Article
Mathematics
Piotr Biler, Alexandre Boritchev, Lorenzo Brandolese
Summary: This paper studies the global existence of the parabolic-parabolic Keller-Segel system in R-d, d >= 2. It is proven that global solutions are obtained for initial data of arbitrary size as long as the diffusion parameter tau is large enough in the equation for the chemoattractant. The analysis improves earlier results and extends them to any dimension d >= 3. Optimal size conditions for the global existence of solutions are discussed, and two toy models are introduced to illustrate the fact, showing finite time blowup for a class of large solutions in a companion paper.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Xiangsheng Xu
Summary: This paper investigates the initial boundary value problem for the Keller-Segel model with nonlinear diffusion. The study shows that nonlinear diffusion can prevent overcrowding and proves that solutions are bounded under certain conditions, expanding the existing knowledge in this area. The results also suggest that the Keller-Segel model can have bounded solutions and blow-up solutions simultaneously.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Mario Bezerra, Claudio Cuevas, Clessius Silva, Herme Soto
Summary: This work investigates the time-fractional doubly parabolic Keller-Segel system in Double-struck capital R-N (N >= 1) and establishes refined results on the large time behavior of solutions. The study involves the well-posedness and asymptotic stability of solutions in Marcinkiewicz spaces, achieved through appropriate estimation of system nonlinearities and analysis based on Duhamel-type representation formulae and the Kato-Fujita framework with a fixed-point argument using a suitable time-dependent space.
SCIENCE CHINA-MATHEMATICS
(2022)
Article
Mathematics
Siming He, Eitan Tadmor
Summary: The study examines the regularity and large-time behavior of species driven by chemo-tactic interactions. Different species interact with the rest of the crowd through different chemical reactions, resulting in a coupled system of parabolic Patlak-Keller-Segel equations. It is shown that the densities of different species diffuse to zero if certain sub-critical conditions for chemical interactions between the species are met.
INDIANA UNIVERSITY MATHEMATICS JOURNAL
(2021)
Article
Biochemical Research Methods
Leon Avery, Brian Ingalls, Catherine Dumur, Alexander Artyukhin
Summary: Collective behaviors in Starved first-stage larvae of the nematode Caenorhabditis elegans are known to produce large-scale organization. A mathematical model was developed to explain how and why the larvae aggregate, focusing on chemotaxis and the role of two chemical signals. Knocking out the sensory receptor gene srh-2 resulted in irregularly shaped aggregates, suggesting that mutant worms moved slower than wild type.
PLOS COMPUTATIONAL BIOLOGY
(2021)
Article
Mathematics, Applied
Kentaro Fujie, Jie Jiang
Summary: This paper investigates the initial Neumann boundary value problem for a degenerate kinetic model of Keller-Segel type, focusing on the boundedness of classical solutions. The study shows that classical solutions are globally bounded in two-dimensional settings when the motility function decreases slower than an exponential speed at high signal concentrations, and boundedness is obtained in higher dimensions when the motility decreases at certain algebraical speed. The proof is based on comparison methods from previous work and a modified Alikakos-Moser type iteration, with new estimations involving weighted energies to establish the boundedness.
ACTA APPLICANDAE MATHEMATICAE
(2021)
Article
Mathematics
Taiki Takeuchi
Summary: We prove the existence and uniqueness of local strong solutions of Keller-Segel system, as well as construct global strong solutions for small initial data. The proof is based on the maximal Lorentz regularity theorem of heat equations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Santanu Saha Ray
Summary: In this paper, the similarity method is employed to solve a fractional Keller-Segel model with a nonlocal fractional Laplace operator, and the results are verified through comparison of the fractional centred difference method and the weighted shifted Gruwald-Letnikov difference method, demonstrating the accuracy and efficiency of the proposed numerical schemes.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Wenbin Lyu
Summary: This paper deals with a class of signal-dependent motility Keller-Segel systems, investigates the existence of global-in-time solutions in a bounded domain, and rules out blow-up phenomenon by superlinear damping.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Xueling Huang, Jie Shen
Summary: We propose a class of scalar auxiliary variable (SAV) schemes with relaxation that unconditionally preserve the physical properties of biological chemotaxis phenomenon at the discrete level. These schemes only require solving decoupled linear systems with constant coefficients at each time step. Extensive numerical results validate the effectiveness of these schemes for simulating chemotactic non-aggregation and aggregation phenomena, as well as investigating blow-up phenomenon.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Mario Fuest, Johannes Lankeit, Yuya Tanaka
Summary: In this paper, we study quasilinear Keller-Segel systems with indirect signal production and prove the existence and boundedness of the solution under certain conditions. We also show that the solution blows up in finite or infinite time under certain conditions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Yuki Naito
Summary: The study focuses on the simplest parabolic-elliptic model of chemotaxis in space dimensions N >= 3, and identifies optimal conditions on the initial data for finite time blow-up and global existence of solutions based on stationary solutions. The argument is based on the analysis of the Cauchy problem for the transformed equation involving the averaged mass of the solution.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Xu Song, Jingyu Li
Summary: We study the large time behaviors of solutions to the Keller-Segel system with logarithmic singular sensitivity in the half space, where biological mixed boundary conditions are prescribed. The existence and asymptotic stability of spiky steady states of this system were proved by Carrillo et al. (2021). In this paper we obtain convergence rate of solutions towards the steady state under appropriate initial perturbations. The proofs are based on a Cole-Hopf type transformation and a weighted energy method, where the weights are artfully constructed.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhen Bin Cao, Xiao Feng Liu, Meng Wang
Summary: This research demonstrates well-posedness and time-decay estimates for the Cauchy problem of the Keller-Segel system in critical scaling-invariant Besov spaces by utilizing Littlewood-Paley analysis and decay estimates of heat kernels.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2021)
Article
Computer Science, Interdisciplinary Applications
Shufen Wang, Simin Zhou, Shuxun Shi, Wenbin Chen
Summary: In this paper, new first-order and second-order accuracy BDF schemes are proposed, allowing for parallel computation of discrete schemes. The standard backward differentiation formula is applied to approximate the original continuous equations with a regularization term added for unconditional energy stability in both first-order and second-order schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Ricardo Ruiz-Baier, Matteo Taffetani, Hans D. Westermeyer, Ivan Yotov
Summary: In this study, a new mixed-primal finite element scheme was proposed to solve the multiphysics model involving fluid flow and consolidation equations without the need for Lagrange multipliers. The research focused on numerical simulations related to geophysical flows and eye poromechanics, exploring different interfacial flow regimes that could help understand early morphologic changes associated with glaucoma in canine species.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Infectious Diseases
Amna Tariq, Tsira Chakhaia, Sushma Dahal, Alexander Ewing, Xinyi Hua, Sylvia K. Ofori, Olaseni Prince, Argita D. Salindri, Ayotomiwa Ezekiel Adeniyi, Juan M. Banda, Pavel Skums, Ruiyan Luo, Leidy Y. Lara-Diaz, Raimund Burger, Isaac Chun-Hai Fung, Eunha Shim, Alexander Kirpich, Anuj Srivastava, Gerardo Chowell
Summary: This study utilizes mathematical models to investigate the transmission dynamics and short-term forecasting of the COVID-19 pandemic in Colombia. The findings show a decline in disease transmission at the national and regional level, but variations in incidence rate patterns across different departments. Additionally, the study examines the relationship between mobility and social media trends and the occurrence of case resurgences, along with the geographic heterogeneity of COVID-19 in Colombia.
PLOS NEGLECTED TROPICAL DISEASES
(2022)
Article
Geochemistry & Geophysics
Jiandong Wang, Aixiang Wu, Zhuen Ruan, Raimund Burger, Yiming Wang, Shaoyong Wang, Pingfa Zhang, Zhaoquan Gao
Summary: Cemented paste backfill (CPB) blended with coarse aggregates (CA-CPB) is a widely used technology for environmental protection and underground goaf treatment. This study investigates the influences of solid concentration, coarse aggregates dosage, and cement dosage on the rheological properties and compressive strength of CA-CPB through experimental methods. The results show that solid concentration and cement dosage have the most significant effects on the rheological properties and compressive strength, respectively. Multiple response optimization is performed using an overall desirability function approach to obtain the optimal parameters for high fluidity and strength, providing valuable information for the CA-CPB process in the Chifeng Baiyinnuoer Lead and Zinc Mine.
Article
Mathematics, Applied
Gabriel N. Gatica, Bryan Gomez-Vargas, Ricardo Ruiz-Baier
Summary: In this paper, the a posteriori error analysis for mixed-primal and fully-mixed finite element methods approximating the stress-assisted diffusion of solutes in elastic materials is developed. Two efficient and reliable residual-based a posteriori error estimators are derived and their performance is confirmed through numerical tests, illustrating the effectiveness of adaptive mesh refinement.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
N. A. Barnafi, B. Gomez-Vargas, W. J. Lourenco, R. F. Reis, B. M. Rocha, M. Lobosco, R. Ruiz-Baier, R. Weber dos Santos
Summary: In this paper, we propose a novel coupled poroelasticity-diffusion model that considers the formation process of extracellular edema and infectious myocarditis under large deformations. The model takes into account the interaction between interstitial flow and the immune-driven dynamics between leukocytes and pathogens. A numerical approximation scheme using five-field finite element method is developed and stability analysis is conducted. The computational tests demonstrate the properties of the model and finite element schemes.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Veronica Anaya, Arbaz Khan, David Mora, Ricardo Ruiz-Baier
Summary: We develop the a posteriori error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. Our methods are robust and valid in 2D and 3D, and for arbitrary polynomial degree. Numerical examples demonstrate the error behavior predicted by the theoretical analysis, and adaptive mesh refinement is performed based on the a posteriori error estimators.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Physiology
Wesley de Jesus Lourenco, Ruy Freitas Reis, Ricardo Ruiz-Baier, Bernardo Martins Rocha, Rodrigo Weber dos Santos, Marcelo Lobosco
Summary: This paper investigates the formation of myocardial edema in acute infectious myocarditis and modifies a model to describe the associated dynamics. Computational methods can provide insights into the relationship between pathogens and the immune system, shedding light on the variations in myocarditis inflammation among different patients.
FRONTIERS IN PHYSIOLOGY
(2022)
Article
Engineering, Multidisciplinary
Raimund Burger, Julio Careaga, Stefan Diehl, Romel Pineda
Summary: This article proposes a numerical scheme for simulating reactive settling in sequencing batch reactors (SBRs) in wastewater treatment plants. The scheme utilizes a complex mathematical model to accurately describe the reactive settling process and has been validated through simulations.
APPLIED MATHEMATICAL MODELLING
(2022)
Article
Engineering, Chemical
Raimund Burger, Julio Careaga, Stefan Diehl, Romel Pineda
Summary: A model of reactive settling is developed for the activated sludge process in wastewater treatment, which accounts for the spatial variability of reaction rates caused by biomass concentration variation. The model includes nonlinear partial differential equations and a numerical scheme for simulating hindered settling, compression, particle dispersion, and fluid dispersion. Experimental data from a pilot plant are used to fit the model, resulting in good predictability of the reactive sedimentation process.
CHEMICAL ENGINEERING SCIENCE
(2023)
Article
Mathematics, Applied
Paulo Amorim, Raimund Burger, Rafael Ordonez, Luis Miguel Villada
Summary: This study investigates the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition. The model considers the diffusion of the predator species towards higher concentrations of a chemical substance produced by the prey, as well as the movement of the prey away from high concentrations of a substance secreted by the predators. The study proves the local existence of nonnegative solutions and provides numerical simulations to discuss the system.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2023)
Article
Mathematics, Applied
Raimund Buerger, Sonia Valbuena, Carlos A. Vega
Summary: A reduced model of blood flow in arteries is proposed as a hyperbolic system of balance laws in one dimension with the unknowns as cross-sectional area and average flow velocity. An entropy stable finite difference scheme is constructed based on the entropy pair property, employing fourth-order entropy conservative flux and sign-preserving reconstruction, as well as a second-order strong stability preserving Runge-Kutta method. The scheme is computationally inexpensive, well-balanced, and numerically validated.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Xi -Yuan Yin, Kai Schneider, Jean-Christophe Nave
Summary: We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the 3D incompressible Euler equations. This method discretizes the flow map associated with the velocity field to evolve advected quantities. By utilizing the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deformations can be accurately computed from short-time submaps on coarse grids. The method extends the CM method for 2D incompressible Euler equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Geochemistry & Geophysics
Fernando Betancourt, Raimund Burger, Stefan Diehl, Leopoldo Gutierrez, M. Carmen Marti, Yolanda Vasquez
Summary: This article investigates the operation model of a froth flotation column. The model is described by a nonlinear convection-diffusion partial differential equation, incorporating solids-flux and drift-flux theories as well as a model of foam drainage. The model predicts the variations of bubble and (gangue) particle volume fractions with height and time. Through comparison with experimental results, operating charts for a modified version of the model are derived for steady-state operations with a stationary froth layer.
Article
Engineering, Chemical
Gezhong Chen, Cuiping Li, Zhuen Ruan, Raimund Burger, Yuan Gao, Hezi Hou
Summary: This study analyzed the bed drainage channel structure and the effect of coarse particles on the bed porosity and pore connectivity in cemented paste backfill technology. The results showed that the pressure and shear significantly influenced the spherical pore diameter, throat channel radius, and throat channel length. The volume and number of coarse particles had a negative impact on the bed porosity and pore connectivity. This research provides guidance for improving mining efficiency and reducing the number of tailings ponds.
MINERALS ENGINEERING
(2023)
Article
Mathematics, Interdisciplinary Applications
Raimund Burger, Stefan Diehl, M. Carmen Marti, Yolanda Vasquez
Summary: This article formulates a triangular system of conservation laws with discontinuous flux to model the one-dimensional flow of two disperse phases through a continuous one. The triangularity is due to the distinction between a primary and a secondary disperse phase, where the movement of the primary disperse phase is independent of the local volume fraction of the secondary phase. The article presents a monotone numerical scheme supported by theoretical arguments, and provides numerical examples and estimations of numerical error and convergence rates.
NETWORKS AND HETEROGENEOUS MEDIA
(2023)
Article
Mathematics, Applied
Xuesong Wang, Yao Cheng
Summary: The local discontinuous Galerkin (LDG) method on a Shishkin mesh is investigated for a one-dimensional singularly perturbed reaction-diffusion problem. Improved pointwise error estimates are derived based on the discrete Green's function in the regular and layer regions. The convergence rates of the pointwise error for both the LDG approximation to the solution and its derivative are analyzed, showing optimal rates in different domains. Moreover, optimal pointwise error estimates are established when the regular component of the exact solution belongs to the finite element space. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arash Goligerdian, Mahmood Khaksar-e Oshagh
Summary: This paper presents a computational method for simulating more accurate models for population growth with immigration, using integral equations with a delay parameter. The method utilizes Legendre wavelets within the Galerkin scheme as an orthonormal basis and employs the composite Gauss-Legendre quadrature rule for computing integrals. An error bound analysis demonstrates the convergence rate of the method, and various numerical examples are provided to validate the efficiency and accuracy of the technique as well as the theoretical error estimate.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
A. Sreelakshmi, V. P. Shyaman, Ashish Awasthi
Summary: This paper focuses on constructing a lucid and utilitarian approach to solve linear and non-linear two-dimensional partial differential equations. Through testing, it is found that the proposed method is highly applicable and accurate, showing excellent performance in terms of cost-cutting and time efficiency.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Shujiang Tang
Summary: This paper investigates the impact of the structure of local smoothness indicators on the computational performance of the WENO-Z scheme. A new class of two-parameter local smoothness indicators is proposed, which combines the classical WENO-JS and WENO-UD5 schemes and appends the coefficients of higher-order terms. A new WENO scheme, WENO-NSLI, is constructed using the global smoothness indicators of WENO-UD5. Numerical experiments show that the new scheme achieves optimal accuracy and has higher resolution compared to WENO-JS, WENO-Z, and WENO-UD5.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xue-Feng Duan, Yong-Shen Zhang, Qing-Wen Wang
Summary: This paper addresses a class of constrained tensor least squares problems in image restoration and proposes the alternating direction multiplier method (ADMM) to solve them. The convergence analysis of this method is presented. Numerical experiments show the feasibility and effectiveness of the ADMM method for solving constrained tensor least squares problems, and simulation experiments on image restoration are also conducted.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Wanying Mao, Qifeng Zhang, Dinghua Xu, Yinghong Xu
Summary: In this paper, we derive, analyze, and extensively test fourth-order compact difference schemes for the Rosenau equations in one and two dimensions. These schemes are applied under spatial periodic boundary conditions using the double reduction order method and bilinear compact operator. Our results show that these schemes satisfy mass and energy conservation laws and have unique solvability, unconditional convergence, and stability. The convergence order is four in space and two in time under the D infinity-norm. Several numerical examples are provided to support the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hui Yang, Shengfeng Zhu
Summary: In this paper, shape optimization in incompressible Stokes flows is investigated based on the penalty method for the divergence free constraint at continuous level. Shape sensitivity analysis is performed, and numerical algorithms are introduced. An iterative penalty method is used for solving the penalized state and possible adjoint numerically, and it is shown to be more efficient than the standard mixed finite element method in 2D. Asymptotic convergence analysis and error estimates for finite element discretizations of both state and adjoint are provided, and numerical results demonstrate the effectiveness of the optimization algorithms.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Reza MohammadiArani, Mehdi Dehghan, Mostafa Abbaszadeh
Summary: Lattice Boltzmann method is a powerful solver for fluid flow, but it is challenging to use it to solve other partial differential equations. This paper challenges the LBM to solve the two-dimensional DKS equation by finding a suitable local equilibrium distribution function and proposes a modification for implementing boundary conditions in complex geometries.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Arijit Das, Prakrati Kushwah, Jitraj Saha, Mehakpreet Singh
Summary: A new volume and number consistent finite volume scheme is introduced for the numerical solution of a collisional nonlinear breakage problem. The scheme achieves number consistency by introducing a single weight function in the flux formulation. The proposed scheme is efficient and robust, allowing easy coupling with computational fluid dynamics softwares.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
H. Ait el Bhira, M. Kzaz, F. Maach, J. Zerouaoui
Summary: We present an asymptotic method for efficiently computing second-order telegraph equations with high-frequency extrinsic oscillations. The method uses asymptotic expansions in inverse powers of the oscillatory parameter and derives coefficients through either recursion or solving non-oscillatory problems, leading to improved performance as the oscillation frequency increases.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Hanen Boujlida, Kaouther Ismail, Khaled Omrani
Summary: This study investigates a high-order accuracy finite difference scheme for solving the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A new compact difference scheme is proposed and the a priori estimates and unique solvability are discussed using the discrete energy method. The unconditional stability and convergence of the difference solution are proved. Numerical experiments demonstrate the accuracy and efficiency of the proposed technique.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Alexander Zlotnik, Timofey Lomonosov
Summary: This paper studies a three-level explicit in time higher-order vector compact scheme for solving initial-boundary value problems for the n-dimensional wave equation and acoustic wave equation with variable speed of sound. By using additional sought functions to approximate second order non-mixed spatial derivatives of the solution, new stability bounds and error bounds of orders 4 and 3.5 are rigorously proved. Generalizations to nonuniform meshes in space and time are also discussed.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Fengli Yin, Yayun Fu
Summary: This paper develops an explicit energy-preserving scheme for solving the coupled nonlinear Schrodinger equation by combining the Lie-group method and GSAV approaches. The proposed scheme is efficient, accurate, and can preserve the modified energy of the system.
APPLIED NUMERICAL MATHEMATICS
(2024)