This study presents a novel and efficient computational framework for numerically solving fractal-fractional Lane-Emden (FFLE) equations using the new generalized Caputo fractal-fractional derivative (NGCFFD). Lane-Emden equations arise naturally in modeling astrophysical phenomena such as stellar structure, polytropic gas spheres, and isothermal gas clouds, as well as in chemical kinetics, thermoelectric processes, and various branches of applied mathematics. The proposed technique employs shifted Legendre polynomials (SLPs) to construct an operational matrix that systematically transforms these complex fractal-fractional differential equations (FFDEs) into tractable systems of algebraic equations, applicable to both linear and nonlinear multi-order FFDEs. Six representative FFLE equations are investigated across twelve computational models with varying fractal-fractional parameters (γ, β, ρ), and the results are rigorously compared against conformable fractional (CF) and modified Riemann–Liouville (mRL) solutions available in the literature. The proposed fractal-fractional models consistently outperform the corresponding standard Caputo models. An error estimation theorem and a convergence analysis confirm that the series solutions converge to the exact solutions in the Hilbert space [0,1]. We think this is the first study to report a numerical solution of fractal-fractional Lane-Emden equations, establishing the fractal order parameter β as an essential, rather than merely supplementary, modeling component for certain classes of singular nonlinear problems.

Sediment loss (SLO) is a critical geological phenomenon characterized by a reduction in mass over time due to the combined effects of controlling variables such as rock competency, weathering, and erosion. This study aims to develop an efficient numerical method for solving SLO fractal-fractional differential equations (SLO-FFDEs) based on shifted Legendre polynomials (SLPs) and to investigate the effects of the fractal, fractional, and erosion parameters on the sediment loss dynamics. The SLO-FFDEs are approximated using the basis vectors of SLPs, and a derivative operational matrix (OM) of SLPs is developed within the framework of the new generalized Caputo fractal-fractional derivative (GCFFD). The proposed method is evaluated for different values of the fractal parameter σ, fractional parameter ϱ, and decaying parameter λ associated with erosion. The numerical results demonstrate that the proposed OM-based framework provides an effective and accurate approach for describing the dynamics of sediment loss under different combinations of fractal and fractional parameters. The results also indicate that variations in the fractal and fractional parameters significantly influence the behavior of the SLO model, while the erosion parameter controls the rate of sediment loss. Therefore, the proposed method provides a flexible computational framework for modeling SLO-FFDEs and can be used to investigate the influence of fractal, fractional, and erosion effects in sediment-loss processes. It is recommended that the proposed framework be further extended to more complex sediment-loss models and validated using experimental or field data

This study introduces novel derivative and integral operators derived from newly formulated generalized Caputo fractional derivatives (GCFDs). A numerical approach isdeveloped utilizing these operators to solve variable-order fractional differential equationswith time-varying delay (VOFDETVD). The approach entails approximating the solutionsof VOFDETVD by employing shifted Legendre polynomials (SLPs) as basis vectors. Toassess the efficiency of the numerical method, tests are conducted on various examples, withdierent values of ϱ specified for the fractional differential operator of the new generalized Caputo. Moreover, the present method is comprehensively evaluated for robustness andeectiveness through comparisons with existing approaches.

The broad applicability of fractional calculus to modeling physical phenomena via fractional differential equations, along with their complexity, has created substantial demand for efficient analytic and semi-analytic techniques for solving them. This paper has derived a numerical approach to solving a specific class of fractal fractional differential equations (FFDEs) involving the Generalized Caputo-type (GC) of a function with respect to another function, or fractal-Ψ-GC, which is shown in this study. The approach is based on an operational matrix (OM) of the fractal fractional derivative of a novel form of orthogonal polynomial. The Ψ-normalized shifted Legendre polynomials (NSLP) and Ψ-shifted Legendre polynomials (SLP) are introduced. The main characteristic of this approach is that it reduces such problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. Examples are provided to portray the efficiency and applicability of this method. Comparison with similar existing approaches is also conducted to demonstrate the accuracy of the proposed approach.

In the past few years, the area of mathematical study has made considerable advances, owing largely to the introduction of artificial intelligence (AI) tools. Among these, artificial neural networks (ANNs) have played an important role in modernizing several mathematical techniques and problem-solving approaches. ANNs have recently become popular as a powerful mathematical research tool, providing an effective alternative to established approaches for solving fractal-fractional differential equations (FFDEs). This paper describes the use of a feed-forward ANN with a hidden layer to address systems resulting from the fractal-fractional Bagley-Torvik differential equation (FFBTDE). In addition, a power series (PS) technique is introduced to increase efficiency. The paper looks at for solving FFBTDE with variable and constant coefficients. The numerical findings show that the suggested strategy not only produces results that closely match exact and reference solutions, but also outperforms existing methods in terms of accuracy.

A highly efficient and accurate numerical method for systems of fractional differential equations (FDEs) with rational order is presented in this paper. Rational power functions and rational Taylor series projection are utilized to obtain approximate solutions. Rational semi-smooth spaces are introduced, and the regularity of solutions in these spaces is established. A series of theoretical results, such as the existence and uniqueness of solutions, properties of the rational Taylor series and its remainder term, and an operational matrix approach, are derived. It is proven that the numerical solution is exact when the exact solution is a rational power series, and the approximate solution is shown to be the rational Taylor series projection of the exact solution. The convergence of the method is analyzed. The efficiency of the proposed method is demonstrated through numerical experiments, which show significant improvements in computational time compared to existing methods.

In this study, we numerically investigate two significant medical models, Ebola Viral Disease (EVD) and Blood Ethanol Concentration (BEC) models-both formulated using Caputo-fractional derivatives. We develop and apply the Modified Fractional Euler Method (MFEM) for their solution, with a specific focus on error analysis. Comparative studies with the classical Runge-Kutta fourth-order method (RK4M) demonstrate that MFEM provides a computationally efficient and accurate alternative for solving such systems. The major features of the given procedure are its ease of application to this type of problem and other systems in various fields, in addition to the absence of numerical errors accumulating. Finally, we can control the increase in the convergence rate and the stability of the simulation process. The convergence examination and error estimation for the suggested scheme are also included. The importance of this study also lies in its contribution to our understanding of the dynamics of these two models in their fractional form. In addition, those numerical investigations demonstrate how control parameters affect specific components within these models.

 We give the approximate solution of the Riccati/Logistic differential equations (RDE/LDE). The suggested approach depends on the homotopy perturbation method developed with the Chebyshev series (CHPM). A study of the convergence analysis of CHPM is presented. The residual error function is calculated and used as a basic criterion in evaluating the accuracy and efficiency of the given numerical technique. We use the exact solution and the Runge-Kutta method of fourth order for comparison with the results of the method used. Through these results, we can confirm that the applied method is an easy and effective tool for the numerical simulation of such models. Illustrative models are given to confirm the validity and usefulness of the proposed procedure. 

In this research, we investigated the Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral constraints. We presented novel sufficient conditions for the uniqueness of the solution. Moreover, we analyzed the continuous dependence of the solution on some functions and parameters. Additionally, we proved the Hyers-Ulam stability of the problem. To demonstrate the applicability of our results, we included several examples. The present study was located in the space L1[0, T]. The techniques of Schauder’s fixed point theorem and Kolmogorov’s compactness criterion were the primary tools utilized in this work. These contributions offer a comprehensive framework for understanding the qualitative behavior of the fractional-order pantograph equation.

This paper presents theoretical proof of the existence of a unique solution to a constrained problem of the Riemann-Liouville fractional differential equation with time delay functions by utilizing the Schauder fixed point theorem. Moreover, we analyzed the continuous dependence of the solution on the initial conditions and other parameters. Further, we investigate the Hyers-Ulam stability of the problem. We introduce some examples and special cases to illustrate our results.