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.

Coxiella burnetii (C. burnetii) is one of the most important bacterial agents that cause abortion (Q fever) in ruminants and is a zoonotic disease. The current study reported the first confirmed detection of C. burnetii in sheep in Libya using real-time polymerase chain reaction coupled with high-resolution melt analysis.

Case Description:

Two flocks of sheep suffered abortion, with the aborted fetuses varying in age from early to late term. Vaginal swabs were collected, and DNA was extracted. DNA samples were analyzed using real-time PCR coupled with high-resolution melting (HRM) analysis, and the results showed infection with the zoonotic Q fever caused by C. burnetii. The animals responded to treatment with the antibiotic enrofloxacin, and the abortion was stopped after treatment.

This study aimed to evaluate the effect of Artemisia herba alba extracts (aqueous, alcoholic and essential oil) on

antibiotic-resistant bacteria. In which four types were tested, namely Staphylococcus aureus (MRSA)

Staphylococcus epidermidis, Klebsiella pneumoniae, Klebsiella oxytoca, using the diffusion disk method and

concentrations (100, 200, 300 mg/ml) for the aqueous and alcoholic extract, while the essential oil was used in

its pure form. The most effective antibacterial activity was observed for the essential oil of plant with an inhibition

diameter from 8.00±0.00 to 11.33±1.52 mm, and its aqueous extract effect on bacteria MRSA and S. epidermidis

was 8.66±1.52 and 8.33±0.57 mm, respectively, at 300 mg/ml. The inhibitory effect of alcoholic extract was

10.00±0.00 mm at 300 mg/ml for MRSA as for S. epidermidis at the same concentration had high inhibition

diameter 13.66±1.15 mm. Phytochemical screention of aqueous extract revealed the presence of phenols and

saponins, whereas, alcoholic extract revealed the presence of phenols, tannins, terpenoids and saponins. The

chemical components of the essential oils were determined by GC-MS device. The main compounds of A. herba

alba were Camphor (30.527%), Thujone (22.471%) and Camphene (10.291%). These results suggest that

essential oil extract can be considered as a potential source of antibacterial compounds.

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 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.

Artemisia herba-alba Asso is a medicinal plant renowned for its therapeutic essential oil. This study characterized the chemical profile and bioactivities of the essential oil from Libyan A. herba-alba. The essential oil, obtained via hydrodistillation with a 2% yield, was analyzed by GC-MS/FID and evaluated for antibacterial (disc diffusion) and anticancer (MTT assay) properties. The oil was dominated by camphor (30.53%) and Thujone (22.47%), classifying it as a camphor-thujone chemotype. It exhibited remarkable antibacterial activity against methicillin-resistant Staphylococcus aureus (MRSA; 17.33±1.52 mm), and demonstrated potent, dose-dependent cytotoxicity against MCF-7 breast cancer cells with an IC₅₀ of 0.59 mg/mL. The results validate the traditional use of this plant and highlight its potential as a source of antimicrobial and anticancer agents

Artemisia herba-alba Asso is a medicinal plant renowned for its therapeutic essential oil. This study characterized the chemical profile and bioactivities of the essential oil from Libyan A. herba-alba. The essential oil, obtained via hydrodistillation with a 2% yield, was analyzed by GC-MS/FID and evaluated for antibacterial (disc diffusion) and anticancer (MTT assay) properties. The oil was dominated by camphor (30.53%) and Thujone (22.47%), classifying it as a camphor-thujone chemotype. It exhibited remarkable antibacterial activity against methicillin-resistant Staphylococcus aureus (MRSA; 17.33 ± 1.52 mm), and demonstrated potent, dose-dependent cytotoxicity against MCF-7 breast cancer cells with an IC₅₀ of 0.59 mg/mL. The results validate the traditional use of this plant and highlight its potential as a source of antimicrobial and anticancer agents.