Mathematical Methods in the Applied Sciences最新文献

In this paper, we present a set of Newton-type inequalities for n-times differentiable convex functions using the Caputo fractional operator, extending classical results into the fractional calculus domain. Our exploration also includes the derivation of Newton-type inequalities for various classes of functions by employing the Caputo fractional operator, thereby broadening the scope of these inequalities beyond convexity. In addition, we establish several fractional Newton-type inequalities by using bounded functions in conjunction with fractional integrals. Furthermore, we develop specific fractional Newton-type inequalities tailored to Lipschitzian functions. Moreover, the paper emphasizes the significance of fractional calculus in refining classical inequalities and demonstrates how the Caputo fractional operator provides a more generalized framework for addressing problems involving non-integer order differentiation. The inclusion of bounded and Lipschitzian functions introduces additional layers of complexity, allowing for a more comprehensive analysis of function behaviors under fractional operations.

In this article, we investigate the existence, uniqueness, and numerical aspects of a one- and two-dimensional nonlinear viscous type Burgers problem defined in a noncylindrical domain. In order to obtain the existence and uniqueness of the solution, the problem with a moving ends is transformed into an equivalent problem in a cylindrical through a diffeomorphism between the domains. The numerical simulation for the one- and two-dimensional cases is performed using Lagrange with degrees 1–3 and cubic Hermite polynomials as base functions for applying the linearized Crank–Nicolson–Galerkin method to obtain an approximate numerical solution. Graphs prove the efficiency of the numerical method along with the order of numerical convergence consistent with the degree of the base polynomial.

In this article, we examine the general space-time fractional diffusion equation for left-invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation, the space-time fractional diffusion equation when diffusion is under the influence of sub-Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well-posedness of the Cauchy problem for the general space-time fractional diffusion equation of the Rockland operator on a graded Lie group in the associated Sobolev spaces and also develop some regularity estimates for it.

This article considers the zero-pressure gas dynamics governed by the conservation laws of mass, momentum, and energy with time-dependent source terms. By adopting the vanishing viscosity method, we propose a viscous system to prove the stability of non-self-similar Riemann solutions including the delta shock wave and vacuum.

We investigate the oscillatory behavior of solutions of second-order linear differential equations with impulses. These equations can be categorized into two types: canonical and non-canonical. This study examines the canonical and non-canonical impulsive differential equations in connection with the famous Leighton-type oscillation theorem. The well-known theorem states that every solution of

This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition $��(�\ast �)�$ fails, that is, when the equation is of non-canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non-canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results.

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem.

In recent years, conventional logistic maps have been applied across various fields including modeling and security, owing to their versatility and utility. However, their reliance on a single modifiable parameter limits their adaptability. This paper aims to explore generalized logistic maps with arbitrary powers, which offer greater flexibility compared to the standard logistic map. By introducing additional parameters in the form of arbitrary powers, these maps exhibit increased degrees of freedom, thus expanding their applicability across a wider spectrum of scenarios. Consequently, the conventional logistic map emerges as a specific instance within the proposed framework. The inclusion of arbitrary powers enriches system dynamics, enabling a more nuanced exploration of system behavior in diverse contexts. Through a series of illustrations, this study investigates the influence of arbitrary powers and equation parameters on equilibrium points, their positions, stability conditions, basin of attraction, and bifurcation diagrams, including the emergence of chaotic behavior.

This paper develops an efficient combined interpolation/finite element approach for solving a three-dimensional Black-Scholes problem with stochastic volatility. The technique consists to approximate the time derivative by interpolation whereas the space derivatives are approximated using the finite element method. Both stability and error estimates of the new algorithm are deeply analyzed in the $��{�L�}^{�\infty �}�$-norm. The proposed method is explicit, unconditionally stable, temporal second-order accurate and fourth-order convergence in space. This result suggests that the constructed scheme is faster and more efficient than a broad range of numerical methods widely studied in the literature for the Black-Scholes models. Some numerical experiments are carried out to confirm the theoretical analysis.

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power-law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non-topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of $��n�$). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach.