Search NASA⌕ Search

SEARCH · Search NASA

Results for “fractionation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2014

Impact of dose size in single fraction spatially fractionated (grid) radiotherapy for melanoma

Purpose: To evaluate the impact of dose size in single fraction, spatially fractionated (grid) radiotherapy for selectively killing infiltrated melanoma cancer cells of different tumor sizes, using different radiobiological models. Methods: A Monte Carlo technique was employed to calculate the 3D dose distribution of a commercially available megavoltage grid collimator in a 6 MV beam. The linear-quadratic (LQ) and modified linear quadratic (MLQ) models were used separately to evaluate the therapeutic outcome of a series of single fraction regimens that employed grid therapy to treat both acute and late responding melanomas of varying sizes. The dose prescription point was at the center of the tumor volume. Dose sizes ranging from 1 to 30 Gy at 100% dose line were modeled. Tumors were either touching the skin surface or having their centers at a depth of 3 cm. The equivalent uniform dose (EUD) to the melanoma cells and the therapeutic ratio (TR) were defined by comparing grid therapy with the traditional open debulking field. The clinical outcomes from recent reports were used to verify the authors’ model. Results: Dose profiles at different depths and 3D dose distributions in a series of 3D melanomas treated with grid therapy were obtained. The EUDs and TRs for all sizes of 3D tumors involved at different doses were derived through the LQ and MLQ models, and a practical equation was derived. The EUD was only one fifth of the prescribed dose. The TR was dependent on the prescribed dose and on the LQ parameters of both the interspersed cancer and normal tissue cells. The results from the LQ model were consistent with those of the MLQ model. At 20 Gy, the EUD and TR by the LQ model were 2.8% higher and 1% lower than by the MLQ, while at 10 Gy, the EUD and TR as defined by the LQ model were only 1.4% higher and 0.8% lower, respectively. The dose volume histograms of grid therapy for a 10 cm tumor showed different dosimetric characteristics from those of conventional radiotherapy. A significant portion of the tumor volume received a very large dose in grid therapy, which ensures significant tumor cell killing in these regions. Conversely, some areas received a relatively small dose, thereby sparing interspersed normal cells and increasing radiation tolerance. The radiobiology modeling results indicated that grid therapy could be useful for treating acutely responding melanomas infiltrating radiosensitive normal tissues. The theoretical model predictions were supported by the clinical outcomes. Conclusions: Grid therapy functions by selectively killing infiltrating tumor cells and concomitantly sparing interspersed normal cells. The TR depends on the radiosensitivity of the cell population, dose, tumor size, and location. Because the volumes of very high dose regions are small, the LQ model can be used safely to predict the clinical outcomes of grid therapy. When treating melanomas with a dose of 15 Gy or higher, single fraction grid therapy is clearly advantageous for sparing interspersed normal cells. The existence of a threshold fraction dose, which was found in the authors’ theoretical simulations, was confirmed by clinical observations.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Scattering problems in the fractional quantum mechanics governed by the 2D space-fractional Schrödinger equation

The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Bidisperse and polydisperse suspension rheology at large solid fraction

At the same solid volume fraction, bidisperse and polydisperse suspensions display lower viscosities, and weaker normal stress response, compared to monodisperse suspensions. The reduction of viscosity associated with size distribution can be explained by an increase of the maximum flowable, or jamming, solid fraction. In this work, concentrated or "dense" suspensions are simulated under strong shearing, where thermal motion and repulsive forces are negligible, but we allow for particle contact with a mild frictional interaction with interparticle friction coefficient of 0.2. Aspects of bidisperse suspension rheology are first revisited to establish that the approach reproduces established trends; the study of bidisperse suspensions at size ratios of large to small particle radii (2 to 4) shows that a minimum in the viscosity occurs for zeta slightly above 0.5, where zeta=phi_{large}/phi is the fraction of the total solid volume occupied by the large particles. The simple shear flows of polydisperse suspensions with truncated normal and log normal size distributions, and bidisperse suspensions which are statistically equivalent with these polydisperse cases up to third moment of the size distribution, are simulated and the rheologies are extracted. Prior work shows that such distributions with equivalent low-order moments have similar phi_{m}, and the rheological behaviors of normal, log normal and bidisperse cases are shown to be in close agreement for a wide range of standard deviation in particle size, with standard correlations which are functionally dependent on phi/phi_{m} providing excellent agreement with the rheology found in simulation. The close agreement of both viscosity and normal stress response between bi- and polydisperse suspensions demonstrates the controlling in influence of the maximum packing fraction in noncolloidal suspensions. Microstructural investigations and the stress distribution according to particle size are also presented.

Pednekar, Sidhant↗
DOE OSTI2014

Retrievals of Cloud Fraction and Cloud Albedo from Surface-based Shortwave Radiation Measurements: A Comparison of 16 Year Measurements

Ground-based radiation measurements have been widely conducted to gain information on clouds and the surface radiation budget; here several different techniques for retrieving cloud fraction (Long2006, Min2008 and XL2013) and cloud albedo (Min2008, Liu2011 and XL2013) from ground-based shortwave broadband and spectral radiation measurements are examined, and sixteen years of retrievals collected at the Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) site are compared. The comparison shows overall good agreement between the retrievals of both cloud fraction and cloud albedo, with noted differences however. The Long2006 and Min2008 cloud fractions are greater on average than the XL2013 values. Compared to Min2008 and Liu2011, the XL2013 retrieval of cloud albedo tends to be greater for thin clouds but smaller for thick clouds, with the differences decreasing with increasing cloud fraction. Further analysis reveals that the approaches that retrieve cloud fraction and cloud albedo separately may suffer from mutual contamination of errors in retrieved cloud fraction and cloud albedo. Potential influences of cloud absorption, land-surface albedo, cloud structure, and measurement instruments are explored.

albedo↗
DOE OSTI2014

On the reliable analysis of indium mole fraction within In x Ga 1-x N quantum wells using atom probe tomography

Surface crystallography and polarity are shown to influence the detection probability of In, Ga, and N ions during atom probe tomography analysis of In x Ga 1-x N m -plane, c -plane, and ($20\overline{21}$) quantum wells. A N deficit is observed in regions of the reconstruction generated from Ga-polar surfaces, and the probability of detecting group-III atoms is lower in In x Ga 1-x N quantum wells than in GaN barrier layers. Finally, despite these artifacts, the detected In mole fraction is consistent throughout a given quantum well regardless of the crystal orientation of the quantum well or the evaporation surface from which the reconstruction was generated.

36 MATERIALS SCIENCE↗
DOE OSTI2019

fPINNs: Fractional Physics-Informed Neural Networks

Physics-informed neural networks (PINNs), introduced, are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE residuals and the mean-squared error in initial-boundary conditions is minimized with respect to the NN parameters. Here we extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we study systematically their convergence, hence explaining both fPINNs and PINNs for the first time. Specifically, we demonstrate their accuracy and effectiveness in solving multidimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box (BB) forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. This approach bypasses the difficulties stemming from the fact that automatic differentiation is not applicable to fractional operators because the standard chain rule in integer calculus is not valid in fractional calculus. To discretize the fractional operators, we employ the Grünwald--Letnikov (GL) formula in one-dimensional fractional ADEs and the vector GL formula in conjunction with the directional fractional Laplacian in two- and three-dimensional fractional ADEs. We first consider the one-dimensional fractional Poisson equation and compare the convergence of the fPINNs against the finite difference method (FDM). We present the solution convergence using both the mean L 2 error as well as the standard deviation due to sensitivity to NN parameter initializations. Using different GL formulas we observe first-, second-, and third-order convergence rates for small size training sets but the error saturates for larger training sets. We explain these results by analyzing the four sources of numerical errors due to discretization, sampling, NN approximation, and optimization. The total error decays monotonically (below 10 –5 for a third-order GL formula) but it saturates beyond that point due to the optimization error. We also analyze the relative balance between discretization and sampling errors and observe that the sampling size and the number of discretization points (auxiliary points) should be comparable to achieve the highest accuracy. As we increase the depth of the NN up to certain value, the mean error decreases and the standard deviation increases, whereas the width has essentially no effect unless its value is either too small or too large. We next consider time-dependent fractional ADEs and compare white-box (WB) and BB forcing. We observe that for the WB forcing, our results are similar to the aforementioned cases; however, for the BB forcing fPINNs outperform FDM. Subsequently, we consider multidimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of 10 –3 ~ 10 –4 . Lastly, we solve several inverse problems in one, two, and three dimensions to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results given proper initializations even in the presence of significant noise.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Comparison of Different Fractionation Schedules Toward a Single Fraction in High-Dose-Rate Brachytherapy as Monotherapy for Low-Risk Prostate Cancer Using 3-Dimensional Radiobiological Models

Purpose: The aim of the present study was the investigation of different fractionation schemes to estimate their clinical impact. For this purpose, widely applied radiobiological models and dosimetric measures were used to associate their results with clinical findings. Methods and Materials: The dose distributions of 12 clinical high-dose-rate brachytherapy implants for prostate were evaluated in relation to different fractionation schemes. The fractionation schemes compared were: (1) 1 fraction of 20 Gy; (2) 2 fractions of 14 Gy; (3) 3 fractions of 11 Gy; and (4) 4 fractions of 9.5 Gy. The clinical effectiveness of the different fractionation schemes was estimated through the complication-free tumor control probability (P{sub +}), the biologically effective uniform dose, and the generalized equivalent uniform dose index. Results: For the different fractionation schemes, the tumor control probabilities were 98.5% in 1 × 20 Gy, 98.6% in 2 × 14 Gy, 97.5% in 3 × 11 Gy, and 97.8% in 4 × 9.5 Gy. The corresponding P{sub +} values were 88.8% in 1 × 20 Gy, 83.9% in 2 × 14 Gy, 86.0% in 3 × 11 Gy, and 82.3% in 4 × 9.5 Gy. With use of the fractionation scheme 4 × 9.5 Gy as reference, the isoeffective schemes regarding tumor control for 1, 2, and 3 fractions were 1 × 19.68 Gy, 2 × 13.75 Gy, and 3 × 11.05 Gy. The optimum fractionation schemes for 1, 2, 3, and 4 fractions were 1 × 19.16 Gy with a P{sub +} of 91.8%, 2 × 13.2 Gy with a P{sub +} of 89.6%, 3 × 10.6 Gy with a P{sub +} of 88.4%, and 4 × 9.02 Gy with a P{sub +} of 86.9%. Conclusions: Among the fractionation schemes 1 × 20 Gy, 2 × 14 Gy, 3 × 11 Gy, and 4 × 9.5 Gy, the first scheme was more effective in terms of P{sub +}. After performance of a radiobiological optimization, it was shown that a single fraction of 19.2 to 19.7 Gy (average 19.5 Gy) should produce at least the same benefit as that given by the 4 × 9.5 Gy scheme, and it should reduce the expected total complication probability by approximately 40% to 55%.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2017

Characterization of upgraded fast pyrolysis oak oil distillate fractions from sulfided and non-sulfided catalytic hydrotreating

Catalytic hydroprocessing of pyrolysis oils from biomass produces hydrocarbons that can be considered for liquid fuel production. This process requires removal of oxygen and cracking of the heavier molecular weight bio-oil constituents into smaller fragments at high temperatures and pressures under hydrogen. A comprehensive understanding of product oils is useful to optimize cost versus degree of deoxygenation. Additionally, a better understanding of the chemical composition of the distillate fractions can open up other uses of upgraded oils for potentially higher-value chemical streams. We present in this paper the characterization data for five well-defined distillate fractions of two hydroprocessed oils with different oxygen levels: a low oxygen content (LOC, 1.8% O, wet basis) oil and a medium oxygen content (MOC, 6.4% O, wet basis) oil. Elemental analysis and 13C NMR results suggest that the distillate fractions become more aromatic/unsaturated as they become heavier. Our results also show that the use of sulfided catalysts directly affects the S content of the lightest distillate fraction. Carbonyl and carboxylic groups were found in the MOC light fractions, while phenols were present in the heavier fractions for both MOC and LOC. PIONA analysis of the light LOC fraction shows a predominance of paraffins with a minor amount of olefins. These results can be used to direct future research on refinery integration and production of value-added product from specific upgraded oil streams.

biofuel↗
DOE OSTI2015

Fractional charge and spin errors in self-consistent Green’s function theory

We examine fractional charge and spin errors in self-consistent Green’s function theory within a second-order approximation (GF2). For GF2, it is known that the summation of diagrams resulting from the self-consistent solution of the Dyson equation removes the divergences pathological to second-order Møller-Plesset (MP2) theory for strong correlations. In the language often used in density functional theory contexts, this means GF2 has a greatly reduced fractional spin error relative to MP2. The natural question then is what effect, if any, does the Dyson summation have on the fractional charge error in GF2? To this end, we generalize our previous implementation of GF2 to open-shell systems and analyze its fractional spin and charge errors. We find that like MP2, GF2 possesses only a very small fractional charge error, and consequently minimal many electron self-interaction error. This shows that GF2 improves on the critical failings of MP2, but without altering the positive features that make it desirable. Furthermore, we find that GF2 has both less fractional charge and fractional spin errors than typical hybrid density functionals as well as random phase approximation with exchange.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2022

Remote Sensing of Aerosol Water Fraction, Dry Size Distribution and Soluble Fraction Using Multi-Angle, Multi-Spectral Polarimetry

A framework to infer volume water fraction, soluble fraction and dry size distributions of fine-mode aerosol from multi-angle, multi-spectral polarimetry retrievals of column-averaged ambient aerosol properties is presented. The method is applied to observations of the Research Scanning Polarimeter (RSP) obtained during two NASA aircraft campaigns, namely the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) and the Cloud, Aerosol, and Monsoon Processes Philippines Experiment (CAMP2Ex). All aerosol retrievals are statistically evaluated using in situ data. Volume water fraction is inferred from the retrieved ambient real part of the refractive index, assuming a dry refractive index of 1.54 and by applying a volume mixing rule to obtain the effective ambient refractive index. The uncertainties in inferred volume water fraction resulting from this simplified model are discussed and estimated to be lower than 0.2 and decreasing with increasing volume water fraction. The daily mean retrieved volume water fractions correlate well with the in situ values with a mean absolute difference of 0.09. Polarimeter-retrieved ambient effective radius for daily data is shown to increase as a function of volume water fraction as expected. Furthermore, the effective variance of the size distributions also increases with increasing effective radius, which we show is consistent with an external mixture of soluble and insoluble aerosol. The relative variations of effective radius and variance over an observation period are then used to estimate the soluble fraction of the aerosol. Daily results of soluble fraction correlate well with in situ-observed sulfate mass fraction with a correlation coefficient of 0.79. Subsequently, inferred water and soluble fractions are used to derive dry fine-mode size distributions from their ambient counterparts. While dry effective radii obtained in situ and from RSP show similar ranges, in situ values are generally substantially smaller during the ACTIVATE deployments, which may be due to biases in RSP retrievals or in the in situ observations, or both. Both RSP and in situ observations indicate the dominance of aerosol with low hygroscopicity during the ACTIVATE and CAMP2Ex campaigns. Furthermore, RSP indicates a high degree of external mixing of particles with low and high hygroscopicity. These retrievals of fine-mode water volume fraction and soluble fraction may be used for the evaluation of water uptake in atmospheric models. Furthermore, the framework allows us to estimate the variation in the concentration of fine-mode aerosol larger than a specific dry radius limit, which can be used as a proxy for the variation in cloud condensation nucleus concentrations. This framework may be applied to multi-angle, multi-spectral satellite data expected to be available in the near future.

Amount of water in particulate matter↗
NASA NTRS2022

Remote Sensing of Aerosol Water Fraction, Dry Size Distribution and Soluble Fraction Using Multi-Angle, Multi-Spectral Polarimetry

A framework to infer volume water fraction, soluble fraction and dry size distributions of fine mode aerosol from multi-angle, multi-spectral polarimetry retrievals of column-averaged ambient aerosol properties is presented. The method is applied to observations of the Research Scanning Polarimeter (RSP) obtained during two NASA aircraft campaigns, namely the Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) and the Cloud, Aerosol, and Monsoon Processes-Philippines Experiment (CAMP2Ex). All aerosol retrievals are statistically evaluated using in situ data. Volume water fraction is inferred from the retrieved ambient real part of the refractive index, assuming a dry refractive index of 1.54 and by applying a volume mixing rule to obtain the effective ambient refractive index. The uncertainties in inferred volume water fraction resulting from this simplified model are discussed and estimated to be lower than 0.2 and decreasing with increasing volume water fraction. The daily mean retrieved volume water fractions correlate well with the in situ values with a mean absolute difference of 0.09. Polarimeter-retrieved ambient effective radius for daily data is shown to increase as a function of volume water fraction as expected. Furthermore, the effective variance of the size distributions also increases with increasing effective radius, which we show is consistent with an external mixture of soluble and insoluble aerosol. The relative variations of effective radius and variance over an observation period are then used to estimate the soluble fraction of the aerosol. Daily results of soluble fraction correlate well with in situ observed sulfate mass fraction with a correlation coefficient of 0.79. Subsequently, inferred water and soluble fractions are used to derive dry fine-mode size distributions from their ambient counterparts. While dry effective radii obtained in situ and from RSP show similar ranges, in situ values are generally substantially smaller during the ACTIVATE deployments, which may be due to biases in RSP retrievals or in the in situ observations, or both. Both RSP and in situ observations indicate the dominance of aerosol with low hygroscopicity during the ACTIVATE and CAMP2Ex campaigns. Furthermore, RSP indicates a high degree of external mixing of particles with low and high hygroscopicity. These retrievals of fine mode water volume fraction and soluble fraction may be used for the evaluation of water uptake in atmospheric models. Furthermore, the framework allows to estimate the variation in the concentration of fine-mode aerosol larger than a specific dry radius limit, which can be used as a proxy for the variation in cloud condensation nucleus concentrations. This framework may be applied to multi-angle, multi-spectral satellite data expected to be available in the near future.

Bastiaan Van Diedenhoven↗
DOE OSTI2015

Single-Fraction Versus 5-Fraction Radiation Therapy for Metastatic Epidural Spinal Cord Compression in Patients With Limited Survival Prognoses: Results of a Matched-Pair Analysis

Purpose: This study compared single-fraction to multi-fraction short-course radiation therapy (RT) for symptomatic metastatic epidural spinal cord compression (MESCC) in patients with limited survival prognosis. Methods and Materials: A total of 121 patients who received 8 Gy × 1 fraction were matched (1:1) to 121 patients treated with 4 Gy × 5 fractions for 10 factors including age, sex, performance status, primary tumor type, number of involved vertebrae, other bone metastases, visceral metastases, interval between tumor diagnosis and MESCC, pre-RT ambulatory status, and time developing motor deficits prior to RT. Endpoints included in-field repeated RT (reRT) for MESCC, overall survival (OS), and impact of RT on motor function. Univariate analyses were performed with the Kaplan-Meier method and log-rank test for in-field reRT for MESCC and OS and with the ordered-logit model for effect of RT on motor function. Results: Doses of 8 Gy × 1 fraction and 4 Gy × 5 fractions were not significantly different with respect to the need for in-field reRT for MESCC (P=.11) at 6 months (18% vs 9%, respectively) and 12 months (30% vs 22%, respectively). The RT regimen also had no significant impact on OS (P=.65) and post-RT motor function (P=.21). OS rates at 6 and 12 months were 24% and 9%, respectively, after 8 Gy × 1 fraction versus 25% and 13%, respectively, after 4 Gy × 5 fractions. Improvement of motor function was observed in 17% of patients after 8 Gy × 1 fraction and 23% after 4 Gy × 5 fractions, respectively. Conclusions: There were no significant differences with respect to need for in-field reRT for MESCC, OS, and motor function by dose fractionation regimen. Thus, 8 Gy × 1 fraction may be a reasonable option for patients with survival prognosis of a few months.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2021

Carbon content of soil fractions varies with season, rainfall, and soil fertility across a lowland tropical moist forest gradient

Tropical forests contain some of the largest soil carbon (C) stocks on Earth, making them broadly relevant to terrestrial-climate feedbacks, yet our understanding of how their soil organic C (SOC) fractions vary over space and time is limited. We studied effects of season, fertility, and mean annual precipitation (MAP) on the C contents of soil fractions across 14 lowland forests in Panama. We measured free-debris, occluded-debris, and mineral-associated SOC fractions, as well as soluble C associated with each fraction. We hypothesized that mineral-associated SOC would be greatest in infertile, strongly weathered soils with large reactive mineral contents. We also hypothesized that the debris SOC fractions would accumulate during the dry season, reflecting seasonal increases in litterfall. To address this, we compared soil fractions in wet and dry seasons from fertile and infertile soils across a range of 1809 – 2864 mm MAP. The C content (mg C / g soil) of all soil fractions varied with fertility and MAP: specifically, free-debris SOC was greatest in wet, high-fertility soils, and occluded-debris SOC was greater in high-fertility than low-fertility soils. The mineral-associated SOC fraction, which contained the majority of bulk soil C, showed increasing C content with greater MAP in infertile sites, presumably driving similar spatial patterns in the bulk soil. Only the free-debris SOC fraction showed strong seasonal variation, increasing in mass during the dry season. Nitrogen behaved similarly to C. In summary, soil C contents increased with MAP in infertile sites but not fertile sites, driven by the mineral-associated SOC fraction. The dry season had greater free-debris SOC, but this seasonal trend was not apparent in bulk soil C, likely because of the small size of the free-debris SOC fraction. Furthermore, changes in the quantity and seasonality of precipitation, which are projected for tropical forests, might shift spatial and temporal patterns of soil C storage, which would in turn influence forest-climate feedbacks for this C-rich biome.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Linear fractional differential equations and eigenfunctions of fractional differential operators

Eigenfunctions associated with Riemann–Liouville and Caputo fractional differential operators are obtained by imposing a restriction on the fractional derivative parameter. Those eigenfunctions can be used to express the analytical solution of some linear sequential fractional differential equations. As a first application, we discuss analytical solutions for the so-called fractional Helmholtz equation with one variable, obtained from the standard equation in one dimension by replacing the integer order derivative by the Riemann–Liouville fractional derivative. A second application consists of an initial value problem for a fractional wave equation in two dimensions in which the integer order partial derivative with respect to the time variable is replaced by the Caputo fractional derivative. The classical Mittag-Leffler functions are important in the theory of fractional calculus because they emerge as solutions of fractional differential equations. Starting with the solution of a specific fractional differential equation in terms of these functions, we find a way to express the exponential function in terms of classical Mittag-Leffler functions. A remarkable characteristic of this relation is that it is true for any value of the parameter n appearing in the definition of the functions, i.e., we have an infinite family of different expressions for e{sup x} in terms of classical Mittag-Leffler functions.

97 MATHEMATICS AND COMPUTING↗