Nonlinear Form Research Articles

Fabrication of a poly(m‑aminophenol)/3-aminopropyl triethoxysilane/graphene oxide ternary nanocomposite for removal of Cu(II) from aqueous solution

Three composites based on Poly (meta-aminophenol) (PmAP), (3-aminopropyl) triethoxysilane (APTES) and graphene oxide (GO) were synthesized with initial GO dispersion of 3.3, 6.6, and 9.9 mg/mL. First, in-situ polymerization of meta-aminophenol monomer on the surface of graphene oxide (GO) was carried out. Then, the hydroxyl groups of both the GO and the polymer were targeted using (3-aminopropyl) triethoxysilane (APTES) to stop the polymer solubility, increase adsorption sites, and bind the two components. The obtained three composites were applied for efficient removal of Cu(II) from polluted water. PmAP/APTES/GO(6.6) composite was the best one for the uptake of Cu(II) with a maximum adsorption capacity of 324.54 mg/g at 40 °C and pH 7 according to Langmuir. PmAP/APTES/GO(6.6) hybrid composite was characterized by different techniques. The adsorption of Cu(II) on this composite was optimized under various experimental conditions. Furthermore, the isotherm data of the uptake of Cu(II) on PmAP/APTES/GO(6.6) were found to agree with the Freundlich and Langmuir model’s linear and nonlinear forms. Chemosorption was suggested by the Dubinin-Radushkevich (D-R) isotherm model as the calculated mean sorption energy exceeds 16 kJ/mol. The thermodynamic analysis of the adsorption process reflects an endothermic, spontaneous process that leads to more disorder at the solid-liquid interface. The chemical interactions between Cu(II) versus oxygen and nitrogen of the functional groups on the surface were demonstrated by X-ray Electron Spectroscopy (XPS). Five cycles of adsorption and desorption of Cu(II) from the prepared composite were carried out with a loss of only 6.3% of its adsorption capacity.

Optimizing Ni (II) adsorption on Carica papaya fiber through experimental and advanced neural network prediction

ABSTRACT Detoxification of Ni (II) ion from the synthetic wastewater by Carica papaya fiber (CPF) has been attempted in batch method. FTIR analysis detected prominent functional groups that helped in binding Ni+2. The most efficient sorption was observed at pH 6 and with the increase in the ratio of biomass to Ni (II) ion strength, the removal efficiency improved. 2.5 g/L CPF powder removed 90.83% Ni (II) from 10 mg/L solution at a contact time of 120 min and at 30°C. Both linear and non-linear forms of Langmuir, Freundlich, and Temkin isotherms were attempted. Langmuir isotherm in both types matched well (R2 = 0.999 and 0.994) and the maximum adsorption capacity was 21.84 mg/g. The spontaneity of the reaction was confirmed by ΔG° values (−12.06 to −13.71 kJ/mol) and was more favorable at higher temperatures. The reaction was endothermic (ΔH° 83.03 kJ/mol) and random (ΔS° 83.07 J/mol/K). The desorption study proposed active reprocessing of CPF up to three cycles. The results were scaled up for use by industries. This study reflects sustainable eco-innovation and a cost-effective approach for locally based small-scale industries for environmental preservation. The feed-forward back propagation neural network technique was applied for efficient prediction of the Ni(II) removal efficiency.

Comment on: ‘Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity’

Abstract The present comment points out all the shortcomings that occurred during the modulational instability analysis in the recent work published by Chahlaoui et al [Phys. Scr. 99 075226] and then methodically presents the investigation of the modulational instability of the steady-state solution of their model equation in the special case of time derivatives in the ordinary sense.

Performance, isotherm, kinetics and mechanism of simultaneous removal of Cr(VI), Cu(II) and F ions by CeO2–MgO binary oxide nanomaterials

This study focuses on the synthesis of a novel Cerium-Magnesium (CeO2-MgO) binary oxide nanomaterials by a simple co-precipitation process and used to remove harmful pollutants such as Cr(VI), Cu(II), and F. The morphology, phase, crystallite size, thermal stability, functional groups, surface area, and porosity of the synthesized nanomaterial were determined by using XRD, SEM, FTIR, TGA/DTA, and BET studies. The prepared nanomaterials showed adsorption selectivity of Cu(II) ≈ F−> Cr(VI) with a high adsorption capacity of 84.3 − 133.3 mg/g for Cu(II), Cr(VI), and F−. The distribution coefficient (Kd) for F− and Cu(II) was found to be in the range of 104 mL/g which was adequate. The adsorption isotherms for Cr(VI), Cu(II), and F− followed the Freundlich isotherm model and the pseudo-second-order kinetic model in linear and nonlinear forms, indicated multilayer adsorption. Maximum removal of Cr(VI), Cu(II), and F− ions was found to be 92.84%, 98.88%, and 95%, respectively, for a high initial concentration of 50 mg/l by 2 g/l dosages of prepared CeO2-MgO binary oxide nanomaterials employed as an adsorbent in this study. The results showed that novel CeO2-MgO binary oxide nanomaterials are promising adsorbent for removing hazardous inorganic contaminants from the water due to their adsorption capability and chemical stability.

Solving fuzzy system of Fredholm integro-differential equations of the second kind by using homotopy analysis method

<p>Certain phenomena with uncertain properties that take the shape of intricate mathematical modeling are known to have fuzzy system integro-differential equations (FSIDEs). The methods used to roughly solve FSIDEs seek to provide open-form solutions that are regarded as solutions for polynomial series. However, for many FSIDEs, the polynomial series solutions are not easily derived, especially in nonlinear forms. Meanwhile, some existing approximate techniques cannot guarantee convergence of the series solution. Nevertheless, to solve second-kind fuzzy Fredholm integro-differential equations (FFSIDEs), there exist perturbation techniques based on the standard Homotopy Analysis Method (HAM) that have the ability to control and rate solution convergence. Therefore, this study focused on modifying new approximate techniques, fuzzy Fredholm HAM (HAMFF), for solving second-kind FFSIDEs subject to initial and boundary value problems. In the theoretical framework modification, the establishment of the series solution convergence was done based on combining some fuzzy sets theory concepts and convergence-control parameters from standard HAM. The HAMFF was not only able to solve linear systems but also difficult nonlinear systems with proper accuracy. The demonstration of the modified technique's performance was shown in comparison to other methods, where HAMFF was individually superior in terms of accuracy for solving linear and nonlinear test problems of FFSIDEs.</p>

The streaming model for the three-point correlation function and its connection to standard perturbation theory

Redshift-space distortions (RSDs) present a significant challenge in building models for the three-point correlation function (3PCF). We compare two possible lines of attack: the streaming model and standard perturbation theory (SPT). The two approaches differ in their treatment of the non-linear mapping from real to redshift space: SPT expands this mapping perturbatively, while the streaming model retains its non-linear form but relies on simplifying assumptions about the probability density function (PDF) of line-of-sight velocity differences between pairs or triplets of tracers. To assess the quality of the predictions and the validity of the assumptions of these models, we measure the monopole of the matter 3PCF and the first two moments of the pair- and triplewise velocity PDF from a suite of N-body simulations. We also evaluate the large-scale limit of the streaming model and determine under which conditions it aligns to SPT. On scales larger than 10 h -1 Mpc, we find that the streaming model for the 3PCF monopole is dominated by the first two velocity moments, making the exact shape of the PDF irrelevant. This model can match the accuracy of a Stage-IV galaxy survey, if the velocity moments are measured directly from the simulations. However, replacing the measurements with perturbative expressions to leading order generates large errors already on scales of 60–70 h -1 Mpc. This is the primary drawback of the streaming model. On the other hand, the SPT model for the 3PCF cannot account for the significant velocity dispersion that is present at all scales, and consequently provides predictions with limited accuracy. We demonstrate that this issue can be approximately addressed by isolating the large-scale limit of the dispersion, which leads to typical Fingers-of-God damping functions. Overall, the SPT model with a damping function provides the best compromise in terms of accuracy and computing time.

Retrieval of optical soliton patterns for time-fractional nonlinear Schrödinger equation integrating Kudryashov’s law of refractive index with dual form of nonlocal nonlinearity

The Schrödinger model, which describes the mobility of a single wave in an optical fiber, is crucial to communication systems. By using the quadrupled-power law and the dual form of nonlocal nonlinearity, this study concentrates on establishing optical solitary solutions for Kudryashov’s law of nonlinear refractive index. The unified Riccati equation method and the new extended auxiliary equation approach are used to derive numerous ranges of Jacobi elliptic, trigonometric, hyperbolic, and dual-wave solitary solution patterns. These findings will eventually lead to the discovery of bright, singular, kink, dark–bright, and singular-periodic soliton solutions, whose 3D, 2D, and density depictions are provided with the help of mathematical software, Mathematica. Moreover, we provide 2D graphical representations for different values of the fractional and time parameters to show how these novel optical solutions behave. Different features of these solitons arise in many physical contexts such as nonlinear optics, fluid dynamics, and laser physics. The derived results show that the proposed framework presents a rich and varied spectrum of wave phenomena and that the methods for solving nonlinear partial differential equations arising in mathematical physics are effective, useful, and dependable for further exploration in the vast field of optical and mathematical physics.

Periodic solutions for nonlinear deformation waves based on new Boussinesq-type equation within strain gradient elasticity and reductive perturbation technique

In this article, we present a detailed variational formulation to study and analyze the geometrically nonlinear behavior of elastic materials at small scales, where the classical theory of elasticity becomes inadequate. The presented technique is based on the variation of the internal energy of the elastic material and the virtual work of the external forces for the dynamical system in the framework of the first strain gradient theory of elasticity. This approach simultaneously determines the equilibrium equations in nonlinear form and the complete set of nonlinear boundary conditions. A new hyperbolic Boussinesq-type equation is derived based on the presented model in one-dimensional space and time to examine the impact of geometrical nonlinearity on wave propagation in an isotropic elastic material. The exponential reductive perturbation technique (ERPT) is employed to obtain an approximate and hierarchical solution for the new nonlinear partial differential equation (NPDE). The obtained results are plotted and analyzed, moreover, the results reveal that geometric nonlinearity significantly influences wave propagation in polycrystalline silicon.

Fazil Say’s «Shahmeran»: the metamodern paradigm of an Eastern myth in the music of a Turkish composer

The relevance of the study The relevance of this study stems from the insufficient scholarly exploration of Turkish academic music, particularly its contemporary manifestations, which integrate Eastern traditions with Western compositional techniques. The musical legacy of Fazıl Say, one of the most prominent representatives of 21st-century Turkish music, showcases a unique synthesis of national identity and global musical trends, reflecting Turkey's broader cultural context as a crossroads of civilizations. The main objective of the study is to examine the stylistic features of Fazıl Say’s symphonic poem «Shahmeran» as a realization of an Eastern myth in the context of global cultural trends of modernity, particularly metamodernism as a contemporary artistic paradigm. The methodologу includes the following approaches: the biographical method (to analyze the impact of the composer’s sociocultural environment on his creativity), the genre-stylistic method (to determine the genre characteristics of the composition and its connection with the traditions of Western instrumental music), the intonational-dramaturgical method (to explore the construction of the musical narrative), as well as cultural-historical and theoretical generalization methods (to define the place of the composition within the contemporary musical discourse). Results and conclusions. The symphonic poem «Shahmeran» by Fazıl Say is analyzed through the lens of the metamodern artistic paradigm as a multilayered composition that synthesizes elements of Eastern mythology, Turkish musical traditions, and contemporary compositional techniques. The central role of the myth of Shahmeran is identified as a tool for reinterpreting cultural heritage and generating new meanings within a globalized world. Emphasis is placed on the capacity of myth to function as a dynamic cultural code, adapting to modern challenges while maintaining its connection to archetypal structures, aligning with the key characteristics of metamodernism. The preconditions for the development of Turkish academic music are outlined, and the main factors that contributed to its rapid growth in the second half of the 20th century are determined. Fazıl Say’s creative method is examined, highlighting the influence of his active career as a pianist and his overseas education. Say’s compositional style is characterized as synthetic, seamlessly integrating Western compositional techniques with the distinctiveness of national musical traditions. The genre status of «Shahmeran» is identified as ambiguous, blending elements of a single-movement symphonic poem and a cello concerto under the framework of a programmatic title. The compositional features of the work are analyzed, including its fragmentary structure and non-linear form. The role of modalintonational and rhythmic specificities in shaping its distinct Eastern modality is emphasized. The significance of oriental instruments (kudüm, waterphone) and unique performance techniques (aleatory glissandi, the use of strings as percussion, and cello timbres approximating traditional instruments) is revealed, highlighting their contribution to the creation of a distinctive sonic landscape. It is demonstrated that the composer employs methods of «new simplicity», presented through the prism of oriental intonations.

A novel approach to construct optical solitons solutions of complex Ginzburg–Landau equation with five distinct forms of nonlinearities

This paper presents a recently introduced innovative approach for analyzing closed-form solutions of nonlinear partial differential equations. While various methods exist for deriving closed-form solutions to many nonlinear evolution equations, additional solutions are still needed to study the various dynamics of physical systems governed by nonlinear partial differential equations. Initially, we give general procedure of the Cham technique for solving nonlinear partial differential equations that yields eight kinds of solutions. This technique is applied to the complex Ginzburg–Landau equation, incorporating five different types of nonlinearities: Kerr law, cubic–quintic law, polynomial nonlinearity, quadratic–cubic law, and parabolic-nonlocal law. With the aid of the proposed strategy, we can obtain a wide array of optical solitons, including bright, breather, kink, periodic, and cusp-shaped solitons, under specific parameter conditions.

Super-resolution imaging of fast morphological dynamics of neurons in behaving animals.

Neurons are best studied in their native states in which their functional and morphological dynamics support animals' natural behaviors. Super-resolution microscopy can potentially reveal these dynamics in higher details but has been challenging in behaving animals due to severe motion artifacts. Here we report multiplexed, line-scanning, structured illumination microscopy, which can tolerate motion of up to 50 μm s-1 while achieving 150-nm and 100-nm lateral resolutions in its linear and nonlinear forms, respectively. We continuously imaged the dynamics of spinules in dendritic spines and axonal boutons volumetrically over thousands of frames and tens of minutes in head-fixed mouse brains during sleep-wake cycles. Super-resolution imaging of axonal boutons revealed spinule dynamics on a scale of seconds. Simultaneous two-color imaging further enabled analyses of the spatial distributions of diverse PSD-95 clusters and opened up possibilities to study their correlations with the structural dynamics of dendrites in the brains of head-fixed awake mice.

Complex quadrupled-power-law nonlinearity form of Radhakrishnan-Kundu-Lakshmanan equation: novel optical soliton analysis

Abstract In this manuscript, the novel optical wave solutions to the newly developed Radhakrishnan-Kundu-Lakshmanan (NRKL) model are studied. Two analytical methods namely, the new extended generalized Kudryashov and the extended modified auxiliary equation mapping methods are used to obtain these novel solutions. Furthermore, the dynamics of the obtained solutions are analyzed thoroughly with the help of different graphical forms such as three-dimensional (3-D), contour, and two-dimensional (2-D) figures. To get the graphical view of the derived solutions, numerical values for the unknown parameters are assigned while balancing the nonlinearity with dispersion. Moreover, the characteristics analysis of the obtained solutions depict anti-kink, periodic, bright, dark-shaped periodic, anti-bell shape, bright-shaped periodic, and dark waveforms Furthermore, the practical implications of these waveforms in optical fiber transmission are profound, with the potential to influence nonlinear signal processing and advanced optical network design.

Entropy optimization in a radiative and chemically reactive EMHD flow of a nanofluid coexisting Ohmic dissipation and multiple slips

PurposeThis research focuses on the controlling irreversibilities in a radiative, chemically reactive electromagnetohydrodynamics (EMHD) flow of a nanofluid toward a stagnation point. Key considerations include the presence of Ohmic dissipation, linear thermal radiation, second-order chemical reaction with the multiple slips. With these factors, this study aims to provide insights for practical applications where thermal management and energy efficiency are paramount.Design/methodology/approachLie group transformation is used to revert the leading partial differential equations into nonlinear ODE form. Hence, the solutions are attained analytically through differential transformation method-Padé and numerically using the Runge–Kutta–Fehlberg method with shooting procedure, to ensure the precise and reliable determination of the solution. This dual approach highlights the robustness and versatility of the methods.FindingsThe system’s entropy generation is enhanced by incrementing the magnetic field parameter (M), while the electric field (E) and velocity slip parameters (ξ) control its growth. Mass transportation irreversibility and the Bejan number (Be) are significantly increased by the chemical reaction rate (Cr). In addition, there is a boost in the rate of heat transportation by 3.66% while 0.05⩽ξ⩽0.2; meanwhile for 0.2⩽ξ⩽1.1, the rate of mass transportation gets enhanced by 12.87%.Originality/valueThis paper presents a novel approach to analyzing the entropy optimization in a radiative, chemically reactive EMHD nanofluid flow near a stagnation point. Moreover, this research represents a significant advancement in the application of analytical techniques, complemented by numerical approaches to study boundary layer equations.

Removal, mechanistic and kinetic studies of Cr(VI), Cd(II), and Pb(II) cations using Fe3O4 functionalized Schiff base chelating ligands.

The Schiff base chelating ligands; (E)-2-(3,3-dimethoxy-2-oxa-7,10-diaza-3-silaundec-10-en-11-yl)phenol (L1), (E)-N-(2-((pyridine-2ylmethylene)amino)ethyl)-3-(trimethoxysilyl)propan-1-amine (L2) and (E)-N-(2-((thiophen-2-ylmethylene)amino)ethyl)-3-(trimethoxysilyl)propan-1-amine (L3) were immobilized on Fe3O4 magnetic nanoparticles (MNPs) and utilized in the extraction of Cr(VI), Cd(II) and Pb(II) metal cations from aqueous solutions. The compounds synthesized, denoted as L1@ Fe3O4, L2@Fe3O4, and L3@Fe3O4, were characterized using FT-IR spectroscopy, TEM-SEM, VSM, and BET/BHJ techniques for analysis of functional groups, surface morphology, magnetic properties, and degree of porosity of the adsorbents, respectively. BET/BHJ technique confirmed the mesoporous nature of the compounds as their pore diameters ranged between 15 and 17nm. The initial optimization conditions of pH, adsorbent dosage, initial metal concentration, and contact time on adsorption were studied using L1@ Fe3O4. The optimum efficiencies recorded were 68% and 46% for Cr(VI) and Cd(II), respectively, obtained at pH 3, and a metal concentration of 20ppm while an efficiency of 99% was recorded for Pb(II) cations at pH 7 and a metal concentration of 100ppm. Compounds L2@Fe3O4 and L3@ Fe3O4 were also used in the extraction of metal cations from aqueous solution and gave efficiencies of 22%, 56%, and 78% for L2@ Fe3O4 and 19%, 90%, and 59% using L3@ Fe3O4 for Cr(VI), Cd(II), and Pb(II), respectively. The maximum adsorption capacities of L1@ Fe3O4 for Cr(VI), Cd(II), and Pb(II) cations were obtained from the Langmuir isotherm as 32.84, 41.77, and 450.45mg/g, respectively. The experimental data was analyzed using pseudo-first-order, pseudo-second-order, intraparticle diffusion, and Elovich kinetic models. Both linear and non-linear forms of kinetic isotherms; Langmuir, Freundlich, Redlich-Peterson, and Temkin were utilized to investigate the nature of adsorption on L1@Fe3O4. The mechanistic studies deduced that the Langmuir isotherm and pseudo-second-order kinetic model better described the adsorption process both yielding high correlation coefficient values (R2 > 0.98).

Detecting damage on engine mounts using hilbert-huang transform vibration analysis

Engine mounts, which are usually composed of elastomeric materials like rubber that can absorb excessive vibrations, are built to withstand vibration sources from engines. Engine mounts may eventually degrade from extended use. When rubber products age or sustain damage, they may grow harder and crack. Engine mounts need to be maintained regularly to ensure optimal performance. Visual examinations and the detection of excessive vibrations can be used to accomplish this. Vibration sensors are mounted on the engine mount in axial, vertical, and horizontal orientations using a vibration detection technique utilizing the Hilbert-Huang Transform (HHT) methodology. Matlab is used to examine the data that is gathered at three distinct rotational speeds: 750 rpm, 2000 rpm, and 3000 rpm. The HHT approach combines two key components: the Hilbert Transform, which analyzes the time-frequency signal of the first decomposition until only residuals remain, and Empirical Mode Decomposition (EMD), which breaks down the signal into Intrinsic Mode Functions (IMFs). According to test data, a damaged mount had an amplitude of 0.00005212 m/s² and a frequency of 8 Hz. On the other hand, in typical circumstances, the highest frequency was 7 Hz with the same amplitude. Five frequency increases were made in the damaged mount throughout this operation. In the damaged mount, the Hilbert Transform showed a frequency of 2124 Hz with an amplitude of 0.007594 m/s², indicating a significant resonance. This illustrates how the Hilbert-Huang Transform's capacity to handle non-stationary and nonlinear signal forms allows it to detect damage in components efficiently

Cnicus Benedictus roots as a low-cost adsorbent for methylene blue dye removal from aqueous solution: kinetic and equilibrium studies

In this research, Cnicus Benedictus roots powder (CBRP) was utilized as a low-cost adsorbent for removing methylene blue (MB) as a cationic dye from aqueous solution. Characterization of CBRP by Boehm titration method, Fourier transform infrared (FT-IR) and environmental scanning electron microscope (ESEM) indicates the presence of different types of functional groups and a very rough surface filled with cavities, suitable for MB adsorption. The effect of various parameters such as CBRP dose (0.2–2 g/1L), contact time (0–180 min), initial pH (1–10), coexisting anions/cations and the initial MB concentration (100–500 mg L−1) on MB adsorption were examined, the results showed that MB uptake depends on the pH of solution, the electrostatic attraction forces, the formation of H-bonds and π–π interactions are the possible CBRP–MB interaction mechanisms. Adsorption kinetic of MB on CBRP was best represented by linear and non-linear form of the pseudo-second order model. The experimental equilibrium data were best described by linear and non-linear forms of Langmuir and Redlich–Peterson isotherm models (R 2> 0.98) with a maximum MB adsorption capacity of 85.47 mg g−1 at 25 °C, the adsorption efficiency R (%) was greater than 90 (%). MB adsorption-desorption experiments showed the reusability of CBRP for five consecutive cycles. According to the obtained results, CBRP can be used as a natural adsorbent material for cationic dyes removal from aqueous solution.

Approximate Solutions of the Boussinesq Equation for Horizontal Unconfined Aquifers During Pure Drainage Phase

In this work, conceptual approximations of the Boussinesq equation were introduced and analyzed, resulting into a very accurate and well-applicable model for horizontal unconfined aquifers during the pure drainage phase, without any recharge and zero-inflow conditions. The model was constructed by employing a variety of methods that included wave solution, variable separation, and series expansion, and its analysis and performance against the Boussinesq equation, at early and later times, providing fruitful insights enlightening the main mechanisms and physical characteristics of the drainage phase. The modeled non-linear forms were finally linearized, concluding with explicit analytical expressions that accurately incorporated most of the basic characteristics regarding the evolution of the water table and the outflow from the exact Boussinesq equation under different initial conditions. The endeavors of this work can be utilized for theoretical and modeling purposes related to this problem.

Reduction of 2,4-dichlorophenoxy acetic acid herbicide toxicity from aqueous media by adsorption: Experimental and theoretical study using DFT method

This work studies the possibility of reducing the toxicity of water laden with 2,4-dichlorophenoxy acetic acid (2,4-D) using a commercial activated carbon (AC). Multiple structural and textural characterization techniques including Fourier transform infrared spectroscopy (FTIR), scanning electron microscopy (SEM/EDS), point of zero charge (pHzpc) determination, Boehm titration, iodine number, Methylene blue index, and BET analysis measurements were performed on the commercial AC. Various parameters affecting adsorption were studied and well-known isotherms models in their linear and non-linear forms were applied for fitting equilibrium data. The kinetic data was analyzed using intraparticle diffusion, pseudo-first and second-order kinetic models. Density functional modeling (DFT) of 2,4-dichlorophenoxyacetic acid was used to investigate the origin of the reactivity. The AC surface area was found to be 1017 m2⋅g−1. A contact time of 30 min, an AC dose of 4 g⋅L−1 and a pH of 6.2 resulted in a maximum adsorption capacity of 235.55 ± 6.43 mg⋅g−1. The thermodynamic study revealed that the adsorption process was physical, spontaneous and endothermic. The adsorption process was governed by second-order kinetics and controlled by intraparticle diffusion.

The Estimating Parameter and Number of Knots for Nonparametric Regression Methods in Modelling Time Series Data

This research aims to explore and compare several nonparametric regression techniques, including smoothing splines, natural cubic splines, B-splines, and penalized spline methods. The focus is on estimating parameters and determining the optimal number of knots to forecast cyclic and nonlinear patterns, applying these methods to simulated and real-world datasets, such as Thailand’s coal import data. Cross-validation techniques are used to control and specify the number of knots, ensuring the curve fits the data points accurately. The study applies nonparametric regression to forecast time series data with cyclic patterns and nonlinear forms in the dependent variable, treating the independent variable as sequential data. Simulated data featuring cyclical patterns resembling economic cycles and nonlinear data with complex equations to capture variable interactions are used for experimentation. These simulations include variations in standard deviations and sample sizes. The evaluation criterion for the simulated data is the minimum average mean square error (MSE), which indicates the most efficient parameter estimation. For the real data, monthly coal import data from Thailand is used to estimate the parameters of the nonparametric regression model, with the MSE as the evaluation metric. The performance of these techniques is also assessed in forecasting future values, where the mean absolute percentage error (MAPE) is calculated. Among the methods, the natural cubic spline consistently yields the lowest average mean square error across all standard deviations and sample sizes in the simulated data. While the natural cubic spline excels in parameter estimation, B-splines show strong performance in forecasting future values.

On the hydrothermal features of MHD nanofluid flow over a differently shaped thin needle influenced by Hall current with nonlinear thermal radiation

PurposeThe investigation has appraised the problem of an incompressible laminar steady magnetohydrodynamic (MHD) nanofluid stream over three distinct slendering thin isothermal needles of paraboloid, cylindrical and cone shapes. Water as a base liquid is assumed in this flow model. The influences of the Hall current and variable sorts of magnetic forces have enriched our investigation. Energy and concentration expressions consist of thermophoresis and Brownian migration phenomena. The analysis of thermal and mass slips of the presumed model has also been performed.Design/methodology/approachA relevant transformation is implemented for the alteration of the leading partial differential equations (PDEs) to the equations with nonlinear ordinary form. Due to the strong nonlinearity of the foremost equations, the problem is solved numerically by embedding the well-known RK-4 shooting practice. The software MAPLE 2017 has been exploited in reckoning the entire computation. To enunciate the investigated upshots, some graphical diagrams have been regarded here. According to technological interest, we measured the engineering quantities like the Sherwood number, the coefficient of drag friction and the Nusselt number in tabular customs.FindingsThe obtained consequences support that Hall current intensifies fluid movement when the needle is in a cone shape, while the superior velocity is noticed for cylindrical-shaped needles. The transference of heat responds inversely along with the growths of thermal and mass slip factors.Originality/valueNo work has been performed on the flow model of radiated nanofluid over a variable-shaped thin needle under Hall current, the variable magnetic field and different slip factors.