subject: continuous solution reconstruction
  topic: keywords
subject: data assimilation
  topic: keywords
subject: sparse observation
  topic: keywords
creator: Qingchun Li
creator: Qinwu Xu
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-10-29
publication date: 2025-10-29
starting page: 204
ending page: 226
doi: 10.4208/eajam.2024-210.060425

A Fast Discontinuous Galerkin Finite Element Method for a Bond-Based Linear Peridynamic Model
type: item
description: A fast discontinuous Galerkin finite element algorithm on a non-uniform mesh for solution of a one-dimensional bond-based linear peridynamic model with fractional noise is developed. It is based on the approximation of the stiffness matrix corresponding to the discontinuous Galerkin finite element method by its hierarchical representation. The fast algorithm reduces the storage requirement for the stiffness matrix from $\mathscr{O} (N^2 )$ to $\mathscr{O} (kN),$ where $k$ is a parameter controlling the accuracy of hierarchical matrices. The computational complexities of assembling the stiffness matrix and the Krylov subspace method for solving linear systems are also reduced from $\mathscr{O} (N^2)$ to $\mathscr{O} (kN).$ Numerical results show the utility of the numerical method.
subject: Articles
  topic: section
subject: Peridynamic model
  topic: keywords
subject: discontinuous Galerkin finite element
  topic: keywords
subject: hierarchical matrix
  topic: keywords
subject: fast algorithm
  topic: keywords
subject: fractional noise
  topic: keywords
creator: Jincheng Dong
creator: Ning Du
creator: Zhiwei Yang
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-10-29
publication date: 2025-10-29
starting page: 129
ending page: 151
doi: 10.4208/eajam.2024-183.020325

Error Analysis of $hp$ Spectral Element Approximation for Optimal Control Problems with Control Constraint
type: item
description: In this paper, an $hp$ spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$-norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.
subject: Articles
  topic: section
subject: Optimal control problem
  topic: keywords
subject: control constraint
  topic: keywords
subject: $hp$ spectral element method
  topic: keywords
subject: a posteriori error estimate
  topic: keywords
creator: Xiuxiu Lin
creator: Yanping Chen
creator: Yunqing Huang
creator: Fangfang Qin
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-05-28
publication date: 2025-05-28
starting page: 1
ending page: 19
doi: 10.4208/eajam.2023-309.010824

Multi-View Data Clustering via Dynamical Optimization of Consensus Laplacian Matrix
type: item
description: Multi-view data analysis has gained increasing popularity, in particular multi-view spectral clustering has attracted much attention for its outstanding performance in mining heterogeneity information in multi-view data. However, most spectral clustering methods exhibit the following disadvantages: firstly, learning consensus representation directly from multi-view data that may contain noise renders a distorted description; secondly, the traditional two-step process may fall into a suboptimal solution. To overcome these disadvantages, a novel multi-view spectral clustering method is proposed by unifying the optimization of consensus Laplacian matrix and the learning and discretization of spectral embedding into one step. We consider that the optimal Laplacian matrix is in the neighborhood of view-specific Laplacian matrix, as the view-specific Laplacian matrix only contains partial information from multi-view data, resulting in certain deviation from the optimal Laplacian matrix. The consensus Laplacian matrix was obtained in a dynamic optimization way with the spectral rotation and embedding information simultaneously determined. Extensive experiments have been conducted to demonstrate the effectiveness and superiority of our proposed method.
subject: Articles
  topic: section
subject: Multi-view
  topic: keywords
subject: dynamical optimization
  topic: keywords
subject: spectral clustering
  topic: keywords
subject: spectral rotation
  topic: keywords
subject: unified framework
  topic: keywords
creator: Senwen Zhan
creator: Hao Jiang
creator: Dong Shen
creator: Wai-Ki Ching
rights: Copyright (c) 2025 East Asian Journal on Applied Mathematics
date: 2025-05-27
publication date: 2025-05-27
starting page: 20
ending page: 44
doi: 10.4208/eajam.2023-314.140824