Studies in Applied Mathematics最新文献

This paper concerns with the global threshold dynamics of a Tuberculosis (TB) model incorporating age-space structure, treatment, and relapse. The original model is converted into a hybrid system comprising two Volterra integral equations and two partial differential equations by integrating along the characteristic line. The well-posedness for the model is demonstrated by using the fixed point theory in conjunction with the induction method. In order to discuss whether a disease is persistent or extinct, we provide the explicit formulation of the basic reproduction number. By analyzing the distribution of the characteristic roots of the characteristic equations and constructing the proper Lyapunov functionals, the local and global stability for the steady states are addressed. Numerical simulations are conducted to confirm the conclusions of our analytical results and reveal that reduction of the TB transmission coefficient, reduction of infectiousness of treated individuals infected with TB, and increasing the treatment rate of infectious class are three feasible measures to control the transmission of TB.

In this research, the (2+1)-dimensional normal biological population model, incorporating the Verhulst law for population growth, is employed to explore species population dynamics. Employing Lie symmetry analysis, we address a nonlinear degenerate parabolic partial differential equation, yielding much-improved results. This analysis includes computing one-dimensional optimal subalgebras, reduced ordinary differential equations, and obtaining invariant solutions with a visual depiction of the physical behavior of the Verhulst biological population model through symmetry group transformations. Additionally, the multiplier method leads to novel conservation laws and potential systems not locally connected to the governing partial differential equation (PDE). These findings have significant implications for understanding and controlling biological populations, offering insights for applications in ecology and the environment.

The Serre–Green–Naghdi equations of water wave theory have been widely employed to study undular bores. In this study, we introduce a modified Serre–Green–Naghdi system incorporating the effect of an artificial term that results in dispersive and dissipative dynamics. We show that the modified system effectively approximates the classical Serre–Green–Naghdi equations over sufficiently extended time intervals and admits dispersive–diffusive shock waves as traveling wave solutions. The traveling waves converge to the entropic shock wave solution of the shallow water equations when the dispersion and diffusion approach zero in a moderate dispersion regime. These findings contribute to an understanding of the formation of dispersive shock waves in the classical Serre–Green–Naghdi equations and the effects of diffusion in the generation and propagation of undular bores.

Within the class of (1+2)-dimensional ultraparabolic linear equations, we distinguish a fine Kolmogorov backward equation with a quadratic diffusivity. Modulo the point equivalence, it is a unique equation within the class whose essential Lie invariance algebra is five-dimensional and nonsolvable. Using the direct method, we compute the point symmetry pseudogroup of this equation and analyze its structure. In particular, we single out its essential subgroup and classify its discrete elements. We exhaustively classify all subalgebras of the corresponding essential Lie invariance algebra up to inner automorphisms and up to the action of the essential point-symmetry group. This allowed us to classify Lie reductions and Lie invariant solutions of the equation under consideration. We also discuss the generation of its solutions using point and linear generalized symmetries and carry out its peculiar generalized reductions. As a result, we construct wide families of its solutions parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation  or one or two arbitrary solutions of (1+1)-dimensional linear heat equations with inverse square potentials.

Forest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed-point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

This paper studies traveling waves with nonzero wave speed (angular traveling waves) of the high-dimensional Boussinesq equation that have not been studied before. We analyze the properties of these waves and demonstrate that, unlike the unique stationary solution, they lack positivity, radial symmetry, and exponential decay. By employing variational and geometric approaches, along with perturbation theory, we establish the orbital (in)stability and strong instability of these traveling waves.

In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number $�R_0$ is defined, and the threshold dynamics are studied. It is shown that the disease-free equilibrium is globally asymptotically stable if $��R_0�<�1�$ and the disease is uniformly persistent if $��R_0�>�1�$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed.

The aim of this paper is to present a survey and a detailed numerical study on a remarkable Boussinesq system describing weakly nonlinear, long surface water waves. In the one-dimensional case, this system can be viewed as a dispersive perturbation of the hyperbolic Saint-Venant (shallow water) system. The asymptotic stability of the solitary waves is numerically established. Blow-up of solutions for initial data not satisfying the noncavitation condition as well as the appearance of dispersive shock waves are studied.

Following publication of the twelve Studies in Applied Mathematics articles which make up the special issue on Integrable Systems and Their Applications in Honor of A. S. Fokas, the article type “Original Article” has been updated to “Special Issue Article”. In addition, the special issue title (Integrable Systems and Their Applications in Honor of A. S. Fokas) has been added with a link to the virtual issue.

Chen, J., Pelinovsky, D.E. (2024). Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation. https://doi.org/10.1111/sapm.12634

Konopelchenko, B.G., Ortenzi, G. (2024). On blowups of vorticity for the homogeneous Euler equation. https://doi.org/10.1111/sapm.12618

Ballew, C., Trogdon, T. A. (2024). Riemann–Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals. https://doi.org/10.1111/sapm.12630

Charlier, C., Lenells, J. (2024). Miura transformation for the “good” Boussinesq equation. https://doi.org/10.1111/sapm.12631

Zhao, Y., Fan, E. (2024). Existence of global solutions to the nonlocal Schrödinger equation on the line. https://doi.org/10.1111/sapm.12636

Alkın, A., Mantzavinos, D., Özsarı, T. (2024). Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: Single-boundary condition case. https://doi.org/10.1111/sapm.12642

Qu, C., Wu, Z. (2024). Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras. https://doi.org/10.1111/sapm.12641

Clarkson, P.A., Dunning, C. (2024). Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials. https://doi.org/10.1111/sapm.12649

Zhang, Y., Ma, R., Feng, B.-F. (2024). KP reductions and various soliton solutions to the Fokas–Lenells equation under nonzero boundary condition. https://doi.org/10.1111/sapm.12654

Biondini, G., Chernyavsky, A. (2024). Whitham modulation theory for the Zakharov–Kuznetsov equation and stability analysis of its periodic traveling wave solutions. https://doi.org/10.1111/sapm.12651

Normatov, B., Smith, D.A. (2024). The Airy equation with nonlocal conditions. https://doi.org/10.1111/sapm.12652

Mao, Y., Mantzavinos, D., Hoefer, M.A. (2024). Long-time asymptotics and the radiation condition with time-periodic boundary conditions for linear evolution equations on the half-line and experiment. https://doi.org/10.1111/sapm.12665

Reaction–diffusion equations (RDEs) model the spatiotemporal evolution of a density field $��u�(�x�,�t�)�$ according to diffusion and net local changes. Usually, the diffusivity is positive for all values of $�u$, which causes the density to disperse. However, RDEs with partially negative diffusivity can model aggregation, which is the preferred behavior in some circumstances. In this paper, we consider a nonlinear RDE with quadratic diffusivity $��D�\left(�u�\right)�=�(�u�-�a�)�(�u�-�b�)�$ that is negative for $��u�\in �(�a�,�b�)�$. We use a nonclassical symmetry to construct analytic receding time-dependent, colliding wave, and receding traveling wave solutions. These solutions are multivalued, and we convert them to single-valued solutions by inserting a shock. We examine properties of these analytic solutions including their Stefan-like boundary condition, and perform a phase plane analysis. We also investigate the spectral stability of the $��u�=�0�$ and