ABSTRACT This paper is concerned with the global classical large solutions for the initial boundary value problem of the radiation hydrodynamics model with viscosity and thermal conductivity. The boundary conditions contain two cases: the periodic boundary conditions and the general boundary conditions: The key point is to derive strictly positive bounds for the density and temperature , which is more complex than the Navier–Stokes equations due to the influence of radiative heat flux. To overcome these difficulties, Zhang and Zhao in (2023) firstly constructed a pointwise estimate between the radiative heat flux and temperature using the method of Fourier analysis. And this pointwise estimate is extremely important to obtain the lower bound of the temperature . However, due to the technical limitations, Zhang and Zhao can only consider the case of periodic boundary conditions and special heat conductivity: (). For the general boundary conditions, we construct a same pointwise estimate between and by analyzing the equation about the radiation heat flux. Besides, we consider a general coefficient of heat conduction: if ; if , which extends the result in Zhang and Zhao in (2023), and also contains some important situations in physics that Zhang and Zhao in (2023) excludes. As a byproduct of our approach, the argument in Wen and Zhu (2013) is also extended to the more general for Navier–Stokes equations with large initial data.

In this paper, we study the initial boundary value problem involving the p-Laplacian parabolic equation \(u_t - \Delta_{p}u + \alpha\vert u \vert^{p-2}u = 0, \quad (x,t) \in \Omega \times ]0,+\infty[,\) with logarithmic boundary condition. By using the potential wells method combined with the Nehari Manifold, we establish the existence of a weak global solution. In addition, we also obtain the decay polynomial of the weak solution. Then, by virtue of the differential inequality technique, we prove that the solutions blow up in finite time under suitable initial values.

Abstract Metric reconstruction is the general problem of parameterizing GR in terms of its two “true degrees of freedom” e.g., by a complex scalar “potential”—in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. 37, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE) on Kerr. Our main innovation is a version of the GHZ integration scheme that is suitable for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a byproduct, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.

Abstract Can a disk orbiting a central body be eccentric, when the disk feels its own self-gravity and is pressureless? Contradictory answers appear in the literature. We show that, to linear order in eccentricity, such a disk can be eccentric, but only if it has a sharply truncated edge: the surface density Σ must vanish at the edge, and the Σ profile must be sufficiently steep at the point where it vanishes. If either requirement is violated, an eccentric disturbance leaks out of the bulk of the disk, into the low-density edge region, and cannot return. An edge where Σ asymptotes to zero but never vanishes, as is often assumed for astrophysical disks, is insufficiently sharp. Similar results were shown by Hunter & Toomre for galactic warps. We demonstrate these results in three ways: by solving the eigenvalue equation for the eccentricity profile; by solving the initial value problem; and by analyzing a new and simple dispersion relation that is valid for any wavenumber, unlike WKB. As a byproduct, we show that softening the self-gravitational potential is not needed to model a flat disk, and we develop a softening-free algorithm to model the disk’s Laplace–Lagrange-like equations. The algorithm is easy to implement and is more accurate than softening-based methods at a given resolution by many orders of magnitude.

This paper investigates a class of initial value problems arising in the modeling of systems with memory and delayed feedback, described by nonlinear fractional differential equations with finite delay, governed by the generalized Caputo-Katugampola fractional derivative. The presence of the parameter [Formula: see text] in this operator allows interpolation between different fractional behaviors and provides additional flexibility in modeling the intensity and scaling of memory effects in delay systems. By employing ρ-Laplace transform techniques, we first derive an equivalent integral formulation of the considered problem. We then establish the existence and uniqueness of solutions to the proposed Cauchy problem by employing the Banach contraction principle and Schauder's fixed point theorem. The use of both fixed-point approaches enables us to address existence and uniqueness under complementary sets of assumptions, thereby enlarging the class of admissible nonlinearities. Moreover, we examine the Ulam-Hyers stability of the solutions under suitable conditions, demonstrating that small variations in the initial data result in proportionally small deviations in the solution. This stability property reflects the robustness of the model with respect to perturbations and is closely related to the contraction condition imposed on the associated operator. To illustrate the theoretical results and confirm the applicability of the method, numerical examples are provided and discussed. The numerical simulations are carried out using the L1 scheme, which is known for its stability and effectiveness in approximating Caputo-type fractional derivatives.

The initial value problem is solved for the excitation of long surface gravity waves in a continuously sheared flow. This reveals the presence of a continuous spectrum alongside the standard normal modes of gravity wave propagation. An analytical similarity solution for the evolution of the free surface displacement from the continuous spectrum is found for the impulse response to surface excitation. It is demonstrated that the continuous spectrum contribution can be a significant fraction of the surface response, with the amplitude of the continuous spectrum exceeding that of the upstream gravity wave mode for Froude numbers of order unity. Thus, the continuous spectrum is shown to be a physically important part of the gravity wave response to forcing in sheared flows. The Landau damped mode description of the continuous spectrum is found to provide a link between methods using dispersion relations for phase speeds within the range of the velocity profile and the variable-shear profiles that do not admit normal modes in this range.

Abstract Fractional iterative differential equations with boundary conditions (FIDEs-wBC) have been mainly studied using the classical Caputo fractional derivative or within the framework of initial value problems. In this paper, we investigate the existence, uniqueness, and Hyers–Ulam stability of solutions for FIDEs-wBC defined by the $\mu$-Caputo fractional derivative. This derivative provides a unified framework that includes several classical fractional derivatives as special cases through suitable choices of the function $\mu$. As a result, this approach allows us to study a broader class of fractional iterative boundary value problems that has not been fully considered in the existing literature. By using the Banach contraction principle and Schauder’s fixed point theorem, we establish results on existence and uniqueness in appropriate function spaces. Several examples are also provided to illustrate the theoretical results.

This study explores and investigates a human respiratory syncytial virus (RSV) infection using a generalized fractional-order susceptible-exposed-infected-recovered (SEIR) model. The model incorporates the recently introduced fractional derivative operator, the ψ-Caputo derivative, defined with respect to an auxiliary function, [Formula: see text]. The formulation allows flexible depiction of memory and genetic effects in disease dynamics, beyond integer-order models. A rigorous mathematical framework proves the existence and uniqueness of solutions to the ψ-Caputo fractional initial-value problem (IVP), proving the model's theoretical well-posedness. We also offer an innovative and efficient numerical approach for solving the fractional model, with verified convergence and a valid error bound. Comprehensive simulations and analyses are conducted to the applicability of the model. In particular, the model represents diverse dynamic behaviors by varying the fractional order α within the range (0,1]. These results indicate that the system's reaction is sensitive to the fractional order α, with classical integer-order dynamics regained when [Formula: see text]. Furthermore, the fractional SEIR model with an optimal control framework uses treatment as a control variable to evaluate intervention options. Simulation results indicate that the fractional ψ-Caputo model, with optimal control, better decreases infectious people than standard integer-order models. These findings demonstrate the modeling and control approach's potential to analyze, predict, and mitigate RSV infections in real-world circumstances.

The Baltic Sea salinity is modelled using a two-box model. The simplistic approach allows for very long integrations where a large part of the phase space of the model can be probed. Particular emphasis is put on the salinity dynamics in the deeper parts of the sea and how they are affected by boundary and initial conditions. Multiple statistically steady states, corresponding to forcing from different years, are examined and the route to them through the model’s phase space is traced out. The model is forced with freshwater fluxes and sea level variations at its boundary. The respective roles of these two forcing terms is investigated using a factorization technique, and it is found the sea level variability is the dominant one for the deep salinity dynamics. The role of natural variability is also examined, and the probability for deep salinity changes for different forcing years is computed.

Abstract In this article, we investigate the existence and uniqueness of solutions for self-adjoint difference equations containing two Hahn difference operators. One is of the first order, and the second is of an $$\alpha $$ α -order with $$0<\alpha \leqslant 1$$ 0 < α ⩽ 1 under certain initial and boundary conditions. For obtaining solutions to fractional Hahn difference equations with boundary conditions, we use the Green function, which is defined by the Cauchy function. The basic and important properties of this function are discussed. The existence of solutions to the considered initial value problems is obtained in terms of the Cauchy function. The solutions to the boundary value problems are established in terms of the Green function. Also, the uniqueness of the solutions is proved by applying Banach’s fixed point theorem. An example is given to illustrate our main results.

Abstract In this paper, we investigate the characteristic initial value problem for the Einstein–Dirac system, a model governing the interaction between gravity and spin- 1 / 2 fields. We apply Luk’s strategy (Luk J 2012 Int. Math. Res. Not. 20 4625) and prove a semi-global existence result for this coupled Einstein–Dirac system without imposing symmetry conditions. More precisely, we construct smooth solutions in a rectangular region to the future of two intersecting null hypersurfaces, on which characteristic initial data are specified. The key novelty is to promote the symmetric spinorial derivatives of the Dirac field to independent variables and to derive a commuted ‘Weyl-curvature-free’ evolution system for them. This eliminates the coupling to the curvature in the energy estimates and closes the bootstrap at the optimal derivative levels. The analysis relies on a double null foliation and incorporates spinor-specific techniques essential to handling the structure of the Dirac field.

This paper presents the development of a mobile application for the Android operating system that does not require an internet connection to obtain the position graph of a mass-spring system. By inputting the necessary parameters for each case, the fourth-order Runge-Kutta (RK4) method is used to solve an initial value problem for a system of differential equations. The corresponding graph is then generated using the MPAndroidChart library. The graphs obtained with this app are compared to those obtained using a Computer Algebraic System (CAS), demonstrating the validity of the results obtained with the application.

ABSTRACT In this paper, we mainly investigate the backward problem for the two‐dimensional time‐fractional diffusion equation by the proper orthogonal decomposition (POD) method, and also apply this method to solve the image deblurring problem. The approach constructs a low‐dimensional basis by solution snapshots to accelerate the reconstruction of the initial field from noisy terminal time measurements. Besides, we give the Tikhonov regularization method and obtain the convergence analysis about the regularization solutions. Finally, we propose several numerical experiments to show the efficiency and accuracy of our method.

ABSTRACT In this study, we transform a triple integral into a third‐order initial value problem and solve it using Euler's method and Richardson's extrapolation. Our objective is to resolve the computational challenges associated with triple integration by reformulating it into an initial value problem. Euler's method serves as a fundamental numerical technique for approximating the solution, establishing a baseline for accuracy. We subsequently improve computational precision using Richardson's extrapolation, which systematically reduces numerical errors. This approach not only illustrates the adaptability of numerical methods in solving intricate mathematical problems, but it also emphasizes the significance of strategic error reduction techniques in enhancing computational outcomes. We demonstrate the efficacy of this method in efficiently solving triple integrals through experimentation and analysis, thereby making a significant contribution to the fields of numerical computation and mathematical modeling.

Abstract The generalized derivative represents the broadest notion of the Hukuhara-type derivative of a fuzzy number-valued function in the literature. It exists for a wide class of fuzzy processes, since the generalized difference exists for any pair of fuzzy numbers. Despite its historical significance, few papers provide theoretical results on the g-derivative, primarily because of its complex analytical behavior. On the other hand, analyzing solutions to fuzzy differential equations from a comparative Hukuhara-type perspective allows establishing features of FDEs whose solutions are exclusively g-differentiable. The study begins by providing a corrected version of the fundamental theorem of calculus via the g-differentiability and Aumann integrability of a fuzzy function. Sufficient conditions over the field of a fuzzy initial value problem for the gH $$*$$ -differentiability of the solutions are presented. Lastly, results on fuzzy processes derived from solutions of FDEs that are g-differentiable, but not gH, and not even gH $$*$$ -differentiable, are given. Examples of population dynamics governed by the Malthusian and Logistic models are provided to illustrate different scenarios for the presented analysis.

In the present paper, time-fractional order linear and nonlinear heat type Emden-Fowler equations are reformulated from existing classical equations by applying Caputo-Fabrizio time-fractional derivative. Then, a semi-analytical scheme, that is an amalgamation of Laplace transformation and Picard’s iterative technique, is exploited to simulate singular initial value problems for corresponding time-fractional order heat type Emden-Fowler equations. Further, the stability of developed scheme is also assessed by exploiting R-stable mapping and Banach contraction principle. Numerical results, error estimation, and comparison of obtained results with exact solutions are presented through graphs and tables to exhibit the efficiency of time-fractional order derivative and implemented semi-analytical scheme.

This is an attempt to construct the well-posed hyperbolic heat conduction model based on the Caputo fractional derivative and to study the corresponding coupled thermoelastic problem. The continuous dependence on initial data and energy supply, and the uniqueness of the solutions are mathematically proved. The general closed-form solution of the time fractional conduction model for the initial Dirichlet boundary value problem is obtained analytically by applying the Laplace transform and finite Fourier sine transform in one-dimensional case. The application of theoretical study for heat propagation in the wire is considered. As a special case, two different examples have been discussed to study the analysis of the temperature distributions in the spatial geometry. The influence of the fractional orders on the speed of heat conductivity in the model is discussed. The physical behavior of the temperature distribution has been graphically represented for different fractional orders. Furthermore, the thermal stress analysis is studied using the coupled thermoelasticity theory. In the Laplace domain, the analytical solutions have been obtained. The Gaver–Stehfest technique was employed to numerically perform time domain inversions of the Laplace transforms, which satisfied Kuznetsov’s convergence theorem.

ABSTRACT This paper introduces an inverse problem for a stochastic damped wave equation, in which the source is driven by a fractional Brownian motion. The well‐posedness of direct problem is obtained by analyzing regularity of the solution for the equivalent stochastic initial value problem in frequency domain. The inverse problem involves recovering two initial values and a random source simultaneously. It is shown that the mean of final observations at two different moments uniquely determines the initial values. Additionally, for inversion of random source, it is demonstrated that its sine modulus can be uniquely determined by the variance of final observations. Based on this sine modulus, recovering the unknown random source can be transformed into a well‐known phase retrieval problem, which can be successfully solved using the methodology of phaselift. Finally, numerical examples are presented to demonstrate the effectiveness of this method.

Spreading dynamics of an SVIRS model.

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient μ(ρ) is a power function of the density with the power larger than 1, that is μ(ρ) = μρα with α > 1, it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and L3-norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional inhomogeneous liquid crystal flows without smallness of velocity.