Fraction Decomposition Research Articles

Pyrolysis and combustion kinetics of thermally treated globe artichoke leaves

There are no data in the literature on the energy valorization of globe artichoke (GA) leaves. Thus, an extensive lab-scale experimental torrefaction, carbonization, and coking study was performed. Operative temperatures of 200 °C–1000 °C with 30–120 min residence times were considered. Nonisothermal thermogravimetric analysis was performed at 10, 20, and 40 °C/min heating rates. Pyrolysis and combustion kinetics of raw and thermally treated samples using the Ozawa–Flynn–Wall (OFW) isoconversional method were investigated. All samples exhibited three-stage thermal decomposition behavior: first, moisture and light volatiles evolution common under air and nitrogen; second, carbohydrate fraction decomposition under nitrogen and volatiles combustion; third, lignin decomposition under nitrogen and char combustion. Average activation energy ranges are 54–223 kJ/mol and 223–503 kJ/mol for combustion and pyrolysis, respectively. Some irregular trends appeared when carbonization exceeded 500 °C due to the occurrence of secondary reactions between residual char and evolved gas and the decomposition of some ash components at temperatures reaching 1000 °C. Negative temperature kinetic coefficient appeared at 800–1000 °C as the temperature approached ash softening/fusing temperatures. SEM images indicated amorphous nature and increased porosity from 600 °C, which explains the pyrolysis and oxidation behavior observed in biochar samples produced over this range. Samples pyrolyzed for 30 min showed better elemental and energy results compared to longer times.

Etkili yöntem kullanan ağ erişim kontrolü modeli için kesirli yaklaşım

In this paper, we find the solution for the system of nonlinear ordinary differential equations having fractional-order arising in network access control using fractional natural decomposition method (FNDM). The consider a model which consists of a system of five nonlinear ordinary differential equations (NODEs), which illustrate the sensor networks are interesting essentials for malicious outbreaks that attack the network with the intention of reducing the integrity, availability and confidentiality. Further, we captured the nature of FNDM results for different value of fractional order in terms of the plots. The considered scheme highly effective and structured while examining nonlinear models and which can be observed and confirm from the obtained results. Further, the conspiracies cited in plots confirm the hired fractional operator and algorithm can help to exemplify the more fascinating properties of the nonlinear system associated real-world problems.

Değişken katsayılı çok boyutlu dalga benzeri denklemler için kesirli yaklaşım üzerine etkili bir metot

In this paper, we study multi-dimensional wave-like equations with variable coefficients within the frame of the fractional calculus using fractional natural decomposition method (FNDM). The considered algorithm is an elegant combination of natural transform and decomposition scheme. Five different cases are considered to illustrate and validate the competence of the projected technique in the present framework. The behaviours of the obtained results have been captured for diverse fractional order. To present the reliability and exactness of the FNDM, the numerical study has been presented. The achieved consequences illuminate that, the projected technique is very effective to analyse and easy to employ to investigate the nature of fractional nonlinear coupled system exemplifying the real-world problems.

New numerical approach for time-fractional partial differential equations arising in physical system involving natural decomposition method

In the present research, we established an efficient and novel algorithm for time fractional multi-dimensional partial differential equations arising from physics and engineering. Taking into account Caputo fractional derivative, this algorithm involves the fractional natural decomposition method and the nonlinearity term decayed by utilizing the aforesaid method. The solution of the model is based on time dependent fractional-order equations such as telegraph equations parabolic equation Sine-Gordon equation wave-like equations. By considering the FNDM algorithm, the analytic solutions of the aforesaid models are analyzed. The suggested approach employs to solve several models of real-world phenomena and the consequences demonstrate that the approach is reliable, explicit and viable. Moreover, closed form solutions are established in many cases, and exact solutions are derived in particular. Numerical simulations were carried out to ensure that the proposed methods are perfect and precise in terms of efficiency and effectiveness, as shown by the exact solutions resolving complex nonlinear problems. The comparative analysis for the projected method reveals innovative attributes of the hybrid fractional derivative in the discussed model.

A novel proof of two partial fraction decompositions

In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.

An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method

We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.

An efficient technique to solve coupled–time fractional Boussinesq–Burger equation using fractional decomposition method

For this work, a novel numerical approach is proposed to obtain solution for the class of coupled time-fractional Boussinesq–Burger equations which is a nonlinear system. This system under consideration is endowed with Caputo time-fractional derivative. By means of the natural decomposition approach, approximate solutions of the proposed nonlinear fractional system are obtained where the exact solutions are presented in the classical case of fractional order at [Formula: see text]. Some numerical examples are given to support the theoretical framework and to point out the role and the effectiveness of the intended method. Our results clearly show the approximate analytical solutions eventually will converge quickly to the already known exact solutions. AMS Classification: 35A22, 35C05, 35C10, 35R11, 44A30.

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.

Two-loop leading colour QCD corrections to qoverline{q} \u2192 \u03b3\u03b3g and qg \u2192 \u03b3\u03b3q

We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

Thermally-induced changes in tropical soils properties and potential implications to sequential nature-based solutions

Some remediation techniques, such as thermal remediation, can significantly change the soil properties. These changes can be beneficial or detrimental to the sequential application of Nature-based Solutions. This work evaluated the effects of thermal remediation on the properties of two tropical soils (Technosol and Oxisol), and discuss how these changes might impact both biotic and abiotic degradation processes. Bench tests using disturbed samples were performed under oxic and anoxic conditions, whereas 3D physical models were used to simulate the heat distribution along undisturbed samples. The changes in soils texture, density, hydraulic conductivity, iron concentration, mineralogy and microbiota were evaluated. The properties of Oxisol were more affected than those of Technosol due to the higher levels in Fe(III), organic carbon and finer texture. When heated in the range of 120 to 300 °C under oxic and anoxic conditions, the Fe(II) content and the magnetism intensity increased in Oxisol, probably due to the formation of magnetite. Under oxic conditions, the burning of Oxisol organic matter promoted an anoxic atmosphere, favoring the formation of Fe(II). However, the continuous increase of the temperature (>300 °C) lead to the decrease of Fe(II) due to the transformation of magnetite to maghemite, and then to hematite. The heating process also promoted some minerals decomposition and cementation of the clay fraction, increasing the soil texture. Bacterial populations were impacted, but showed ability to recover at 60 °C. However, above 100 °C no culturable cells were recovered and at temperatures above 270 °C soil sterilization occurred. The changes observed, especially in Oxisol samples, indicated that mild heating (between 120 and 240 °C), in turn, can increase the potential for abiotic degradation of some contaminants, such as chlorinated solvents. Therefore, heating conditions up to 240 °C during thermal remediation can be defined as to promote beneficial changes in soil properties, increasing its potential for natural attenuation by abiotic processes even when the microbiota is affected, and improving its sustainability.

A POWERFUL ITERATIVE APPROACH FOR QUINTIC COMPLEX GINZBURG–LANDAU EQUATION WITHIN THE FRAME OF FRACTIONAL OPERATOR

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology.

Dynamics of leaf litter decomposition in the timberline zone of western Himalaya

Litter decomposition plays a vital role in carbon and nutrient budget of forest ecosystems, but our understanding of this process for high altitude ecosystems, especially the timberline zone, is not very clear. We studied leaf litter decomposition of Betula utilis, which is the main timberline species of Western Himalaya. We placed 144 litterbags (each containing 10 g of oven dried litter) at three ‘spatially well separated sites’ in Oct 2014, to study the direction and magnitude of change in key variables associated with leaf litter decomposition and to test for generality of decomposition process across these spatially distant sites. The results revealed that initial nutrient concentration and lignin content of litter was significantly different for the sites. Further, magnitude of mass loss and decay rate was found to be greater for up to 34 weeks (19.03 ± 4.55% and 0.35 ± 0.06 yr−1 respectively), which reached 47.01 ± 1.98% and 0.17 ± 0.01 yr−1 respectively, after 4 years. N, P and K concentrations showed sharp decease for up to 34 weeks, after which N consistently increased with incubation time, whereas P and K concentrations were relatively stable. The lignin and cellulose contents decreased consistently with incubation time (from 12.29 ± 1.21% to 5.21 ± 0.50% and from 16.87 ± 1.01% to 8.25 ± 0.50% respectively). There were significant increases in C:N and lignin:N ratios for up to 34 weeks, followed by a continuous decrease thereafter. The change in various variables associated with litter decomposition was not consistent among the sites and the magnitude of change was greater during initial stages due to leaching of labile components. The increase in N and decreases in C:N and lignin:N ratios with incubation time might be due to microbial immobilization of N and are important for nutrient conservation in timberline ecosystems, as such changes are related to faster decomposition of recalcitrant fraction of litter, N mineralization and building of soil nutrient pool.

Analysis of fractional‐order Schrödinger–Boussinesq and generalized Zakharov equations using efficient method

In this paper, we find the solution for an interesting family of coupled Schrödinger equations exemplifying simulating consequences. The fractional Schrödinger–Boussinesq (FSB) and fractional generalized Zakharov (FGZ) equations are analyzed in the present framework usingfractional natural decomposition method(FNDM). The hired technique is graceful amalgamations of natural transform technique with Adomian decomposition scheme. Three different cases are considered, one with FSB equations and two cases are associated with FGZ equations with different initial conditions. To validate and illustrate the proficiency of the projected solution procedure, we analyzed the projected model in terms of fractional order. Further, we captured the nature of FNDM results for different values of fractional order in terms of the plots. The considered scheme is highly effective and structured while examining nonlinear models and which can be observed and confirmed from the obtained results. Moreover, the plots show that the hired fractional operator and algorithm can help to exemplify the more fascinating properties of the complex system associated with real‐world problems.

Chip Appearance Inspection Method for High-Precision SMT Equipment

In order to meet the defect-detection requirements of chips in high-precision surface mount technology (SMT) equipment widely used in the electronic industry, a chip appearance defect-detection method based on multi-order fractional discrete wavelet packet decomposition (DWPD) is proposed in this paper. First, lead and body regions were extracted from chip images using the image segmentation algorithm with asymmetric Laplace mixture model and connected-component labelling algorithm; then, the texture feature of the region to be inspected was extracted with the multi-order fractional DWPD algorithm and the geometric and gradient features were combined to form image features of the region to be inspected before the subset of features was selected from image features with the feature selection algorithm based on the variational Bayesian Gaussian mixture model; and finally, the support vector machine was used to determine whether the region to be inspected was defective. An experiment was conducted on a data set captured in high-precision SMT equipment. The accuracy of the proposed chip appearance defect-detection method is about 93%, which is more accurate than existing ones.

Soil organic matter formation is controlled by the chemistry and bioavailability of organic carbon inputs across different land uses

Soil organic matter (SOM) formation involves microbial transformation of plant materials of various quality with physico-chemical stabilisation via soil aggregation. Land use and vegetation type can affect the litter chemistry and bioavailability of organic carbon (OC), and consequently influence the processing and stabilisation of OC into SOM. We used 13C nuclear magnetic resonance (13C NMR) and hot-water extraction to assess the changes in chemical composition and labile OC fractions during the transformation processes from leaf to litter to SOM depending on land use and vegetation type. The hot-water-extractable OC (HWEOC) decreased from leaf (43–65 g kg−1) to litter (19–23 g kg−1) to SOM (8–16 g kg−1) similar in four land use types: grassland, sugarcane, forest and banana. These trends demonstrated the uniform converging pathways of OC transformation and increasing stability by SOM formation. The preferential decomposition and decrease of labile OC fractions (∑% di-O-alkyl, O-alkyl and methoxyl) from leaf (54–69%) to SOM (41–43%) confirmed the increasing stability of the remaining compounds. Despite differences in the biochemical composition of the leaf tissues among the vegetation types, the proportions of labile OC fractions in SOM were similar across land uses. The OC content of soil was higher in forest (7.9%) and grassland (5.2%) compared to sugarcane (2.3%) and banana (3.0%). Consequently, the HWEOC per unit of soil weight was higher in forest and grassland (2.0 and 1.2 g kg−1 soil, respectively) compared to sugarcane and banana (0.3 and 0.4 g kg soil−1, respectively). The availability of labile SOM is dependent on the quantity of SOM not the chemical composition of SOM. In conclusion, labile OC fractions in SOM, as identified by 13C NMR, were similar across land use regardless of vegetation type and consequently, SOM formation leads to convergence of chemical composition despite diversity of OC sources.

Some extensions for the several combinatorial identities

In this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers.

Generalized Exponential Time Differencing Schemes for Stiff Fractional Systems with Nonsmooth Source Term

Many processes in science and engineering are described by fractional systems which may in general be stiff and involve a nonsmooth source term. In this paper, we develop robust first, second, and third order accurate exponential time differencing schemes for solving such systems. Rather than imposing regularity requirements on the solution to account for the singularity caused by the fractional derivative, we only consider regularity requirements on the source term for preserving the optimal order of accuracy of the proposed schemes. Optimal convergence rates are proved for both smooth and nonsmooth source terms using uniform and graded meshes, respectively. For efficient implementation, high-order global Pade approximations together with their fractional decompositions are developed for Mittag–Leffler functions. We present numerical experiments involving a typical stiff system, a fractional two-compartment pharmacokinetics model, a two-term fractional Kelvin–Viogt model of viscoelasticity, and a large system obtained by spatial discretization of a sub-diffusion problem. Demonstrations of the efficiency of the rational approximation implementation technique and the newly constructed high-order schemes are provided.

On the stability of the zero solution of differential equation of the second order under critical case

Differential equation of the form x¨+x2sgnx = Y (t, x, x˙), is considered, where the right-hand side is a small periodic perturbation of t, a sufficiently differentiable function in the origin neighborhood with variables x, x˙. It is assumed that X perturbation is of order smallness not lower than the fifth, if x is assigned the second order, and x˙ is assigned the third order. Periodic functions are introduced that are the solution of the equation above with zero righthand side. Since differentiability of the quadratic part is bounded, then differentiability of the introduced functions is also bounded. These functions are used to transfer from the initial equation to the system of equations in coordinates similar to polar. This system with the help of the polynomial replacement is reduced to a system with Lyapunov constants. Replacement coefficients are found by the partial fraction decomposition. The conclusion about the nature of stability of the zero solution is made by a sign of the first non zero constant. Due to the bounded differentiability of the introduced functions, the degree of the polynomial replacement must be limited. The system of differential equations for the replacement coefficients is solved recursively. The number of found Lyapunov constants is also bounded. This article considers the case when all found constants are zero. To study this problem there used the method of isolating the main part of the introduced functions and their combinations as a result of the expantion of the latter in the Fourier series. The remainder of the series is assumed to be sufficiently small and it is shown that its presence can be neglected. The transition to the main parts instead of functions allows to compensate for the lack of differentiability of the introduced functions. Considering such systems, polynomial replacement can be again used and the Lyapunov constant for each main part can be found. It is shown that the sign of the constant for any main part is preserved. Sufficient conditions for stability and instability are indicated.

Schrödinger’s tridiagonal matrix

Abstract In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

Investigation of Electrohydraulic Effect on Physicochemical Characteristics of High-Resinous Oil “Karazhanbas”

In order to change the viscosity of high-resinous oil of the “Karazhanbas” field (Kazakhstan), the effect of electrohydraulic action on it was studied. The effect of adding an organic solvent xylene on the rheological properties of oil is investigated. A comparative study of the hydrocarbon composition of oil before and after electrohydraulic impact was carried out by the method of gas chromatographymass spectrometry. Fragment composition of oil hydrocarbons before and after electrohydraulic treatment was determined by 1H and 13C NMR spectroscopy. It is shown that the conversion of heavy oil fractions to light ones begins with a five-time electrohydraulic impact. It is determined that 20‒25 electric discharge pulses are sufficient for the quantitative process of splitting hydrocarbons. It was found that the addition of xylene to high-viscosity oil leads to an increase in the conversion of hydrocarbons under electrohydraulic action. The content of paraffins and naphthalenes in high-viscosity oil is slightly reduced during electrohydraulic processing. In oil, after electrohydraulic action, a decrease in the proportion of protons of long alkyl terminal СН3-groups of hydrocarbons is observed, which indicates the process of decomposition of heavy oil fractions into light fractions.