The objective of this study is to investigate the inhomogeneous nonlinear generalized Hartree equation within the mass-critical regime. Precisely, we give a threshold of global existence versus blow-up of energy solutions depending on the size of the mass compared with the $L^{2}$ norm of the associated ground state. This complements the existing literature to the case of a non-local source term.

Fractional-order partial differential equations (PDEs) have become an essential tool in modeling complex phenomena across various scientific disciplines due to their ability to describe processes involving memory and hereditary effects. This paper explores the application of the Caputo fractional derivative in solving FOPDEs and systems, presenting a novel numerical method that combines the Laplace transform with the Least-Squares Residual Power-Series Method. This approach enables the efficient approximation of solutions to fractional differential equations, even under complex boundary conditions. Additionally, the paper addresses the Kolmogorov equation, commonly used in stochastic processes, by integrating fractional derivatives to model nonlocal and anomalous diffusion phenomena. The results demonstrate the effectiveness of the proposed method in solving both FOPDEs and Kolmogorov equations, providing valuable insights for future research in fractional calculus and its diverse applications in engineering, physics, and finance.

This research delves into the intriguing dynamics of a magneto-Sisko fluid within a two-dimensional realm, shaped by the influence of a nonlinearly stretchable sheet nestled in a porous medium. The investigation embraces the impacts of a steady magnetic field, thermal radiation, and heat generation, while artfully weaving in the phenomena of Brownian motion and thermophoresis diffusion. By applying a similarity transformation, the governing nonlinear partial differential equations morph into ordinary differential equations, which are deftly tackled using the Newton–Raphson shooting method in tandem with the Runge–Kutta–Fehlberg algorithm. The outcomes indicate that an increase in the Sisko fluid parameter amplifies velocity profiles, concurrently diminishing both temperature and concentration distributions. For example, elevating the Sisko parameter from 1.0 to 2.0 propels the velocity profile upward by approximately 20%, while the temperature and concentration profiles witness reductions of 10% and 8%, respectively. Moreover, escalating values of the power-law exponent and the nonlinear stretching coefficient contribute to a decline in velocity, temperature, and concentration. In particular, when the power-law index ascends from 1.2 to 1.8, the velocity experiences a decrease of 14%, and temperature and concentration diminish by 12% and 5%, respectively. The introduction of thermal radiation enhances the temperature profile by nearly 20%, underscoring its pivotal role in the transport of energy. These results resonate with prior research, highlighting the intricate dance of magnetic fields, viscosity fluctuations, and heat diffusion mechanisms in shaping the fluid’s behavior. The findings furnish invaluable perspectives for refining industrial applications such as polymer extrusion, metallurgical endeavors, and chemical engineering systems.

This study presents a local meshless method based on radial basis functions for the numerical solution of the nonlinear time-fractional Burgers’ equation (TFBE) involving the fractional derivatives of Caputo (CFD) and Caputo–Fabrizio (CFFD). Time is discretized using the implicit finite difference scheme with θ=1, while radial basis functions (RBFs), which do not require a mesh to approximate the solution, are used for spatial discretization. The Rubin–Graves technique is used to linearize nonlinear terms. Approximation of the existing spatial derivatives is done using central and finite difference methods, and the temporal derivatives are approximated by using the definition of Caputo and Caputo–Fabrizio derivatives. With the mentioned techniques, a system of algebraic equations is established. By comparing the results obtained from solving this system with the previous results, it is clear that the presented technique provides accurate, stable, and convergent results.

The existence of regular and singular bound state solutions to △pu+f(u)=0,r∈Rn∖{0} is considered. Our result concerns the solution according to its behavior as r→0 and r→∞. Under the assumption that f is supercritical for small u>00$\\end{document}]]> and is subcritical for large u>00$\\end{document}]]>, we show the existence of various types of solutions. The Pohozaev identity plays a crucial role in our investigation.

Fractals are of immense importance across various branches of mathematics, science, and integral inequalities, as their intricate, self-similar structures can model complex natural phenomena and enhance the precision of mathematical descriptions. In this article, we explore generalized Milne-type integral inequalities within the framework of generalized m-convex functions on fractal sets. To achieve this, we introduce a novel fractal integral identity involving differentiable generalized functions. Utilizing this new identity, we derive several contemporary fractal Milne-type integral inequalities and provide specific inequalities for bounded functions. Additionally, we offer illustrative examples and applications, including additional inequalities for generalized special means and various error estimates for the generalized Milne-type quadrature formula.

This study introduces a novel implicit Jungck-type iterative scheme for solving nonlinear functional integral equations with double delays. By incorporating weak contractive conditions, the proposed scheme enhances computational efficiency and stability while achieving convergence equivalent to its explicit counterpart. Compared to existing Jungck-type explicit schemes, the implicit iteration exhibits superior stability and faster convergence.This work addresses key limitations in previous studies (Alam, Rohen in J. Appl. Math. Comput., https://doi.org/10.1007/s12190-024-02134-z, 2024; Copur et al. in Math. Comput. Simul. 215:476–497, 2024; Sharma et al. in Contemp. Math. 5:743–760, 2024) by:1. Establishing fixed-point results for a broader class of mappings with guaranteed existence and uniqueness of fixed point,2. Refining the stability framework for more reliable theoretical outcomes,3. Removing restrictive conditions on control sequences to improve convergence, and4. Deriving sharper estimates for data dependence, ensuring greater accuracy.Theoretical analysis within the weak w2-stability framework is supported by numerical experiments, including boundary value problems, confirming the scheme’s enhanced stability and efficiency. By refining and extending recent developments in fixed-point and coincidence-point theory, this study provides a robust framework for solving complex nonlinear problems.

In this work, the revised conceptions of single-valued and set-valued Meir-Keeler-type contraction are defined, together with fixed point findings for such contractions in fuzzy metric space (FMS). Extensive examples are provided to validate the outcomes. The findings demonstrated in this work generalize a number of intriguing findings in existing metric spaces. Our novel thoughts and results are also applied to the folding of an elastic band restrained at both ends by taking into account the two-point boundary issue involving differential equations of fourth order.