Mathematical Methods in the Applied Sciences最新文献

In this work, we introduce a mathematical model for diabetes that uses time-delay and variable-order fractional derivatives, aiming to better reflect the complex and memory-dependent behavior of glucose and insulin dynamics. The model is built using the Caputo definition of variable-order derivatives. We explore the system's equilibrium points and examine their stability to understand how the system's behavior changes with different parameters. We also study the positivity and boundedness of the proposed system. To solve the model numerically, we design an effective method that combines a nonstandard finite difference scheme with the Grünwald–Letnikov operator. We analyze the proposed scheme and prove that the approximated solutions remain nonnegative and bounded. Through numerical simulations and comparisons, we demonstrate the reliability and practical advantages of our approach. The results highlight the crucial impact of time-delay and variable-order fractional dynamics on diabetes progression and treatment. Delayed insulin response and memory effects in glucose–insulin interaction are effectively modeled. This enhances the realism and personalization of blood sugar regulation analysis.

This research contributes to the field of fractional calculus in theoretical sense. We investigate a new operator, proposed by combing the already known Hilfer operator and weighted operator with respect to functions, in terms of some fundamental properties. These properties include composition properties, boundedness, convergence, and mapping in two different function spaces. Lastly, we discuss fractional differential equations involving these operators and establish conditions for existence, uniqueness, and stability by using concepts from fixed point theory.

The dynamics of the electrons interacting with its self-consistent electrostatic field as well as its grazing collisions with a fixed background of ions can be described by Vlasov–Poisson–Landau (VPL) system. In this paper, we focus on the Cauchy problem for the inhomogeneous VPL system with hard potentials in the whole space $��{�ℝ�}^{�3�}�$. In a perturbation framework, the unique global solution is established in the spatially Besov space. Our proof mainly depends on some trilinear estimates of the nonlinear collision term through the Littlewood–Paley theory and the relationship between homogeneous and nonhomogeneous Chemin–Lerner spaces, which lead to the global energy estimates.

In this paper, we study a stochastic variant of the classical Keller–Segel system on a two-dimensional domain, where the leading diffusion term is replaced by a porous media operator and the dynamics are perturbed by a pair of independent Wiener processes. The model describes the interaction between the cell density $��u�$ and the concentration of a chemoattractant $��v�$, incorporating nonlinear diffusion, chemotactic sensitivity, production and damping effects, together with multiplicative stochastic perturbations of strengths $��{\sigma }_u�$ and $��{\sigma }_v�$. Since the randomness is intrinsic, the stochastic terms are interpreted in the Stratonovich sense. To construct solutions, we introduce an integral operator and establish its continuity and compactness properties in a suitable Banach space. This leads to a stochastic analogue of the Schauder-Tychonoff-type fixed point theorem tailored to our framework, which ensures the existence of a martingale solution. Furthermore, we establish pathwise uniqueness, uniqueness in law, and the existence of strong solutions. The uniqueness results, however, require additional assumptions on the chemoattractant noise and the initial condition of $��v�$.

This study presents a new numerical approach specifically developed to solve the two-dimensional fractional mobile/immobile equation (FMIE). These equations have various applications in areas such as subsurface pollutant transport, groundwater flow, and heat transfer in porous media. The suggested method utilizes the compact higher order finite difference scheme (CHFDS) and incorporates the Crank–Nicolson (C–N) method. This combination results in a unique combination of unconditional stability and a high convergence order of $��O�(�{�\tau �}^{�2�-�\beta �}�+�{�\kappa �}^{�4�}�)�$. The method achieves accuracy and efficiency by utilizing the C–N fourth-order finite difference scheme for spatial variables and the Caputo derivative for temporal fractional derivatives. The proposed approach effectively manages complex FMIEs in real-world scenarios, optimizing computational efficiency, minimizing simulation time, and enhancing robustness. The results confirm the enhanced precision and efficiency compared to conventional methods, hence reinforcing the practical significance of the CHFDS methodology.

In this work, we mainly consider a class of multivalued distribution-dependent stochastic differential equations (MDDSDEs, in short). With some assumptions, we prove that the solutions of MDDSDEs can be approximated by the solutions of associated averaged multivalued SDEs in two senses of convergence in mean square and convergence in probability, respectively.

This study uses critical point theory to investigate the existence of periodic solutions for second-order partial difference equations with local superlinear conditions. It is specifically noted that the nonlinear component taken into consideration in the research displays mixed nonlinear characteristics at the origin, which can be either superlinear or asymptotically linear, but only satisfies local superlinear criteria at infinity. Due to the local superlinear condition, classical method cannot be applied. We develop new idea to address this issue and obtain at least one nonconstant periodic solution. The conclusions drawn not only enrich the relevant research on periodic solutions of second-order difference equations but also provide a new perspective for exploring the existence of timespace periodic solutions for two-dimensional discrete Schrödinger equations with local nonlinear growth conditions.

Timoshenko system with its various dissipative mechanisms has been thoroughly examined in the literature. In this paper, we consider the Timoshenko beam model resting on Winkler foundation coupled with Kelvin–Voigt damping. For the classical case, we establish the well-posedness of the system using $��{�C�}_{�0�}�$-semigroup theory. Without assuming the equal wave speeds condition, we prove that the $��{�C�}_{�0�}�$-semigroup established with the fully viscoelastic system, where both the bending moment and the shear stress exhibit viscoelastic damping, is analytic. Consequently, exponential stability is acquired. Otherwise, the partial viscoelastic systems are not exponentially stable, no matter the value of the coefficients. Then, the two systems are shown to be polynomially stable using Borichev and Tomilov's theorem. Then, we investigate the case where the system is free of second spectrum, as we showed the well-posedness of the system using Faedo–Galerkin technique. Furthermore, exponential stability was proved using the energy method. Finally, a numerical scheme is proposed and analyzed, and numerical simulations are provided to illustrate and validate the theoretical findings, confirming the predicted decay rates.

In this study, we introduce and explore the fractional powers of Meijer $��K�$-transformation of order $��\mu �$ (where $��-�\frac{�1�}{�2�}�\le �\text{Re}��\mu �\le �\frac{�1�}{�2�}�$) with a specific parameter $��\alpha �$, applying them to certain generalized functions. We examine the properties of these fractional powers in detail, establishing a theoretical foundation for their behavior. As a practical application of our results, we demonstrate how these concepts can be utilized to solve partial differential equations involving Kepinski-type operator $��{�\Delta �}_{�\mu �,�\alpha �}^{�\ast �}�$, thereby illustrating the theoretical results in a tangible context.