Search Results (46)

The coupled nonlinear system of fractional Boussinesq–Burger equations that may be utilized to model the propagation of shallow water waves is solved in this study using a novel numerical approach. The fractional derivatives in Caputo–Fabrizio and Atangana–Baleanu manner are executed in the system under consideration. The exact solutions of the proposed nonlinear fractional system are shown in the classical scenario of fractional order at $\mathrm{ß}=1$, whereas the approximate solutions are derived using the natural decomposition method. The series solution is generated such that it is simple to compute. Our results are compared with the exact results which clearly show that the suggested approach solutions quickly converge to the known accurate results. We acquire some analysis of the absolute error by comparing the approximate values with their corresponding precise solutions throughout the provided computations. Numerical and graphical simulations are used to confirm the usefulness of the suggested approach, and the outcomes are compared with well-known methods like the fractional decomposition method (FDM) and Laplace residual power series method (LRPSM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of the proposed system. The obtained outcomes ensure that the suggested approach is more effective and examines the highly nonlinear problems arising in engineering and science. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article

Triaxial suction-controlled relaxation tests were performed on unsaturated reticulated red clay from a highway cut slope to quantify the coupled influence of matric suction (50–200 kPa), net confining pressure (100–300 kPa), and axial strain (2–8%) on time-dependent stress decay. The results reveal that 60–80% of deviatoric stress dissipates instantaneously, with the remaining loss evolving nonlinearly toward a stable residual; higher suction or confinement raises residual capacity but enlarges absolute relaxation, whereas increasing strain accelerates damage and intensifies stress drop. A parsimonious three-element fractional Poynting–Thomson (FPTh) model that embeds Caputo-derived Koeller dashpot and the exponential damage variable of the viscous coefficient was formulated. The proposed model demonstrates a superior performance compared with the Merchant, Burgers, and Nishihara models (R² > 0.99 and RMSE < 3.5). The FPTh model faithfully reproduces the rapid and attenuating relaxation phases, offering a robust predictive tool for the long-term stability assessment of unsaturated clay slopes. Full article

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods. Full article

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the $L1-2$ formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the $L_2$ norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of $O({\tau }^{1+\alpha }+{\tau }^2{t}_n^{\alpha -2})$ for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations. Full article

Background: Vegetables are consumed worldwide in various forms, including raw, as green leaves in salads, and as ingredients in a wide range of dishes, such as curries, sauces, and burgers. They are rich in carbohydrates and dietary fiber (DF), and also provide moderate amounts of protein, fat, oils, essential micronutrients, minerals, vitamins, and phytochemicals. Among their carbohydrate components, simple sugars such as monosaccharides/hexoses significantly influence postprandial blood glucose responses. The glycemic index (GI) is critical for managing chronic conditions, such as diabetes, obesity, hyperglycemia, and other metabolic diseases. The influence of individual carbohydrate fractions, such as hexoses, on GI and glycemic load (GL) has not been extensively investigated. Methods: This retrospective study analyzed the carbohydrates in vegetables (n = 65), focusing on hexoses and fibers, their carbohydrate-to-fiber ratio, and their effect on the GI and GL. Carbohydrate data were obtained from publicly accessible databases, including the U.S. Department of Agriculture (USDA), FooDB, European and Australian food databases, and PubMed. The study assessed total carbohydrates (TC), hexoses, dietary starch (DS), total sugars (TS), and DF, and examined their correlations with GI using regression analysis. Results: Our analysis revealed that fiber ratios are a more reliable predictor of GI than conventional net carbohydrate measures. Among the carbohydrates analyzed, TC exhibited the highest positive correlation with GI, both in absolute terms and when normalized to fiber, while TS showed a weak correlation. Among the ratios studied, TC demonstrated a stronger correlation with the GI, followed by DS. Conclusions: Comparative evaluation revealed that DF exerts a buffering effect on glycemic response (GR) and supports the use of fiber ratios as a more stable and intrinsic parameter for predicting GI than standard estimation methods. Traditional approaches that rely on net carbohydrates may overlook important factors affecting glycemic impact, particularly the buffering effects of dietary fiber. This study advocates for the incorporation of carbohydrate-to-fiber ratios into GI estimation models. Our research may help evaluate the carbohydrate content in vegetables for further in vitro and in vivo studies aimed at clarifying the mechanisms and validating these metrics in glycemic regulation. Full article

This work investigates how fractionality affects the dynamical behavior of dust-acoustic shock waves that arise and propagate in a depleted-electron complex plasma. This model consists of inertial negatively charged dust grains and inertialess nonextensive distributed ions. Initially, the fluid model equations that govern the propagation of nonlinear dust-acoustic shock waves are reduced to the integer Burgers-type equations using the reductive perturbation method. Thereafter, the integer Burgers-type equations are converted to the fractional cases using a suitable transformation. For analyzing this fractional family, both the Tantawy technique and the new iterative method are implemented within the Caputo sense framework. These methods can produce highly accurate analytical approximations without necessitating stringent assumptions or intricate computational processes, in contrast to other similar methods. Numerical examples and the calculation of the absolute error demonstrate the efficacy of the suggested methodologies, emphasizing their superior precision and swift convergence. 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

The paper consists of various types of wave solutions for the truncated M-fractional Bateman–Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the ${exp}_a$ function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman–Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective. Full article

This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via $H^1$ regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is $\alpha >0$ and the stability is of order O($\delta$) with respect to observation noise ($\delta$); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions. Full article

The rheological behavior of firefighting foam is the basis for analyzing foam flow and foam spreading. This experimental study investigates the complex rheological behavior of rapidly aging firefighting foams, specifically focusing on alcohol-resistant aqueous film-forming foam. The primary objective is to characterize the time-dependent viscoelasticity, yielding, and viscous flow of firefighting foam under controlled shear conditions, addressing the significant challenge posed by its rapid structural evolution (drainage and coarsening) during measurement. Using a cylindrical Couette rheometer, conductivity measurements for the liquid fraction, and microscopy for the bubble size analysis, the study quantifies how foam aging impacts key rheological parameters. The results show that the creep and relaxation response of the firefighting foam in the linear viscoelastic region conforms to the Burgers model. The firefighting foam shows ductile yielding and significant shear thinning, and its flow curve under slow shear can be well represented by the Herschel–Bulkley model. Foam drainage and coarsening have competitive effects on the rheology of the firefighting foam, which results in monotonic and nonmonotonic variations in the rheological response in the linear and nonlinear viscoelastic regions, respectively. The work reveals that established empirical relationships between rheology, liquid fraction, and bubble size for general aqueous foams are inadequate for firefighting foams, highlighting the need for foam-specific constitutive models. 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

The research on nano titanium dioxide (nano-TiO₂)-modified asphalt has received increasing attention. However, further studies are required in order to ascertain the influence of the phenomenon under discussion on the rheological characteristics and ability to resist deformation of bitumen. In the present study, modified bitumen was formulated by adding nano titanium dioxide. Physical property tests, temperature scanning tests, frequency scanning tests, repeated creep recovery tests, bending creep stiffness tests, and long-term aging performance experiments were carried out on the specimen of asphalt that had undergone the process of modification in order to assess the rheological characteristics and ability to resist unrecoverable deformation of the modified bitumen at different temperatures, both high and low. The outcomes of the repeated creep recovery experiment were analyzed using Burgers and fractional derivative models. The microstructure of nano-TiO₂ high-viscosity modified asphalt was observed by Scanning Electron Microscope(SEM). In order to ascertain the manner in which base bitumen and nano-TiO₂ interact, Fourier transform infrared spectroscopy (FTIR) was utilized in the study. The results show that the thermal stability and prolonged aging resistant properties of the modified bitumen binder improved, but nano-TiO₂ made the asphalt binder weaker and more likely to crack at lower temperatures. Taking into account the variation in the road performance of the bitumen binder, 6% is recommended as the optimal amount of nano-TiO₂. Nano-TiO₂ was mainly uniformly distributed in asphalt and nano-TiO₂ was physically mixed with asphalt. In comparison with the Burgers model, the present fractional derivative empirical creep model can fit the creep test data of the asphalt binder well with the advantages of high accuracy and few parameters. The research results provide a reference for promoting the implementation of modified bitumen incorporating nano-TiO₂. Full article

The long-term effects of dry–wet cycles induced by seasonal rainfall significantly influence the creep behavior of sliding zone soft rocks, contributing to landslide occurrence. Understanding this aspect is crucial for predicting and mitigating long-term slope instability. This study investigates the Mohuandang landslide, conducting shear creep tests on carbonaceous shale under dry–wet cycles. A quantitative approach was introduced, incorporating a fractional derivative to modify the Burgers model and develop an improved creep equation. Model validity was verified through experimental data. The key findings are as follows: (1) At low deviatoric stress levels (within the viscoelastic stage), creep deformation exhibits a nonlinear increase under dry–wet cycles, leading to a progressive reduction in long-term strength. (2) The modified creep model effectively captures the creep behavior of the sliding zone under the influence of dry–wet cycle-induced damage. (3) The damage evolution characteristics exhibit clear physical significance. These results provide theoretical insights and practical guidance for landslide prediction and risk management in regions subjected to dry–wet cycles induced by seasonal rainfall. Full article

In this work, we introduce an innovative analytical–numerical approach to solving nonlinear fractional differential equations by integrating the homotopy perturbation method with the new integral transform. The Kawahara equation and its modified form, which is significant in fluid dynamics and wave propagation, serve as test cases for the proposed methodology. Additionally, we apply the fractional new integral transform–homotopy perturbation method (FNIT-HPM) to a nonlinear system of coupled Burgers’ equations, further demonstrating its versatility. All calculations and simulations are performed using Mathematica 12 software, ensuring precision and efficiency in computations. The FNIT-HPM framework effectively transforms complex fractional differential equations into more manageable forms, enabling rapid convergence and high accuracy without linearization or discretization. By evaluating multiple case studies, we demonstrate the efficiency and adaptability of this approach in handling nonlinear systems. The results highlight the superior accuracy of the FNIT-HPM compared to traditional methods, making it a powerful tool for addressing complex mathematical models in engineering and physics. Full article