Fraction Decomposition Research Articles

LinApart: Optimizing the univariate partial fraction decomposition

We present LinApart, a routine designed for efficiently performing the univariate partial fraction decomposition of large symbolic expressions. Our method is based on an explicit closed formula for the decomposition of rational functions with fully factorized denominators. We provide an implementation in the Wolfram Mathematica language, which can lead to very significant performance gains over the built-in Apart command. Furthermore, a C language library implementing the core functionality and suitable for interfacing with other software is also provided. Both codes are made available at https://github.com/fekeshazy/LinApart.Program summaryProgram title:LinApartCPC Library link to program files:https://doi.org/10.17632/v8vg2mjjnb.1Developer's repository link:https://github.com/fekeshazy/LinApartLicensing provisions: MIT licenseProgramming language:Wolfram Mathematica and CNature of problem: Analytic computations in quantum field theory often produce large expressions involving complicated rational functions. In many cases, such as symbolic integration, performing partial fraction decomposition of the latter is vital. Even though univariate partial fraction decomposition is conceptually well-understood, the complexity of state-of-the-art calculations is such that use of readily available tools, such as the Apart command in Mathematica, becomes highly impractical due to time and memory constraints. Here we present LinApart, a routine developed for performing univariate partial fraction decomposition in a highly efficient manner. Improvements in timing and memory usage of up to five orders of magnitude are achieved for rational expressions with as few as ten denominator factors. As such, the routine allows to perform decomposition problems that were previously intractable.Solution method: The LinApart routine implements a simple closed form expression for univariate partial fraction decomposition of rational functions with fully factored denominators based on the residue theorem. The Mathematica implementation makes use of the highly efficient built-in differentiation routine to evaluate this formula. The C language implementation employs a version of the formula that does not require symbolic differentiation but is rather based on the computation of multinomial coefficients.Additional comments including restrictions and unusual features: The routine is based on a formula which assumes that the denominator of the rational function to be decomposed is in a fully factorized form, i.e., it is a product whose factors are linear in the decomposition variable. For maximum efficiency, the Mathematica implementation by default does not perform any factorization and non-linear denominators, if present, are simply disregarded during the decomposition. Thus, in such cases the output will not be in a fully decomposed form. This issue does not arise in the C implementation, which only accepts linear denominator factors of the form (x−a)m as input.

A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass

Abstract In this article, we study the following fractional Kirchhoff-type problems with critical and sublinear nonlinearities: a + b ∬ R N × R N ∣ u ( x ) − u ( y ) ∣ 2 ∣ x − y ∣ N + 2 s d x d y ( − Δ ) s u = λ u q − 1 + u 2 s * − 1 , u > 0 , in Ω , u = 0 , in R N \ Ω , ∫ R N u 2 d x = c 2 , \left\{\begin{array}{l}\left(a+b\mathop{\iint }\limits_{{{\mathbb{R}}}^{N}\times {{\mathbb{R}}}^{N}}\frac{{| u\left(x)-u(y)| }^{2}}{{| x-y| }^{N+2s}}{\rm{d}}x{\rm{d}}y\right){\left(-\Delta )}^{s}u=\lambda {u}^{q-1}+{u}^{{2}_{s}^{* }-1},\hspace{1em}u\gt 0,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0\left,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2},\end{array}\right. where ( − Δ ) s {\left(-\Delta )}^{s} is the fractional Laplacian, Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, 0 < s < 1 , 2 s < N < 4 s 0\lt s\lt 1,2s\lt N\lt 4s , 1 < q < 2 1\lt q\lt 2 , λ > 0 , a > 0 , b > 0 , c > 0 \lambda \gt 0,a\gt 0,b\gt 0,c\gt 0 . First, we prove that the bounded Palais-Smale sequence has a profile decomposition in the fractional Laplacian setting. Then, by utilizing decomposition techniques and variational methods, we acquire that there are two positive normalized solutions for the aforementioned problems.

Control of landscape position on organic matter decomposition via soil moisture during a wet summer

Sustainable cropland management requires preservation of soil organic matter (SOM). In spite of in depth understanding gained from ample field and laboratory studies, we have a poor understanding of landscape scale spatial variation of fresh organic matter (OM) decomposition and its conversion into soil organic carbon (SOC). Particularly, local topographic position may be expected to co-control these processes via soil hydrology. In this study, we sought to identify if such control is significant by setting up a field experiment with two contrasting positions across 10 gently sloping cropland fields covering three different soil texture groups, i.e. loamy sand, (sandy) loam and silt loam. We wanted to link OM decomposition to within-field differences in soil moisture, whilst keeping variation in other soil and management factors minimal. Specifically, mesocosms with 13C enriched ryegrass (the OM source) were incorporated in the fields for ten weeks and afterwards, soil was separated into > 500 µm, 53 – 500 µm and < 53 µm sized fractions. Overall, we found that lower located positions were wetter than higher positions with average differences of 11 %, 20 % and 16 % in water-filled pore space for the loamy sand, (sandy) loam and silt loam soil, respectively. Mineralization of added OM was surprisingly independent of landscape position, even though moisture conditions appeared wetter than optimal at the low but not at the high landscape positions. Remaining ryegrass residues > 500 µm did follow local topography-driven gradients in soil moisture with higher amounts in low landscape positions. In other words, drier conditions at high landscape positions improved coarse OM decomposition, with consequently more ryegrass-carbon (C) ending up in finer soil fractions (< 500 µm). Additionally, soil texture affected decomposition of the smallest fraction (< 53 µm) with a stabilizing effect for finer-textured (silt loam) soils. We conclude that, despite significant contrasts in moisture conditions between landscape positions, within-field spatial variability of OM mineralization was overall limited during the observed wet summer period. Nevertheless, landscape position affected the quality of remnant unmineralized C, with relatively more conversion of freshly added OM into OM associated with silt and clay at the drier higher positions, potentially improving the long-term stability of SOM. Likewise observations under different weather conditions are needed to evaluate the necessity of precise modelling of local soil hydrology for predicting SOC stock evolution on the landscape scale.

A comparative study on the slow pyrolysis of Miscanthus (Miscanthus×giganteus Greef et Deu.) cultivated on agricultural and contaminated soils: Assessment of distribution of final products

This study aims to investigate the slow pyrolysis behavior of uncontaminated (MSC-I - reference) and heavy metals contaminated (MSC-R) samples with heavy metals from the tailings of a Pb-Zn-Cu flotation mine, consisting of whole stems of Miscanthus. Physicochemical properties of raw materials were investigated through instrumental characterization techniques (Atomic Absorption Spectrometry (AAS), Scanning Electron Microscopy (SEM), Fourier Transform Infrared (FTIR) spectroscopy, and X-ray diffraction (XRD)), while the pyrolysis process was monitored by simultaneous thermal analysis techniques (thermogravimetry (TG) – derivative thermogravimetry (DTG)), coupled with Mass spectrometry (MS), for evolved gas analysis. After determination of the lignocellulose content (cellulose, hemicellulose, and lignin) and extractive of the MSC-I and MSC-R samples, it was found that Pb, Zn, Fe, and Mn in MSC-R lead to very fast decomposition of extractive fraction, which facilitates their distribution in formed bio-char, together with lignin char-participation, acting catalytically. It was established that a much greater fraction of extractives decomposition products and non-volatile heavy metals are incorporated in MSC-R bio-char increasing its yield (23 %), compared to MSC-I bio-char yield (21.5 %). It was identified that MSC-R has increased production of H2, CO, and CO2, while decreased production of CH4, influenced by Fe (Fe has a significant positive effect on CO2 evaluation during MSC-R pyrolysis, enhancing decarboxylation process). It was established that CH4 reforming reactions catalyzed by iron additionally affect methane reduction, and increase production of H2, compared to the reference sample. Isoconversional kinetic analysis showed that pyrolysis reactions profile was strongly conditioned by the presence of heavy metals, such as Cd, Pb, and Zn, because they affect the modification of biomass lignocellulosic structure. Also, it was found that small amounts of Cd present in MSC-R can increase pyrolysis activation energy of three pseudo-components, and inhibit their deoxygenation, thus increasing yield of bio-char.

Research on fractional wavelet transform combined with parameter adaptive PCNN for infrared and visible image fusion algorithm

Infrared and visible image fusion holds significant value across various fields due to its ability to provide complementary information. However, existing fusion algorithms often suffer from susceptibility to external physical conditions of source images, leading to inconsistent fusion quality and limited adaptive capability. In this paper, we propose a fractional wavelet transform fusion algorithm for infrared and visible images. This algorithm uses fractional wavelet domain image decomposition to more accurately locate the image structures of different scales. Initially, we perform image decomposition using a non-subsampled shearlet transform (NSST) with multi-scale and multi-directional decomposition. Subsequently, to effectively fuse detailed information, we employ discrete fractional wavelet transform (DFRWT) to decompose low-frequency subbands while selecting an appropriate p-value. High-frequency subbands are fused using a parameterized adaptive pulse-coupled neural network (PCNN) model. To validate the superiority of our algorithm, we conduct visual and quantitative comparative analyses against several existing algorithms. The results confirm that our proposed algorithm exhibits robustness against external physical conditions and surpasses these existing algorithms, making it highly applicable for infrared–visible fusion tasks.

The impact of struvite presence on the thermal decomposition of sewage sludge, including formation of NOx emissions

Monoincineration of sewage sludge (SS) is becoming a prevailing type of SS treatment due to its contamination and the ability to achieve enriched concentration of valuable inorganics in the end ash. In order to improve the recovery potential, particularly of inorganic, phosphorous (P) bearing species, thermal transformations of organic as well as inorganic SS compounds have to be considered. While important gains in predictive capability have already been made through extensive description of organic part decomposition, P is mostly present in the inorganic part of SS, together with notable amount of nitrogen (N). Current models consider these species in a simplified manner, making the ability to predict P and N fate during thermal treatment limited. The present work focuses on development and implementation of a one step reaction kinetics mechanism for struvite, a common P and N bearing compound in SS, which also presents an important route for P and N recovery. The new mechanism is implemented into an existing SS organic fraction decomposition mechanism and validated using 0D multiphase reactor simulations in Chemkin software. Results reveal up to 10 s faster temperature increase and decomposition of SS with the presence of struvite due to emission of NH3 and H2O occurring already from 350 °C onwards. The NH3 emissions are especially important as they lead to higher NOx formation. The work presents large potential for improving thermal decomposition prediction as well as emission mitigation strategies if struvite is included in the thermal decomposition models of SS. The developed methodology allows efficient implementation of also other P and N bearing species in the inorganic part of kinetic mechanism.

Photovoltaic system faults detection using fractional multiresolution signal decomposition

Photovoltaic system faults detection using fractional multiresolution signal decomposition

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α&gt;1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

Fungal decomposition and transformation of molecular and colloidal fractions of dissolved organic matter extracted from boreal forest soil

Dissolved organic matter (DOM) plays a central role in soil carbon (C) dynamics, serving as both a substrate for microbial decomposers and a source of material stabilised via physical protection in molecular aggregates and associations with mineral particles. It is well established that soil microorganisms play a key role in mineral-associated C aggregates; however, their impacts on molecular aggregates is not clearly understood. Here, we examined the ability of an ectomycorrhizal fungus (Paxillus involutus) and a saprotrophic fungus (a strain of Gloeophyllum), two major functional groups of fungal decomposers in forest ecosystems, to decompose and process the molecular and colloidal size fractions of DOM. DOM was extracted by water from boreal forest soil, and the chemical composition and colloidal properties were followed over 11 days using nuclear magnetic resonance (NMR) spectroscopy and small-angle light and X-ray scattering techniques. Both fungi decompose various organic compounds into their molecular fractions in the presence of an energy source (i.e. glucose). The decomposition rate was significantly higher for Gloeophyllum than for P. involutus. When glucose was depleted, Gloeophyllum continued to decompose more complex carbohydrates, whereas the decomposition activity of P. involutus almost stopped. A large proportion of the C in the DOM was found in organic colloids. At later stages, Gloeophyllum but not P. involutus, significantly affected the colloids by promoting the formation of larger aggregates. Thus, saprotrophic fungi activity can significantly influence the colloidal properties of DOM. Our results support the view that ectomycorrhizal fungi decompose some of the soil organic C however, their overall capacity for DOM decomposition and transformation is significantly lower than that of saprotrophic fungi.

Fractional Fischer decompositions by inframonogenic functions

Inframonogenic functions can be viewed as a non-commutative version of the more traditional harmonic functions. In this paper we obtain a new Fischer decomposition for homogeneous polynomials in Rm in terms of (φ,ψ)-inframonogenic homogeneous polynomials. The latter being a natural generalization arising when structural sets φ, ψ are considered instead of the standard orthonormal basis of Rm. Moreover, we extend our results to the fractional context by means of the Caputo derivative and Weyl relations.

A Eulerian Monte Carlo method for the numerical solution of the multivariate population balance equation

Statistical descriptions of particulate phases and granular media based on the notion of a property distribution are particularly advantageous for dense, many-particle dispersions since they support the direct evaluation of Eulerian property statistics, permit a kinematically simple treatment of particle collisions and feature a model complexity that is independent of the number of physical particles. On the minus side, the property distribution is governed by a high-dimensional population balance equation (PBE) whose moment reduction is accompanied by a challenging closure problem and engenders intricate dynamical nonlinearities. Circumventing both the curse of dimensionality and the moment closure problem, we present a Eulerian Monte Carlo (EMC) solution method that harnesses a representation of the property distribution in terms of an ensemble of property samples and is based on a statistical enforcement of the spatially and temporally discretized PBE. After application of a kinetic finite volume scheme, the PBE is subjected to a fractional steps decomposition that separates spatial transport from transport in property space. Conceptually, spatial transport is emulated by assembling the ensemble in one spatial grid cell from samples drawn from the ensembles in neighbouring cells at the previous time point. In this regard, we propose a two-level sampling scheme that is sensitive to large-scale features of the marginal velocity distribution and effectively reduces the instantaneous sampling error. The property transport step, by contrast, is based on the method of characteristics and involves the propagation of individual samples as property characteristics in time. While the sampling error afflicting the property statistics needs to be controlled using a top-level time or ensemble averaging, the EMC method remains well-conditioned for an arbitrary number of samples, is easy to implement and involves a computational expense that scales only linearly with the number of particle properties. Based on two benchmark examples, we analyze the spatial and statistical convergence properties of the EMC method and demonstrate its capacity to capture multiple crossings of particle trajectories. A third example involves the two-way coupled transport of size-polydispersed inertial particles in an inhomogeneous flow field.

Promotion of levoglucosan production from biomass pyrolysis by hydrogen peroxide pre-oxidation

Due to the complexity of biomass structures, the conversion of raw biomass into value-added chemicals is challenging and often requires efficient pretreatment of the biomass. In this paper, a simple and green pre-oxidation method, which was conducted under the conditions of 2 wt% H2O2, 80 min, and 150 °C, was reported to significantly increase the production of levoglucosan (LG) from biomass pyrolysis. The result showed that the LG yield significantly increased from 2.3 wt% (without pre-oxidation) to 23.1 wt% when pine wood was employed as a sample for pyrolysis at 400 °C, resulting from the removal of hemicellulose fraction and the in-situ acid catalysis of lignin carboxyl groups formed during the pre-oxidation. When the conditions for pre-oxidation became harsher than the above, the LG yield reduced because the decomposition of cellulose fraction in biomass. The study supplies an effective method for utilization of biomass as chemicals.

A novel chaotic image encryption is based on fractional wavelet decomposition and quantum transform model

A new chaotic image encryption scheme based on fractional order wavelet decomposition is proposed in this paper. Initially, the image undergoes a three-stage fractional-order wavelet decomposition, resulting in high-frequency and low-frequency components. High-frequency components are dislocated using a chaotic system, while low-frequency components are dislocated employing a fractional matrix model. Subsequently, the scrambled image undergoes quantum encoding followed by the Arnold transform to yield the final scrambled result. Ultimately, the encryption process involves iteratively acquiring the M4 key matrix and diffusing the quantum image through permutation to obtain the final encryption result. Simulation experiments and numerical analyses demonstrate the high security level of the proposed encryption method.

An Efficient New Technique for Solving Nonlinear Problems Involving the Conformable Fractional Derivatives

In this paper, an efficient new technique is used for solving nonlinear fractional problems that satisfy specific criteria. This technique is referred to as the double conformable fractional Laplace‐Elzaki decomposition method (DCFLEDM). This approach combines the double Laplace‐Elzaki transform method with the Adomian decomposition method. The fundamental concepts and findings of the recently suggested transformation are presented. For the purpose of assessing the accuracy of our approach, we provide three examples and introduce the series solutions of these equations using DCLEDM. The results show that the proposed strategy is a very effective, reliable, and efficient approach for addressing nonlinear fractional problems using the conformable derivative.

LOCAL FRACTIONAL SUMUDU DECOMPOSITION METHOD TO SOLVE FRACTAL PDEs ARISING IN MATHEMATICAL PHYSICS

In this paper, we investigate solutions of telegraph, Laplace and wave equations within the local fractional derivative operator by using local fractional Sumudu decomposition method. This method is coupled by the Sumudu transform and decomposition method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

A Numerical Scheme for Time-Space Fractional diffusion Models

Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transform technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing to the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial fraction decomposition of these approximations. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary conditions. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Fractal dynamics and computational analysis of local fractional Poisson equations arising in electrostatics

In this paper, the local fractional natural decomposition method (LFNDM) is used for solving a local fractional Poisson equation. The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution. Numerical examples with computer simulations are presented in this paper. The obtained results show that LFNDM is effective and convenient for application.

Functional Diversity Accelerates the Decomposition of Litter Recalcitrant Carbon but Reduces the Decomposition of Labile Carbon in Subtropical Forests

The biodiversity of litter can regulate carbon and nutrient cycling during mixed decomposition. It is common knowledge that the decomposition rates of mixed litters frequently deviate from those predicted for these component litter species. However, the direction and magnitude of the nonadditive effects on the degradation of mixed litters remain difficult to predict. Previous studies have reported that the different carbon fractions of leaf litters responded to litter mixture differently, which may help to explain the ambiguous nonadditive effect of diversity on bulk litter decomposition. Therefore, we conducted decomposition experiments on 32 litter mixtures from seven common tree species to test the responses of different carbon fractions to litter diversity in subtropical forests. We found that the overall mass loss of the mixed litter was faster than that estimated from single species. The relative mixing effects (RMEs) of different carbon fractions exhibited different patterns to litter diversity and were driven by different aspects of litter functional dissimilarity. Soluble carbon fractions decomposed more slowly than expected from single species, while lignin fractions decayed more quickly. Moreover, we found that the RMEs of bulk litter decomposition may be determined by the lignin fraction decomposition. Our findings further support that distinguishing the response of different carbon fractions to litter diversity is important for elucidating the nonadditive effects of total litter decomposition.

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.