Mathematical Methods in the Applied Sciences最新文献

In this paper, the stochastic Hartree equation with multiplicative noise is considered, and the global well-posedness is obtained for the 1D defocusing mass-subcritical nonlinearities.

There are dozens of languages spoken worldwide, but due to linguistic shifts and rivalry, many of them are in danger of becoming extinct. A vital component, therefore, is linguistic sustainability because cultural sustainability depends on language preservation. To represent linguistic sustainability, which traditional models do not adequately present, this work presents a novel fractional-order method that uses both constant and variable fractional-order derivatives. The significance of the arbitrary order derivative and the theoretical conclusions we got were validated by implementing a recently suggested Toufik–Atangana numerical technique with Caputo (C), Caputo–Fabrizio (CF), and Atangana Balaneau Caputo (ABC) fractional derivatives. Additionally, a comparison between the Mathematica NDSolve approach and the integer order Toufik-Atangana method was done. This innovative method provides insightful information for language preservation plans, allowing linguists and policymakers to create more potent interventions to save endangered languages.

This paper aims to derive novel forms of trapezoid and midpoint inequalities within the context of fractional extended Riemann–Liouville integrals (FERLIs). The approach relies on the foundational Jensen–Mercer inequality to develop the arguments. Central to the development are several identities involving FERLIs and $��h�$-convex functions that form the basis of the obtained results. Relationships with earlier studies concerning both classical and extended Riemann–Liouville fractional integrals are examined.

This paper investigates a one-dimensional dissipative porous–elasticity system that circumvents the detrimental effects of the second spectrum, as highlighted in recent work by Ramos et al. (Applied Mathematics Letters 101 (2020): 106061). We first establish the global well-posedness of the system using the Faedo–Galërkin method. Subsequently, by employing multiplier techniques and analyzing the memory-driven viscoelastic damping, we derive explicit exponential decay rates for the system's energy.

In this study, we deal with the perturbed nonlinear Schrödinger equation (PNLSE). This equation is an important model for describing optical soliton propagation in nonlinear optical fibers with the Kerr law nonlinearity. Two efficient techniques are employed: the generalized Abel equation method (GAEM) with variable coefficients and the collective variable (CV) approach. As a consequence of GAEM, we derive a periodic-type soliton solution and enhance its physical interpretation through graphical simulations. The CV method, carrying a fourth-order Runge–Kutta scheme and the Gaussian ansatz, is used to model the dynamics of soliton parameters such as amplitude, width, chirp, and frequency, revealing periodic fluctuations influenced by propagation distance. Both methods provide a comprehensive understanding of soliton behaviors in optical fibers, offering valuable outcomes for advancing optical communication systems.

In this paper, we propose a review of the free boundary formulation for BVPs defined on semi-infinite intervals. The main idea and theorem are illustrated, for the reader convenience, by using a class of second-order BVPs. Moreover, we are able to show the effectiveness of the proposed approach using two examples where the exact solution both for the BVPs and their free boundary formulations are available. Then, we describe the free boundary formulation for a general class of BVPs governed by an $��n�$-order differential equation. In this context, we report three problems solved using the free boundary formulation. The reported numerical results, obtained by the iterative transformation method or Keller's second-order finite difference method, are found to be in very good agreement with those available in the literature. The last result of this research is that, in order to orient the interested reader, we provide an extensive bibliography. Of course, we may aspect further and more interesting applications of the free boundary formulation in the future.

In view of the advantages of the solution of the reduced biquaternion matrix equation, which was originally employed to solve the color image deblurred problems, we study the minimal norm Hermitian solution, pure imaginary Hermitian solution and pure real Hermitian solution of the reduced biquaternion matrix equation $��B�Y�=�E�$ by presenting a new real representation method and establish the necessary and sufficient conditions for the existence of the above solutions. It is demonstrated that our real representation method can also obtain the minimal norm least squares Hermitian solution, the minimal norm least squares pure imaginary Hermitian solution and the minimal norm least squares pure real Hermitian solution of the studied reduced biquaternion matrix equation when it is inconsistent. Finally, we provide some numerical examples to illustrate the feasibility and validity of our method in comparison with the complex representation method, especially in terms of computing error and CPU time. Experimental results demonstrate that the proposed method is feasible and much more effective for color image restoration than the complex representation method.

In this study, we delve into the intricate dynamics of the ($��3�+�1�$) dimensional $��q�$-deformed tanh-Gordon equation, exploring its fractional form through the lens of the Caputo fractional derivative. By employing the innovative controlled Picard technique combined with Laplace transforms, we derive highly accurate approximate solutions, ensuring both convergence and reliability in the realm of fractional nonlinear equations. Our rigorous analysis of the solution's existence and uniqueness provides a robust theoretical foundation, while two- and three-dimensional graphical representations vividly illustrate the profound impact of fractional and $��q�$-deformed parameters on the system's behavior. The findings not only highlight the efficacy of the controlled Picard technique in tackling complex fractional models but also offer valuable insights into their numerical treatment. This research opens new avenues for exploring higher dimensional and coupled fractional systems, paving the way for future advancements in nonlinear dynamics and fractional calculus.