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The coupled Burgers model, a system of nonlinear partial differential equations, is widely used to describe complex interactions in various fields, including fluid dynamics, traffic flow, and biological systems. It provides insights into phenomena such as shock waves, turbulence, pattern formation, and transitions from order to chaos. This study applies the He-Laplace-Carson (HLC) scheme to solve the (3 + 1) dimensional coupled Burgers model under fuzzy-fractional conditions. The HLC scheme, which combines the homotopy perturbation method with the Laplace-Carson transform, is utilized to obtain accurate numerical solutions for these highly nonlinear systems. The model is analyzed in both fractional and fuzzy-fractional frameworks, enabling a comprehensive examination across lower and upper bounds, as well as the traditional crisp case. Two benchmark problems are investigated to evaluate the influence of Caputo fractional derivatives, with residual error analysis validating the accuracy of the results. Detailed two and three-dimensional visualizations, along with innovative contour plots, provide deeper physical interpretations of the model’s behavior. The findings demonstrate the robustness and effectiveness of the proposed methodology in addressing complex mathematical models that arise in diverse scientific applications.

Fractional calculus was born in 1695 on September 30 due to a very deep question raised in a letter of L’Hospital to Leibniz. The prophetical answer of Leibniz to that deep question encapsulated a huge inspiration for all generations of scientists and is continuing to stimulate the minds of contemporary researchers. During 325 years of existence, fractional calculus has kept the attention of top level mathematicians, and during the last period of time it has become a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering. In this short manuscript, we briefly review the tremendous effect that the main ideas of fractional calculus had in science and engineering and briefly present just a point of view for some of the crucial problems of this interdisciplinary field.

Cancer is a complex and heterogeneous condition marked by the unchecked growth and dissemination of abnormal cells, posing significant challenges in detection, treatment, and patient care. As one of the leading global causes of death, a deep understanding of its underlying biological mechanisms is essential for advancing therapeutic strategies and improving clinical outcomes. Mathematical modeling serves as a crucial tool in capturing the multifaceted dynamics of cancer initiation, progression, and response to interventions. By simulating critical aspects of cancer within a controlled computational framework, mathematical models enable scientists to explore innovative treatment strategies, investigate the disease’s underlying biological dynamics, and identify novel therapeutic targets. This study presents a fuzzy-fractional differential model of tumor-immune interaction, incorporating tumor cells, immune effector cells, and the concentration of chemotherapy agents in the bloodstream, modeled using Caputo-type time-fractional derivatives. To better capture the uncertainty associated with patient-specific factors—particularly psychological impacts such as depression—triangular fuzzy numbers are integrated into the initial conditions, thereby enhancing the model’s realism and predictive capability. The current model is addressed using the proposed modified He–Laplace–Carson algorithm for solution and analysis by creating multiple homotopies related to the perturbation method. The model’s solution trajectories for chemotherapy levels, immune effector cells, and tumor cell populations are examined to evaluate the bidirectional interaction between immune response and tumor growth. Additionally, the simulation provides insight into the dynamic behavior of chemotherapy concentration over the duration of treatment, offering a clearer understanding of its therapeutic progression. An extensive graphical analysis is conducted by varying a range of parameters, including effect of depression, death rate of immune cells due to malignant cells attachment, maximum growth rate factors, and fuzzy parameters introduced in the cancer system. It was observed that as the fractional parameter increased, all profiles rose, with effector cells showing a more notably faster growth than tumor cells. Furthermore, all fuzzy and non-fuzzy parameters generally showed a strong positive influence on effector cells, with tumor cell growth remaining comparatively subdued. The fractional parameter is analyzed under diverse conditions using 2D/3D visualizations and contour gradients, confirming the method’s reliability in handling uncertainty and highlighting its adaptability to broader fuzzy-fractional systems. Such applications have the capability to significantly enhance the understanding of ongoing challenges faced in oncology.

Cholera is a waterborne disease that is mostly spread by taking tarnished food and water. This disease is brought on by bacteria Vibrio cholerae and causes infection in humans. The current manuscript proposes a fuzzy-fractional SEIHRD modeling framework for cholera disease outbreak in Angola using epidemiological data taken from World Health Organization. The Caputo fractional derivatives are used to capture the memory effects, while involved parameters are fuzzified using triangular fuzzy numbers for incorporating uncertainties involved in real-world data. The stability of the proposed model is examined to look at the circumstances in persistence and eradication of disease. For finding important factors influencing the dynamics of cholera transmission, sensitivity analysis is also carried out in this study. The effect of fractional and fuzzy parameters on the proposed model is analyzed via contour diagrams. The hybrid of fractional and fuzzy calculus yields a more realistic depiction of cholera dynamics, as depicted in numerical simulations. The analysis provides valuable results for planning public health initiatives for predicting and controlling cholera outbreaks.

In this article, two numerical approaches are presented to solve a system of three fractional differential equations that express the pollution of lakes. In our recent study, a new class of hat functions, called QHFs, are constructed. The proposed approaches utilize Modified Hat Functions (MHFs) and QHFs. Fractional-order operational of MHFs and QHFs are used to build algorithms that transform the main problem into a system of six equations with six unknowns and three equations with three unknowns, respectively. Absolute errors of obtained approximate solutions and convergence analysis of the utilized approach will be studied. Finally, three examples are provided to illustrate the capabilities of these algorithms. The pollution monitoring results are reported in some tables and figures for different values of a.

This paper introduces an efficient numerical algorithm for solving a significant class of linear and nonlinear time-fractional partial differential equation governed by Fredholm–Volterra operator in the sense of Robin conditions. A direct approach based on the normalized orthonormal function systems that fitted from the Gram–Schmidt orthogonalization process is utilized to transcribe the problem under study into appropriate Hilbert space. Some functional analysis theories such as upper error bound and convergence behavior under some assumptions which give the hypothetical premise of the proposed calculation are likewise talked about. Mathematical properties of the numerical results obtained are analyzed as well as general features of many numerical solutions have been identified. At long last, the used outcomes demonstrate that the present calculation and mimicked toughening give a decent planning procedure to such models.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin.

The ability of the so-called Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) operators to create suitable models for real data is tested with real world data. Two alternative models based on the CF and AB operators are assessed and compared with known models for data sets obtained from electrochemical capacitors and the human body electrical impedance. The results show that the CF and AB descriptions perform poorly when compared with the classical fractional derivatives.

Inverse problems involving time-fractional differential equations have become increasingly important for modeling systems with memory-dependent dynamics, particularly in biotransport and viscoelastic materials. Despite their potential, these problems remain challenging due to issues of stability, non-uniqueness, and limited data availability. Recent advancements in Physics-Informed Neural Networks (PINNs) offer a data-efficient framework for solving such inverse problems, yet most implementations are restricted to integer-order derivatives. In this work, we develop a PINN-based framework tailored for inverse problems involving time-fractional derivatives. We consider two representative applications: anomalous diffusion and fractional viscoelasticity. Using both synthetic and experimental datasets, we infer key physical parameters including the generalized diffusion coefficient and the fractional derivative order in the diffusion model and the relaxation parameters in a fractional Maxwell model. Our approach incorporates a customized residual loss function scaled by the standard deviation of observed data to enhance robustness. Even under 25% Gaussian noise, our method recovers model parameters with relative errors below 10%. Additionally, the framework accurately predicts relaxation moduli in porcine tissue experiments, achieving similar error margins. These results demonstrate the framework’s effectiveness in learning fractional dynamics from noisy and sparse data, paving the way for broader applications in complex biological and mechanical systems.

This work is about the mass and heat transfer flow for adhesive fluid between two upright plates pulled apart by a distance, d. Fractional model of the considered problem is developed after making governing equations dimensionless. Laplace transform technique is utilized to acquire analytical solutions and some graphics are presented to see the physical behavior of embedded parameters.