Search Results (113)

Using the newly introduced, by the author, basic complex and hybrid complex rheologic models of the fractional type, the dynamics of a series of mechanical rheologic discrete dynamical systems of the fractional type (RDDSFT) of rheologic oscillators (ROFTs) or creepers (RCFTs), with corresponding independent generalized coordinates (IGCs) and external (IGCEDF) and internal (IGCIGF) degrees of freedom of movement, were studied. Laplace transformations of solutions for independent generalized coordinates (IGCs), as well as external (IGCEDFs) and internal (IGCIDF) degrees of freedom of system dynamics, were determined. On the studied specimens, it was shown that rheologic complex models of the fractional type introduce internal degrees of freedom into the dynamics of rheologic discrete dynamical systems. New challenges appear for mathematicians, such as translating the Laplace transformations of solutions for independent generalized coordinates (LTIGCs) into the time domain. A number of translations of Laplace transformations of solutions into the time domain were performed by the author of this paper. A series of characteristic surfaces of elongations of Laplace transformations of independent generalized coordinates (IGCs) of the dynamics of rheologic discrete dynamic systems of the rheologic oscillator type, i.e., the rheologic creeper type, is shown as a function of fractional order differentiation exponent and Laplace transformation parameter. This manuscript presents the scientific results of theoretical research on the dynamics of rheologic discrete dynamic systems of the fractional type that was conducted using new models and a rigorous mathematical analytical analysis with fractional-order differential equations (DEFOs) and Laplace transformations (LTs). These results can contribute to new experimental research and materials technologies. A separate section presents the theoretical foundations of the methods and methodologies used in this research, without the details that can be found in the author’s previously published works. Full article

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation matrix (SGDM), while other terms are computed with precise quadrature rules. By integrating advanced techniques such as the shifted Gegenbauer pseudospectral method (SGPS), fractional derivative and integral approximations, and barycentric integration matrices, the HSG-IPS method achieves spectral accuracy. Numerical results show it reduces average absolute errors (AAEs) by up to 99.99% compared to methods like Crank–Nicolson linearized difference scheme (CNLDS) and finite integration method using Chebyshev polynomial (FIM-CBS), with computational times as low as 0.04–0.05 s. The method’s stability is improved by avoiding ill-conditioned high-order derivative approximations, and its efficiency is boosted by precomputed matrices and Kronecker product structures. Robust across various fractional orders, the HSG-IPS method offers a powerful tool for modeling wave propagation and nonlinear phenomena in fractional calculus applications. Full article

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied. Full article

This study combines artificial intelligence (AI) with mathematical modeling to improve the forecasting of the water cycle mechanism. The proposed model simulates the development of global temperature, precipitation, and water availability, integrating key climate parameters that control these dynamics. Using a system of fractional-order differential equations in the fractal–fractional sense of derivatives, the model captures interactions between solar radiation, the greenhouse effect, evaporation, and runoff. The deep learning framework is trained on extensive climate datasets, allowing it to refine predictions and identify complex patterns within the water cycle. By applying AI techniques alongside mathematical modeling, this procedure provides valuable insights into climate change and water resource administration. The model’s predictions can contribute to assessing future climate states, optimizing environmental policies, and designing sustainable water management strategies. Furthermore, the hybrid methodology improves decision-making by offering data-driven solutions for climate adaptation. The findings illustrate the effectiveness of AI-driven models in addressing global climate challenges with improved precision. Full article

Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework $\theta$-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued mappings, thereby broadening the scope of fuzzy analysis. Furthermerefore, we have for aore, we apply one of our key results to derive conditions, ensuring the existence and uniqueness of a solution to Hadamard $\Psi$-Caputo tempered fuzzy fractional differential equations, particularly in the context of the SIR dynamics model. These theoretical advancements are expected to open new avenues for research in fuzzy fixed-point theory and its applications to hybrid models within $\theta$-fuzzy metric spaces. Full article

In this study, we introduce an innovative approach for addressing fractional partial differential equations (fPDEs) by combining Monte Carlo-based physics-informed neural networks (PINNs) with the cuckoo search (CS) optimization algorithm, termed PINN-CS. There is a further enhancement in the application of quasi-Monte Carlo assessment that comes with high efficiency and computational solutions to estimates of fractional derivatives. By employing structured sampling nodes comparable to techniques used in finite difference approaches on staggered or irregular grids, the proposed PINN-CS minimizes storage and computation costs while maintaining high precision in estimating solutions. This is supported by numerous numerical simulations to analyze various high-dimensional phenomena in various environments, comprising two-dimensional space-fractional Poisson equations, two-dimensional time-space fractional diffusion equations, and three-dimensional fractional Bloch–Torrey equations. The results demonstrate that PINN-CS achieves superior numerical accuracy and computational efficiency compared to traditional fPINN and Monte Carlo fPINN methods. Furthermore, the extended use to problem areas with irregular geometries and difficult-to-define boundary conditions makes the method immensely practical. This research thus lays a foundation for more adaptive and accurate use of hybrid techniques in the development of the fractional differential equations and in computing science and engineering. Full article

This paper addresses a coupled system of hybrid fractional differential equations governed by non-local Hadamard-type boundary conditions. The study focuses on proving the existence, uniqueness, and stability of the system’s solutions. To achieve this, we apply Banach’s fixed point theorem and the Leray–Schauder alternative, while the stability is verified through the Ulam–Hyers framework. Additionally, a numerical example is presented to illustrate the practical relevance of the theoretical findings. Full article

An innovative approach is utilized in this paper to solve the fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional Fokker–Planck–Levy (FFPL) equation numerically. Two scenarios of the FFPL equation are considered by varying the value of (i.e., 1$.75, 1.85$). Moreover, three cases of each scenario are numerically studied for different discretized domains with $100, 200,$ and $500$ points in $x\in [-1, 1]$ and $t\in [0, 1]$. For the FFPL equation, solutions are obtained via the FDM-APINN technique via $1000, 2000,$ and $5000$ iterations. The errors, loss function graphs, and statistical tables are presented to validate our claim that the FDM-APINN is a better alternative intelligent technique for handling fractional-order partial differential equations with complex terms. The FDM-APINN can be extended by using nongradient-based bioinspired computing for higher-order fractional partial differential equations. Full article

The emerging concept of hybrid nanofluids has grabbed the attention of researchers and scientists due to improved thermal performance because of their remarkable thermal conductivities. These fluids have enormous applications in engineering and industrial sectors. Therefore, the present research study examines thermal and mass transportation in hybrid nanofluid past an inclined linearly stretching sheet using the Maxwell fluid model. In the current problem, the hybrid nanofluid is engineered by suspending a mixture of aluminum oxide $A{l}_2{O}_3$ and copper $Cu$ nanoparticles in ethylene glycol. The fluid flow is generated due to the linear stretching of the sheet and the sheet is kept inclined at the angle $\zeta =\pi /6$ embedded in porous medium. The current proposed model also includes the Lorentz force, solar radiation, heat generation, linear chemical reactions, and permeability of the plate effects. Here, in the current simulation, the cylindrical shape of the nanoparticles is considered, as this shape has proven to be excellent for the thermal performance of the nanomaterials. The governing equations transformed into ordinary differential equations are solved using MATLAB bvp4c solver. The velocity field declines with increasing magnetic field parameter, Maxwell fluid parameter, volume fractions of nanoparticles, and porosity parameter but increases with growing suction parameter. The temperature drops with increasing magnetic field force and suction parameter values but increases with increasing radiation parameter and volume fraction values. The concentration profile increases with increasing magnetic field parameters, porosity parameters, and volume fractions but reduces with increasing chemical reaction parameters and suction parameters. It has been noted that the purpose of the inclusion of thermal radiation is to augment the temperature that is serving the purpose in the current work. The addition of Lorentz force slows down the speed of the fluid and raises the boundary layer thickness, which is visible in the current study. It has been concluded that, when heat generation parameters increase, the temperature field increases correspondingly for both nanofluids and hybrid nanofluids. The increase in the volume fraction of the nanoparticles is used to enhance the thermal performance of the hybrid nanofluid, which is evident in the current results. The current results are validated by comparing them with published ones. Full article

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. Full article

The study investigates the performance of fluid flow, thermal, and mass transport within a cavity, highlighting its application in various engineering sectors like nuclear reactors and solar collectors. Currently, the focus is on enhancing heat and mass transfer through the use of ternary hybrid nanofluid. Motivated by this, our research delves into the efficiency of double-diffusive natural convective (DDNC) flow, heat, and mass transfer of a ternary hybrid nanosuspension (a mixture of Cu-CuO-Al₂O₃ in water) in a quadrantal enclosure. The enclosure’s lower wall is set to high temperatures and concentrations (${\mathrm{T}}_{\mathrm{h}}$ and ${\mathrm{C}}_{\mathrm{h}}$), while the vertical wall is kept at lower levels (${\mathrm{T}}_{\mathrm{c}}$ and ${\mathrm{C}}_{\mathrm{c}}$). The curved wall is thermally insulated, with no temperature or concentration gradients. We utilize the finite element method, a distinguished numerical approach, to solve the dimensionless partial differential equations governing the system. Our analysis examines the effects of nanoparticle volume fraction, Rayleigh number, Hartmann number, and Lewis number on flow and thermal patterns, assessed through Nusselt and Sherwood numbers using streamlines, isotherms, isoconcentration, and other appropriate representations. The results show that ternary hybrid nanofluid outperforms both nanofluid and hybrid nanofluid, exhibiting a more substantial enhancement in heat transfer efficiency with increasing volume concentration of nanoparticles. Full article

In this paper, a new class of coupled hybrid systems of proportional sequential $\psi$-Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel’ski$\breve{\mathrm{i}}$’s fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential $\psi$-Hilfer fractional differential equations. Full article

Fractional-order differential equations have been proved to have great practical application value in characterizing the dynamical peculiarity in biology. In this article, relying on earlier work, we formulate a new fractional oxygen–plankton model with delay. First of all, the features of the solutions of the fractional delayed oxygen–plankton model are explored. The judgment rules on non-negativeness, existence and uniqueness and the boundedness of the solution are established. Subsequently, the generation of bifurcation and stability of the model are dealt with. Delay-independent parameter criteria on bifurcation and stability are presented. Thirdly, a hybrid controller and an extended hybrid controller are designed to control the time of onset of bifurcation and stability domain of this model. The critical delay value is provided to display the bifurcation point. Last, software experiments are offered to support the acquired key outcomes. The established outcomes of this article are perfectly innovative and provide tremendous theoretical significance in balancing the oxygen density and the phytoplankton density in biology. Full article