Article
Mathematics, Applied
Abu Bakr Elbukhari, Zhenbin Fan, Gang Li
Summary: The manuscript investigates the existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space. No assumptions are made about the compactness of a function or the Lipschitz continuity of a nonlinear function for the nonlocal item and the nonlinear function, respectively. However, continuity of the nonlocal item, continuity of the nonlinear term, and satisfaction of specified assumptions for the nonlinear term are assumed, as well as compactness of the associated semigroup. The theorems are proved using approximate techniques, semigroup methods, and fixed point theorem, which address the noncompactness of operators caused by specified assumptions in the paper. The results obtained in this manuscript improve upon existing knowledge. Finally, two examples are presented to illustrate the main results.
JOURNAL OF FUNCTION SPACES
(2023)
Article
Mathematics
Harsh Prasad, Vivek Tewary
Summary: We prove the local boundedness of variational solutions to the double phase equation under certain restrictions on s, s', p, q, and the non-negative function a(x, y).
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Lun Guo, Qi Li
Summary: In this paper, a Choquard equation with Kirchhoff operator is studied. By using the classical linking theorem and global compactness theorem, it is proved that the equation has at least one bound state solution if the norm of V in L^(N/2) is small. Furthermore, a novel feature of Kirchhoff problems is covered, allowing the parameter a to be zero.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Juan E. Santos, Jose M. Carcione, Jing Ba
Summary: The study demonstrates the existence of a unique solution for wave propagation in linear thermo-poroelastic isotropic media, showing regularity in both space and time variables.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Ke Wu, Guangze Gu
Summary: This paper studies the problem of the Kirchhoff equation involving fractional Laplacian in R-N. By reducing the equation to an equivalent system, the existence and uniqueness of a positive solution with general nonlinearities are obtained. It is concluded that there exist infinitely many sign-changing solutions if the function g is odd. It is also found that a small value of b is a necessary condition for the existence of nontrivial solutions of (K) in the case where 0 < s <= N/4. The method used in this paper works well for the degenerate case a = 0.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics
Minh Le
Summary: This paper investigates the classical solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary conditions. It shows the existence and uniqueness of nonnegative global-in-time classical solutions under certain parameter conditions, and also extends the similar result to the parabolic-parabolic chemotaxis system.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Sun -Sig Byun, Jihoon Ok, Kyeong Song
Summary: We prove local boundedness and Holder continuity for weak solutions to nonlocal double phase problems. Sharp assumptions on the modulating coefficient and the powers are identified, analogous to those for local double phase problems.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2022)
Article
Mathematics, Applied
Cuiying Li, Rui Wu, Ranzhuo Ma
Summary: This paper investigates the existence and uniqueness of solutions for nonlinear quadratic iterative equations in the sense of the Caputo fractional derivative with different boundary conditions. It demonstrates the existence and uniqueness of a solution for the boundary value problems of Caputo fractional iterative equations with arbitrary order by applying the Leray-Schauder fixed point theorem and topological degree theory. It also establishes the well posedness of the control problem of a nonlinear iteration system with a disturbance and guarantees the existence of solutions for a neural network iterative system.
Article
Mathematics
Abbas Moameni, K. L. Wong
Summary: By utilizing a new variational principle, the study proves the existence of weak solution for a nonlocal semilinear elliptic problem, particularly focusing on supercritical cases, and utilizing fractional Sobolev spaces for analysis. This new variational principle allows effective handling of problems beyond standard weakly compact structure. Instead of working on the entire appropriate Sobolev space, this principle enables dealing with the problem on appropriate convex weakly compact subsets.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Mathematics, Applied
Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas
Summary: This paper discusses the existence, uniqueness, and stability of boundary value problems for psi-Hilfer fractional integro-differential equations with mixed nonlocal boundary conditions. The uniqueness result is proved using Banach's contraction mapping principle, and the existence results are established using the Krasnosel'skii's fixed point theorem and the Leray-Schauder nonlinear alternative. Further, four different types of Ulam's stability are studied, and some examples are provided to demonstrate the application of the main results.
Article
Automation & Control Systems
Yadong Shu, Bo Li
Summary: This paper discusses the properties of solutions for uncertain nonlinear switched systems with finite-time horizon and proposes an existence and uniqueness theorem. The research fills the gap in the study of such type of nonlinear switched systems.
Article
Mathematics, Applied
Xueyan Tao, Zhong Bo Fang
Summary: This work studies a generalization of a chemotaxis system and proves the existence of global classical solutions under certain conditions. It also takes into account a borderline case and shows that a global classical solution exists with a suitably large parameter.
APPLIED MATHEMATICS AND OPTIMIZATION
(2023)
Article
Mechanics
M. A. Agwa, A. Pinto da Costa
Summary: This work focuses on the occurrence of multiple solutions in frictional contact problems involving flexible bodies, specifically addressing the quasi-static incremental problem with rectilinear obstacles in two dimensions. The conditions for multiple solutions are presented for different criteria, with a proposed simplified criterion to avoid exponential complexity. An algorithm is suggested for computing all solutions and verifying the accuracy of frictional coefficient estimates.
EUROPEAN JOURNAL OF MECHANICS A-SOLIDS
(2021)
Article
Mathematics, Applied
Fengfei Jin, Baoqiang Yan
Summary: This paper focuses on the existence, uniqueness, and long-time behavior of solutions for a parabolic equation with nonlocal diffusion, even in cases where the reaction term is not Lipschitz-continuous at 0 and grows superlinearly or exponentially at + infinity. Special sub-supersolution pairs are provided and the method of sub-supersolution is established. Using this method, the existence, uniqueness, and long-time behavior of positive solutions are proven, with numerical experiments presented for validation.
Article
Mathematics, Applied
Xue Xu, Sen-Zhong Huang
Summary: The study focuses on the finite time blow-up of solutions to general RD-systems with chemotaxis for multi-species. The blow-up is shown to be equivalent to the blow-up of the L(R)-norms of the solutions for r exceeding a critical value r(c), and an estimation of r(c) is provided under loose conditions, based on a variant of Gagliardo-Nirenberg inequality and a bootstrap method similar to the Alikakos-Moser iteration procedure.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Engineering, Biomedical
Ruirui Liu, Kathryn A. Higley, Maciej H. Swat, Mark A. J. Chaplain, Gibin Powathil, James A. Glazier
Summary: The paper discusses the importance of understanding and designing clinical radiation therapy in oncological treatments. A computational platform was introduced to build a sophisticated multicellular model to simulate how radiation affects living tissue biology. By coupling Geant4 and CompuCell3D, the study developed a tool to simulate the dynamics of biological tissue in the presence of ionizing radiation for quantifying the biological consequences of radiation therapy.
PHYSICS IN MEDICINE AND BIOLOGY
(2021)
Article
Biology
Chiara Villa, Mark A. J. Chaplain, Alf Gerisch, Tommaso Lorenzi
Summary: Mechanical and mechanochemical models of pattern formation in biological tissues, considering different stress-strain constitutive equations for the ECM, reveal that fluid-like constitutive models such as Maxwell and Jeffrey models have higher pattern formation potential compared to solid-like models like Kelvin-Voigt and standard linear solid models. This finding suggests the importance of acquiring detailed quantitative information on the mechanical properties of ECM components in various biological tissues to improve the accuracy of mechanical models in representing tissue rheology.
BULLETIN OF MATHEMATICAL BIOLOGY
(2021)
Article
Biology
Sara Hamis, James Yates, Mark A. J. Chaplain, Gibin G. Powathil
Summary: The study successfully simulated the treatment responses of LoVo cells to the anti-cancer drug AZD6738 by combining a systems pharmacology approach with an agent-based modelling approach, showing the potential of agent-based models in bridging the gap between in vitro and in vivo research in preclinical drug development.
BULLETIN OF MATHEMATICAL BIOLOGY
(2021)
Article
Biology
Linnea C. Franssen, Nikolaos Sfakianakis, Mark A. J. Chaplain
Summary: A three-dimensional hybrid atomistic-continuum model is developed to describe the invasive growth dynamics of individual cancer cells in tissue, accounting for phenotypic variation and transitions between epithelial-like and mesenchymal-like cell phenotypes. The model consists of partial and stochastic differential equations considering matrix-degrading enzyme concentrations and extracellular matrix density, calibrated to an in vitro invasion assay experiment of oral squamous cell carcinoma cells through parameter estimation and sensitivity analysis. This model provides a new theoretical basis for studying the invasion mechanisms of cancer cells.
JOURNAL OF THEORETICAL BIOLOGY
(2021)
Editorial Material
Biology
Philip K. Maini, Mark A. J. Chaplain, Mark A. Lewis, Jonathan A. Sherratt
BULLETIN OF MATHEMATICAL BIOLOGY
(2022)
Article
Mathematics, Applied
Piotr Gwiazda, Blazej Miasojedow, Jakub Skrzeczkowski, Zuzanna Szymanska
Summary: This paper investigates the EBT algorithm (a particle method) for the nonlocal equation with a discontinuous interaction kernel. The main challenge lies in the low regularity of the kernel, which prevents the use of standard arguments. Instead, the radial symmetry of the problem is utilized and transformed using spherical coordinates. The resulting equation has a Lipschitz kernel with a singularity at zero. A new weighted flat norm is introduced, and the convergence of the particle method is proved in this norm. The two-dimensional case is also discussed, which requires the application of the theory of measure spaces on general metric spaces, and numerical simulations are provided to support the theoretical results.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Biology
Chiara Villa, Alf Gerisch, Mark A. J. Chaplain
Summary: The formation of new vascular networks is crucial for tissue development and regeneration. Cluster-based vasculogenesis, a new mechanism involving the mobilization of cells from the bone marrow, plays a key role in connecting distant blood vessels in vivo. We propose a mathematical model to study the dynamics of cluster formation and investigate the effects of endogenous chemotaxis and matrix degradation through numerical and parametric analysis.
JOURNAL OF THEORETICAL BIOLOGY
(2022)
Article
Mathematics, Applied
Purnedu Mishra, Dariusz Wrzosek
Summary: This study investigates the role of predator evasion mediated by chemical signaling in a prey-predator diffusive model, considering or not considering prey taxis. The existence and stability of solutions, as well as the emergence of complex space-time patterns, are analyzed through mathematical modeling and numerical simulations.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Multidisciplinary Sciences
Zuzanna Szymanska, Jakub Skrzeczkowski, Blazej Miasojedow, Piotr Gwiazda
Summary: This paper aims to improve cancer models by proposing a new non-local model of cell proliferation, performing Bayesian inference for unknown parameters, and providing proof of stability of posterior distributions. Further research on well-posedness and convergence of the EBT algorithm is provided in a companion paper.
ROYAL SOCIETY OPEN SCIENCE
(2021)
Article
Mathematics, Applied
Miroslaw Lachowicz, Henryk Leszczynski, Krzysztof A. Topolski
Summary: In this paper, Euler-type approximations along characteristics were studied for a class of kinetic equations describing swarm formations with variable interaction rates. The proposed numerical schemes preserve essential properties of the kinetic equations and approximate the solution almost to the point of blow-ups, which indicate self-organization swarm behavior. Additionally, a class of exact solutions known as TWES, traveling wave-type equilibrium solutions, were defined.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Purnedu Mishra, Dariusz Wrzosek
Summary: This study extends Schoener's model to examine the impact of indirect prey and indirect predator taxis, showing that a sufficiently large value of taxis sensitivity parameter disturbs the stability of the coexistence steady state, leading to pattern formation governed by the Hopf bifurcation.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
N. Bellomo, F. Brezzi, M. A. J. Chaplain
Summary: This editorial proposes modeling and simulation of mutating virus pandemics in a globally connected world. It is divided into three parts: a general framework that goes beyond deterministic population dynamics, the contents of the papers in this issue, and a critical analysis of research perspectives.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Biology
Dimitrios Katsaounis, Mark A. J. Chaplain, Nikolaos Sfakianakis
Summary: This paper addresses the question of identifying the migratory pattern and spread of individual cancer cells or small clusters of cancer cells when the macroscopic evolution of the cancer cell colony is determined by a specific partial differential equation (PDE). The authors demonstrate that the traditional understanding of the diffusion and advection terms of the PDE as responsible for random and biased motion of solitary cancer cells, respectively, is imprecise. Instead, they show that the drift term of the correct stochastic differential equation scheme should also take into account the divergence of the PDE diffusion. Numerical experiments and computational simulations are used to support their claims.
JOURNAL OF MATHEMATICAL BIOLOGY
(2023)
Article
Mathematics
Purnedu Mishra, Dariusz Wrzosek
Summary: We studied a diffusive predator-prey model that incorporates prey-taxis and indirect predator taxis. This model extends the Rosenzweig MacArthur model by including intraspecific competition among predators. Our results showed the existence of global-in-time classical solutions for space dimension n <= 3, which is not expected in the Rosenzweig MacArthur model.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jose Antonio Carrillo, Martin Parisot, Zuzanna Szymanska
Summary: This study introduces a new model for tendon healing, describing the rearrangement of collagen fibers during the healing process. The simplified model allows for parameter estimation and showcases the qualitative properties as well as the time for tendon fibers to align. The research suggests the potential importance of tendon cell size in patient recovery based on numerical experiments.
KINETIC AND RELATED MODELS
(2021)
Article
Mathematics, Applied
Kevin J. Painter, Thomas Hillen, Jonathan R. Potts
Summary: The use of nonlocal PDE models in describing biological aggregation and movement behavior has gained significant attention. These models capture the self-organizing and spatial sorting characteristics of cell populations and provide insights into how animals perceive and respond to their surroundings. By deriving and analyzing these models, we can better understand biological movement behavior and provide a basis for explaining sociological phenomena.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
Article
Mathematics, Applied
Nicola Bellomo, Massimo Egidi
Summary: This paper focuses on Herbert A. Simon's visionary theory of the Artificial World and proposes a mathematical theory to study the dynamics of organizational learning, highlighting the impact of decomposition and recombination of organizational structures on evolutionary changes.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
Article
Mathematics, Applied
Tayfun E. Tezduyar, Kenji Takizawa, Yuri Bazilevs
Summary: This paper provides an overview of flows with moving boundaries and interfaces (MBI), which include fluid-particle and fluid-structure interactions, multi-fluid flows, and free-surface flows. These problems are frequently encountered in engineering analysis and design, and pose computational challenges that require core computational methods and special methods. The paper focuses on isogeometric analysis, complex geometries, incompressible-flow Space-Time Variational Multiscale (ST-VMS) and Arbitrary Lagrangian-Eulerian VMS (ALE-VMS) methods, and special methods developed in connection with these core methods.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2024)
