Theoretical Techniques Research Articles

Chromatic M-polynomial of Mobius Function Graphs of Product Groups

Objective: To investigate the chromatic M-polynomials of the Mobius function graph of product groups, specifically focusing on the direct product of finite groups. Methods: We used algebraic and number theoretical techniques to construct a Mobius function graph of product groups and derived the chromatic M-polynomials using graph theoretic methods. Findings: Chromatic M polynomials of Mobius function graphs are introduced with a focus on deriving their significant properties and providing a practical approach for computation. Novelty: This research introduces a novel approach to studying the chromatic M-polynomials of the Mobius function graph of product groups, providing insights into the interplay between group theory, graph theory, and number theory. Our findings contribute connection between chromatic M polynomials and coloring problems, followed by a discussion and understanding of graph polynomials. Keywords: Mobius function graph of groups, Chromatic number, Chromatic M-polynomials, Product group, Dihedral Groups

The Role of Protonation in the PfMATE Transporter Protein Structural Transitions.

Multi-antimicrobial extrusion (MATE) transporter membrane proteins provide drug and toxin resistivity by expelling compounds from cells. MATE proteins can be pictured as V-shaped. To regulate its functioning, the protein structure can switch between outward-facing (OF) and inward-facing (IF). Pyrococcus furiosus MATE (PfMATE) is the only member of the multidrug/oligosaccharidyl-lipid/polysaccharide (MOP) superfamily that has available both the IF and OF crystal structures. With the availability of both the IF and OF structures, we are able to perform computational investigations to determine how protonation of specific amino acids causes a cascade of changes in the protein conformation that allow PfMATE to change its state from OF to IF in order to regulate its antiporter function. Using a variety of computational and theoretical techniques, we investigated four different systems of IF and OF PfMATE along with the native archaeal lipid bilayer, without or with protonation at the experimentally determined locations within the protein. We performed molecular dynamics (MD) simulations to investigate the flexibility of the four different PfMATE structures and also performed targeted molecular dynamics (TMD) simulations, during which we observed occluded conformations. Our analysis of hydrogen bond changes, potential of mean force, dynamic network analysis, and transfer entropy analysis provides information on how protonation can induce cascading structural changes responsible for the transition between the IF and OF configurations.

Exploration of quantum oscillation in antiferromagnetic Weyl semimetal GdSiAl.

GdSiAl single crystal has been investigated by means of magnetic and magneto-transport measurements and compared with ab-initio density functional theory (DFT) calculations. Significant non-saturating magnetoresistance reaching ∼ 18% at 12T and 2K was observed, alongside the presence of Shubnikov-de Haas oscillations with the fundamental frequencies 22.09T and 77.33T. Shubnikov-de Haas oscillations provide the information about the nontrivial π Berry phase in GdSiAl with the Fermi surface areas of 0.00211 Å-2and 0.00739 Å-2. Angle-dependent magnetoresistance shows anisotropy with θ, exhibiting a maximum at 180°. The magnetic susceptibility data for H ∥ c and H ⊥ c reveals that the magnetic moments of Gd3+ions orders antiferromagnetically below 32K along with an another transition occurs at ∼ 8K, which is consistent with the heat capacity measurements where a distinct λ-shaped anomaly has been observed near antiferromagnetic ordering temperature 32K. The high value of Debye temperature indicates the contribution of acoustic phonons. Electronic structure calculations suggest the existence of nested Fermi surface pockets characterized by nesting wave vectors that closely align with the observed magnetic ordering wave vector. Furthermore, DFT calculations reveal the presence of Weyl nodes in close proximity to the Fermi surface. Our findings from combined experimental and theoretical techniques indicate GdSiAl to be a potential candidate for an antiferromagnetic topological Weyl semimetal.

Optimizing Multi-Channel Green Supply Chain Dynamics with Renewable Energy Integration and Emissions Reduction

In response to the global imperative of mitigating greenhouse gas emissions (GHGs) and the shifting landscape of business models toward multi-channel structures, this study delves into the intricacies of a green supply chain. Operating through both online and traditional channels with uncertain demands, the producer’s distribution strategy prompts an exploration of supply chain dynamics. Utilizing an integer programming model, this study calculates optimal prices, optimizes total profit, and minimizes transportation costs to curtail carbon dioxide emissions, depending on the transportation mode. Additionally, this study incorporates renewable energy sources into the production and transportation processes to further minimize carbon dioxide emissions. The integration of renewable energy not only supports environmental goals, but also contributes to the overall profitability of the supply chain by reducing energy costs. Employing a theoretical technique for linearization, the model, resolved through the Jimenez and TH methods, demonstrates efficacy in reconciling economic and environmental goals. The Jimenez method enables the transformation of fuzzy parameters into deterministic equivalents, allowing for a more reliable optimization during uncertainty, while the TH method provides an interactive fuzzy multi-objective approach, aligning the model’s dual objectives for both economic and environmental goals. Notably, when transportation costs to both markets are equal, the model prioritizes devices with a lower environmental impact, showcasing adaptability. Furthermore, the proposed solution empowers decision makers to influence pricing and enhance the entire supply chain’s profitability. In conclusion, this research offers nuanced insights, strategically aligning economic viability with environmental sustainability in the discourse on green supply chains.

Faradically Dominant Pseudocapacitive Graphitic Carbon Nitride Nanosheets Decorated with Strontium Tungstate Nanospheres for Supercapattery Device and Hydrogen Evaluation Reaction

Transition metal oxides are promising for hydrogen evolution reaction (HER) and hybrid energy storage due to their excellent redox properties, inherent electrochemical activity, and abundant electroactive sites. A significant challenge limiting their broader application is their intrinsic low electrical conductivity and reduced electrochemical stability. For hybrid energy storage devices and HER, a highly electrochemical active material is designed from 2D graphitic carbon nitride nanosheet (g-C3N4) networks anchored with strontium tungstate nanospheres (SrWO4/g-C3N4). The excellent performance observed can be attributed to several factors: multiple electro-active sites, well-defined electronic structures, and interaction between SrWO4 nanosphere on the surface of g-C3N4 nanosheets surface. The supercapattery device exhibited superior energy density (65.4 W h/kg) and power density (1240.5 W/kg) in comparison. In addition, the theoretical technique was utilized to provide a detailed analysis of the experimental findings. In addition, the SrWO4/g-C3N4 material demonstrates a low overpotential of 129 mV at -10 mA/cm2, along with Tafel slope values of 67 mV/dec for the HER, and it exhibits excellent cyclic stability. This study presents an advanced method for designing SrWO4/g-C3N4-based supercapacitors and HER platforms with nanoscale structures and optimized interface arrangements.

Discrete element modelling and mechanical properties and cutting experiments of Caragana korshinskii Kom. stems.

The forage crop Caragana korshinskii Kom. is of high quality, and the biomechanical properties of its plant system are of great significance for the development of harvesting equipment and the comprehensive utilisation of crop resources. However, the extant research on the biomechanical properties of Caragana korshinskii Kom. is inadequate to enhance and refine the theoretical techniques for mechanised harvesting. In this study, we established a discrete element model of CKS based on the Hertz-Mindlin bonding contact model. By combining physical experiments and numerical simulations, we calibrated and validated the intrinsic and contact parameters. The Plackett-Burman design test was employed to identify the significant factors influencing bending force, and the optimal parameter combination for these factors was determined through response surface analysis. When the shear stiffness per unit area was 3.56×109 Pa, the bonded disk scale was 0.93 mm, the normal stiffness per unit area was 9.68×109 Pa, the normal strength was 5.62×107 Pa, the shear strength was 4.27×107 Pa, the discrete element numerical simulation results for three-point bending, radial compression, axial tension, and shear fracture exhibited a maximum failure force error of 3.32%, 4.37%, 4.87% and 3.74% in comparison to the physical experiments. In the cutting experiments, a smaller radial angle between the tool edge and the stem resulted in less damage to the cutting section, which was beneficial for the smoothness of the stubble after harvesting and the subsequent growth of the stem. The discrepancy in cutting force between the physical and numerical simulations was 3.89%, and the F-x (force versus displacement) trend was consistent. The multi-angle experimental validation demonstrated that the discrete element model of CKS is an accurate representation of the real biomechanical properties of CKS. The findings offer valuable insights into the mechanisms underlying crop-machine interactions.

C-H Activation of Pyridines by Boryl Pincer Complexes: Elucidation of Boryl-Directed C-H Oxidative Addition to Ir and Discovery of Transition Metal-Assisted Reductive Elimination from Boron at Rh.

Experimental and theoretical techniques were used to investigate the mechanism of pyridine C-H activation by diarylboryl/bis(phosphine) PBP pincer complexes of Ir. The critical intermediate (PBP)IrCO (4) contains a three-coordinate, Ir-bound boron that retains Lewis acidity in the perpendicular direction. Coordination of pyridine to this boron center in 4 leads to fast insertion of Ir into the 2-CH bond of pyridine, providing a different topology of direction than the conventional directed C-H activation where both the directing group coordination and C-H activation happen at the same metal center. Beyond this critical sequence, the system possesses significant complexity in terms of possible isomers and pathways, which have been thoroughly explored. Kinetic and thermodynamic preferences for the activation of differently substituted pyridines were also investigated. In experimental work, the key intermediate 4 is accessed via elimination of benzene from a phenyl/hydride containing precursor (PBPhP)IrHCO (3). Density functional theory (DFT) investigations of the mechanism of benzene loss from 3 revealed the possibility of a genuinely new type of mechanism, whereby the Ph-H bond is made in a concerted process that is best described as C-H reductive elimination from boron, assisted by the transition metal (TMARE). For Ir, this pathway was predicted to be competitive with the more conventional pathways involving C-H reductive elimination from Ir, but still higher in energy barrier. However, for the Rh analog 3-Rh, TMARE was calculated to be the preferred pathway for benzene loss and this prediction was experimentally corroborated through the study of reaction rates and the kinetic isotope effect.

Short Lysine-Containing Tripeptide as Analgesic Substance: The Possible Mechanism of Ligand–Receptor Binding to the Slow Sodium Channel

A possible molecular mechanism of the ligand–receptor binding of Ac-Lys-Lys-Lys-NH2 (Ac-KKK-NH2) to the NaV1.8 channel that is responsible for nociceptive signal coding in the peripheral nervous system is investigated by a number of experimental and theoretical techniques. Upon Ac-KKK-NH2 application at 100 nM, a significant decrease in the effective charge carried by the NaV1.8 channel activation gating system Zeff is demonstrated in the patch-clamp experiments. A strong Ac-KKK-NH2 analgesic effect at both the spinal and supraspinal levels is detected in vivo in the formalin test. The distances between the positively charged amino groups in the Ac-KKK-NH2 molecule upon binding to the NaV1.8 channel are 11–12 Å, as revealed by the conformational analysis. The blind docking with the NaV1.8 channel has made it possible to locate the Ac-KKK-NH2 binding site on the extracellular side of the voltage-sensing domain VSDI. The Ac-KKK-NH2 amino groups are shown to form ionic bonds with Asp151 and Glu157 and a hydrogen bond with Thr161, which affects the coordinated movement of the voltage sensor up and down, thus modulating the Zeff value. According to the results presented, Ac-KKK-NH2 is a promising candidate for the role of an analgesic medicinal substance that can be applied for pain relief in humans.

Unveiling the secrets of lotus seed longevity: insights into adaptive strategies for extended storage.

Seed longevity is crucial for long-term storage, but prolonged unfavorable conditions can lead to viability loss. This study integrated theoretical and experimental techniques to elucidate the inherent mechanisms underlying the unique ability of lotus seed capacity to maintain stable viability even after enduring years. Transcriptome analysis and microscopy revealed the sturdy structure of the lotus seed pericarp, which predominantly expressed cellulose synthase genes involved in cell wall biogenesis. The cotyledon serves as a nutrient source for seeds during long-term storage. Additionally, the inactivation of chlorophyll degradation pathways may allow for the retention of chlorophyll in the lotus seed plumule, potentially enhancing the environmental adaptability of lotus seedlings. While the reduced abundance of transcripts corresponding to heat shock protein genes could impact protein processing and consequently diminish the vitality of aging lotus seeds. Moreover, an expansion in the number of seed maturation and defense response genes was observed in the lotus genome compared to other 11 species, which might represent an adaptive strategy against long-term adverse storage conditions. Overall, these findings are crucial for understanding the mechanisms underlying lotus seed longevity and may inform future improvements in the extended storage periods of seed crops.

Advancing Optoelectronics with Benzonitriles: Theoretical Understanding, Experimental Realization, and Single‐Molecule White Light Emission

AbstractBenzonitrile derivatives (BDs) are very promising for applications in materials science, photonics, and optoelectronics due to their intriguing electronic and optical properties. This study comprehensively investigates BDs, aiming to uncover their fundamental characteristics and potential applications. Using advanced theoretical techniques like Gaussian software, it is gained unprecedented insights into the photoinduced isomerization phenomenon for all‐optical switching in these compounds. The theoretical framework clarifies molecular transition states and explores a range of properties, providing a comprehensive understanding of BDs. Empirical data on the emission properties of BDs, from fluorescence analyses in liquid solutions to light amplification in solid‐state PMMA thin films, complement the theoretical examinations. Notably, white light emission from a single benzonitrile compound is achieved, showcasing its potential for data transmission through Li‐Fi technology. Finally, the all‐optical switching phenomenon in BDs using 3rd order nonlinear optical effects is experimentally validated. This complementary and comprehensive study advances understanding of BDs and demonstrates their potential for practical applications in emerging photonic and optoelectronic technologies.

Structural modified coumarin fused triazole analogues for Fe3+ metal ion detection: A comprehensive photophysical and fluorescent chemosensor study

This study investigates the potential of coumarin fused thiadiazole (CTR-1 and CTR-2) derivatives as chemosensors for Fe3+ ions detection. The photophysical and solvatochromic properties were examined using spectroscopic and theoretical techniques. Key parameters such as the energy bandgap, chemical hardness, softness, electronegativity, chemical potential, and electronic charge of electrophilic groups were computed using frontier molecular orbitals and compared with experimental results. Molecular electrostatic potential map studies indicated a strong electrophilic site in the molecules. Fluorescence quenching techniques demonstrated that both CTR-1 and CTR-2 exhibit strong, selective, and sensitive interactions with Fe3+ ions, with detection limits of 4.15 μM for CTR-1 and 5.02 μM for CTR-2, and rapid detection times of less than 6.5 min. Intramolecular interactions between the chemosensors and Fe3+ ions were analyzed using density functional theory and spectroscopic techniques, revealing a 1:1 binding ratio. These findings suggest that CTR-1 and CTR-2 are effective chemosensors for detecting Fe3+ions.

Absolute mass spectrometry: A theoretical analytical technique for quantification

Mass spectrometry (MS) is an analytical chemistry technique that requires calibration with standards in order to quantify an analyte in a sample. In many cases the reference standards may be difficult to obtain or unavailable. In fact, when MS is applied to study reactions that yield multiple products, their quantification requires several standards and involves time-consuming and tedious calibration procedures. In other applications like Mass Spectrometry Imaging (MSI), quantification becomes even more tedious when on-tissue standards or mimetic tissue models are required. In this work, we introduce a new measurement technique that we called the Absolute Mass Spectrometry (AbMS), which allows the absolute quantification of an analyte without the need for calibration standards. This technique is based on the “Influence Method” developed in 2015 for nuclear measurements which, when applied to MS, produces this improvement for analyte quantification. Two possible applications of this technique are presented. One is the conceptual development of a mass spectrometer with electronic ionization that does not require calibration to determine the amount of analyte. The other one describes how to apply this technique in a mass spectrometer with other ionization sources like MALDI, DESI, and SIMS, which are used in MSI. AbMS technique uses the conceptual basis of Influence Method and two consecutive measurements to estimate the amount of analyte. Specifically, AbMS can be directly employed as an absolute method for quantification in MSI. Therefore, it constitutes an attractive complement for research applications in pharmaceutical and health sciences.

Synthesis and structural identification of 3,6-Bis(3,5- dimethylpyrazol-1-yl)-1,2,4,5-tetrazine (BDT)

3,6-Bis(3,5-dimethylpyrazol-1-yl)-1,2,4,5-tetrazine (BDT) has emerged as a key intermediate for accessing other promising tetrazine-based high-energy-density compounds. The study presented the step-by-step process of synthesizing BDT while thoroughly analyzing and investigating its crystal structure. The chemical structure of BDT was characterized using spectroscopic techniques such as FT-IR, 1H-NMR, 13C-NMR, and data from PXRD and CCDC 254069. BDT was analyzed using various theoretical techniques, such as 2D fingerprint and Hirshfeld surface. The morphology of the BDT crystals was evaluated through optical microscope images. The results show that BDT crystallizes in a monoclinic space group P21/c, with four molecules per unit cell (Z=4) with the calculated lattice parameters as follows: a = 11.37 Å, b = 16.67 Å, c = 7.74 Å. The proportion of interactions of N...H and H...H of the total Hirshfeld surface area of BDT were 31.9% and 45.7%, respectively. ESP's maximum negative and positive values are -212.9 kJ.mol-1 and 65.6 kJ.mol-1, respectively.

Optical transmission, dielectric constants, and gamma shielding performance of Se95-xIn5Prx metallic alloys: Radiation protection applications

Due to the high density of metals, metallic materials such as alloys have the potential to have high gamma-radiation-absorbing abilities. This would make them relevant in a radiation environment as either a shielding material or a radiation sensor. These would, however, depend on their gamma radiation responses. Research aimed at evaluating the gamma response parameters of different glassy alloys that have technical applications is scarce compared to the traditional characterization with respect to optical features, structural evolution, and mechanical strength. In this paper, the optical and radiation shielding abilities of Se95-xIn5Prx (x = 2, 4, and 6) glassy alloys were investigated using standard theoretical techniques. The mass attenuation coefficient (MAC) of the materials was computed using the XCOM software. Reflection loss values are 0.20218, 0.22634, and 0.22908 for SIPr1, SIPr2, and SIPr3, respectively, while the optical transmittance values are 0.66365, 0.63087, and 0.62723 in the same order. The metallization criterion declined when Pr increased in the alloy. The values of MAC ranged from 0.0317 cm2/g to 98.2099 cm2/g for SIPr1, 0.0319 cm2/g to 97.3798 cm2/g for SIPr2 and 0.0322 cm2/g to 96.5372 cm2/g for SIPr3. The Higher attenuation and absorption coefficients were found for Pr-rich alloys. Pr therefore increase the ability of the SIPr-alloy to attenuate and absorb photon energy. The optical parameters of the investigated alloys could be used to identify their optical applications. SIPr3 had comparatively higher shielding ability compared to some known shields and could thus be used for radiation control and protection purposes.

Analisis Peran Humas Melalui Guest Experience Leader Dalam Meningkatkan Pelayanan Di Mcdonald’s Bengkulu

This research aims to find out the role of Public Relations through the Guest Experience Leader (GEL) in improving service at McDonald's Bengkulu. This research is qualitative research. To analyze the researcher used Ruslan Efendi's theoretical analysis technique (Communicator, Relationship, Back Up Management and Corporate Image). The research results show that there is a role for GEL in improving service at McDonald's Bengkulu. GEL plays a role as a communicator who can ensure customers get a good impression during their visit to McDonald's Bengkulu, GEL's role as a Relationship Shop GEL must be able to build positive relationships through a good approach to customers and mutual benefits, GEL's role as Back Up Management GEL must be able to inspire, motivating and providing team training if necessary so they can work better, as Corporate Image GEL plays a role in maintaining the good name of the company as best as possible

Double Tseng’s Algorithm with Inertial Terms for Inclusion Problems and Applications in Image Deblurring

In this research paper, we present a novel theoretical technique, referred to as the double Tseng’s algorithm with inertial terms, for finding a common solution to two monotone inclusion problems. Developing the double Tseng’s algorithm in this manner not only comprehensively expands theoretical knowledge in this field but also provides advantages in terms of step-size parameters, which are beneficial for tuning applications and positively impact the numerical results. This new technique can be effectively applied to solve the problem of image deblurring and offers numerical advantages compared to some previously related results. By utilizing certain properties of a Lipschitz monotone operator and a maximally monotone operator, along with the identity associated with the convexity of the quadratic norm in the framework of Hilbert spaces, and by imposing some constraints on the scalar control conditions, we can achieve weak convergence to a common zero point of the sum of two monotone operators. To demonstrate the benefits and advantages of this newly proposed algorithm, we performed numerical experiments to measure the improvement in the signal–to–noise ratio (ISNR) and the structural similarity index measure (SSIM). The results of both numerical experiments (ISNR and SSIM) demonstrate that our new algorithm is more efficient and has a significant advantage over the relevant preceding algorithms.

High‐performance vanadium oxide‐based aqueous zinc batteries: Organic molecule modification, challenges, and future prospects

AbstractAqueous Zn‐vanadium batteries have been attracting significant interest due to the high theoretical capacity, diverse crystalline structures, and cost‐effectiveness of vanadium oxide cathodes. Despite these advantages, challenges such as low redox potential, sluggish reaction kinetics, and vanadium dissolution lead to inferior energy density and unsatisfactory lifespan of vanadium oxide cathodes. Addressing these issues, given the abundant redox groups and flexible structures in organic compounds, this study comprehensively reviews the latest developments of organic‐modified vanadium‐based oxide strategies, especially organic interfacial modification, and pre‐intercalation. The review presents detailed analyses of the energy storage mechanism and multiple electron transfer reactions that contribute to enhanced battery performance, including boosted redox kinetics, higher energy density, and broadened lifespan. Furthermore, the review emphasizes the necessity of in situ characterization and theoretical calculation techniques for the further investigation of appropriate organic “guest” materials and matched redox couples in the organic‐vanadium oxide hybrids with muti‐energy storage mechanisms. The review also highlights strategies for Zn anode protection and electrolyte solvation regulation, which are critical for developing advanced Zn‐vanadium battery systems suitable for large‐scale energy storage applications.

Crystal Growth, Structural Elucidation, Chemical Reactivity, Topology Exploration (ELF, LOL, AIM, RDG) and NLO Activity of 4-Cumyl Phenol Crystal Using Quantum Chemical Calculation

The main objective of the study is to analyze the structural behavior and NLO activity of 4-cumyl phenol by experimental and theoretical spectroscopic techniques. Its computational result is compared with the cumene compound. The optimized structural parameters of the title compound were computationally obtained at the B3LYP/6-31G (d, p) basis set. The fundamental modes of vibration were examined by experimental FT-IR, and FT-Raman techniques. The distribution of the vibrational bands was carried out with the help of normal coordinate analysis (NCA). The resulting harmonic wavenumbers were scaled by using NCA method. Topological analysis, such as Electron localization function (ELF), natural bond orbital analysis (NBO), reduced density gradient (RDG) and atoms in molecule (AIM) analysis, molecular electrostatic potential (MEP) have been used to evaluate the intermolecular interaction, especially the hydrogen bonds. UV-visible spectrum of the compound was recorded in the region 200–900 nm and the electronic properties and HOMO-LUMO energies were calculated by time dependent density functional theory approach.

RFOR-DQHFEM: Hybrid relaxed first-Order reliability and differential quadrature hierarchical finite element method for multi-physics reliability analysis of conical shells

In this current work, a hybrid reliability analysis and theoretical frequency technique are suggested for the reliability response of conical shells. Two levels of analyses are proposed as the main loop of the reliability method for finding the failure probability and the second level applied in the main loop for giving the performance function of frequency applied in conical shell structures with multi-physics vibration analysis. A dynamical adjusting procedure is proposed for computing the relaxed factor using the enough descent condition inside the reliability method. The superior convergence rate is considered for selecting the relaxed factor of the proposed first-order reliability method named RFORM. An elastic-electro-mechanical model based on the Higher-Order Shear Deformation Theory (HSDT) is extended for frequency analysis of conical shells. The innovative numerical procedure named Differential Quadrature Hierarchical Finite Element Method (DQHFEM) as a robust framework for giving the vibration behavior of studied mechanical structures is applied for solving motion equations. The developing DQHFEM and RFORM are applied for the laminated, nanocomposite, and piezoelectric conical shell structures with multi-source uncertainties. Increasing the volume percentage of nanoparticles from 0% to 10% significantly enhances the reliability index, with carbon nanoparticles showing a 132% increase, silica nanoparticles showing a 97% increase, and other nanoparticles showing an approximate 40% increase. Also, as moisture content increases from 0% to 30%, the reliability index for a thickness-to-large-radius ratio of 0.2 drops by about five times. Excessive moisture levels (above 20%) result in a negative reliability index, indicating a hazardous condition.

Theoretical and Numerical Analysis of Singular Perturbation Problems in Ordinary Differential Equations

Background: Singular perturbation troubles in everyday differential equations (ODEs) involve a small parameter ϵ that causes answers to show off behaviors on multiple scales, main to massive challenges in both theoretical and numerical evaluation. Objective: This paper at targets to comprehensively look at the theoretical and numerical techniques for reading and solving singular perturbation troubles in ODEs. Focus is located on know-how the behaviors precipitated through the small parameter and developing strong numerical techniques to accurately seize these behaviors. Methods: Theoretical approaches employed include matched asymptotic expansions and multiple scale analysis. These methods decompose the solution domain into regions with different scales, constructing and matching approximate solutions to ensure smooth transitions. Numerical techniques such as finite difference methods, finite element methods, and spectral methods are utilized, with particular emphasis on adaptive mesh refinement to handle boundary layers effectively Results: Theoretical evaluation demonstrates the effectiveness of matched asymptotic expansions and multiple scale analysis in presenting correct approximations and easy transitions among one of a kind solution regions. Numerical techniques showed high accuracy and performance, particularly whilst blended with adaptive techniques. Finite distinction strategies with non-uniform grids and adaptive mesh refinement, finite detail techniques with adaptive mesh strategies, and spectral strategies with special handling of boundary layers all proved successful. Stability and convergence analyses showed the reliability of those techniques. Conclusions: This comprehensive evaluation highlights the strengths and applicability of each theoretical and numerical strategies in tackling singular perturbation issues in ODEs. The mixed use of these techniques lets in for correct and green answers, presenting treasured equipment for applications in diverse clinical and engineering fields.