Estimating soil hydraulic conductivity by Inverse Laplace Transform

Abstract Milk is one of the most widely consumed staple foods worldwide. Its increasing commercial importance both increases the need for quality classification and creates a breeding ground for fraudulent practices, such as adding water to dilute. In this work, some milk varieties, including raw (unheated), skimmed, semi-skimmed, and fat (whole) milk, as well as boiling raw milk, were analyzed using the Time Domain Nuclear Magnetic Resonance (TD-NMR) methodology in parallel with the Microwave (MW) dielectric method. The study also examined the effects of adding water to certain kinds of milk. Spin lattice (T 1 ) and spin–spin (T 2 ) relaxation times and dielectric permittivity constants Ɛ 1 and Ɛ 2 were used to classify milk, and the Inverse Laplace Transform (ILT) method was used to examine these relaxation times further. Furthermore, the effects of temperature categorization on milk were examined using temperature-dependent T 1 and T 2 measurements. As a result, the relaxation times of different types of milk are listed from longest to shortest: skimmed milk, raw milk, semi-skimmed milk, fat (whole) milk, and raw milk (boiled). It was found that T 1 and T 2 values increased with the increase in temperature. And also, skimmed milk has a greater Ɛ 1 value compared to fat (whole) milk. Furthermore, it has been observed that adding water to milk increases the dielectric constant (Ɛ 1- Ɛ 2 ) as the water concentration increases. Overall, our results demonstrate that the TD- NMR and MW approach is a sensitive and efficient method for identifying various milk kinds and detecting the presence of water adulteration.

The majority of MR-based brain imaging methods provides macroscopic information averaged over the entire imaging voxel. Yet tissue composition and microstructure are heterogeneous within the cubic millimeter-sized MRI voxel that contains numerous distinct water pools at mesoscopic, microscopic, and nanoscopic length scales. Accurately measuring their individual characteristics in live human brain has the potential to reveal hidden salient meso/micro-structural features and uncover subtle changes that may occur in development, neurological disorders, trauma, etc. Nevertheless, because of many technical and scientific challenges, there is a dearth of robust, quantitative methods to probe tissue water dynamics at these subvoxel length scales. Here we present a novel empirical spectroscopic diffusion MRI method that estimates the probability density function (pdf) of diffusion tensors, i.e., the diffusion tensor distribution (DTD) in the human brain in-vivo . Our method entails performing a multi-dimensional Inverse Laplace Transform (ILT) which is generally an ill-posed and ill-conditioned problem. However, we overcome these obstacles using a hierarchy of lower-dimensional marginal distributions of the DTD estimated from diffusion weighted (DW) signals obtained from single, double, and triple pulsed-field gradient (PFG) experiments. Iteratively applying this hierarchy of marginal distribution progressively shrinks the space of admissible solutions. We extensively vet this framework with simulated DWI data obtained from realistic DTD motifs that mimic different cell and tissue properties seen in the brain. We then experimentally test our approach in vivo in brains of healthy normal human subjects. We segment the reconstructed DTD within a voxel to identify signatures of different tissue and cell types, and cluster these DTDs to identify various water pools. We use the high dimensional spectrum to robustly remove the free water compartment that often confounds tissue microstructure. We take ensemble averages of invariants of the micro-diffusion tensors, and measure and map their distributions to visualize salient intrinsic mesoscopic features. Since DTD MRI subsumes DTI, we also compute the family of DTI-derived quantitative imaging biomarkers from the moments of the distributions of the mean diffusivity and FA derived from the DTD. Our approach has great translational potential, revealing new microstructural features not observed previously observed in in vivo MRI.

ABSTRACT Based on the complex variable method and the corresponding principle of viscoelasticity, viscoelastic solutions for the stress and displacement of a lined non‐circular tunnel subjected to in‐situ stresses and internal water pressure is derived. The basic equations for solving the analytic functions are established according to the stress boundary condition along the inner boundary of the lining and the stress and displacement continuity conditions along the rock‐lining interface. The analytic functions are expressed as Laurent series and the Laplace transformation is performed on the basic equation. Herein, the power series method is applied to obtain the linear equations which are expressed by the analytic function coefficients in the Laplace domain. The stress and displacement solutions of tunnel in Laplace domain can be addressed by solving the equations, and then the viscoelastic solutions are obtained through Laplace Inverse transformation. Subsequently, an example for the horseshoe‐shaped tunnel is performed. The example used the generalized Kelvin model to simulate the rheological properties of surrounding rock mass. The obtained solution is compared with the numerical solution. The influences of the lateral pressure coefficient and the internal water pressure on the stresses and displacements of lining are analyzed.

Analytical models of transdermal drug delivery (TDD) often represent deeper skin layers using ideal sink assumptions or phenomenological interfacial resistances. While mathematically convenient, these approaches obscure the physical role of the dermis and hypodermis in controlling molecular transport. Here, we develop an impedance-based analytical model for diffusion across multilayer skin membranes, in which the epidermal barrier is dynamically coupled to a finite diffusive backing layer representing the dermis-hypodermis composite. Diffusion impedance links transport conductivity, storage capacity, and layer thickness, while preserving continuity of concentration and flux at all interfaces. Closed-form expressions in the Laplace domain describe concentration fields and interfacial fluxes, and cumulative drug uptake is computed in the time domain via inverse Laplace transformation. The model identifies distinct short- and long-time transport regimes. Commonly used Dirichlet and Robin boundary conditions emerge as limiting cases but cannot reproduce the regime-dependent behavior of a backing layer. In particular, Robin formulations reduce the backing layer to a constant effective resistance, neglecting its storage capacity and time-dependent impedance. By replacing ad hoc boundary conditions with a physically grounded impedance framework, this approach provides a unified and extensible method for analyzing multilayer transport systems, including extensions to anomalous or memory-dependent diffusion.

DyF3 nanoparticle formation in a 5 mm NMR tube is studied. Real-time 1H NMR spectroscopy measurements of coprecipitation reaction in the NMR tube in a magnetic field of 0.6T were performed on a time scale up to 40 h. As it is known, the nanoparticle formation by the coprecipitation method occurs instantly. The chemical reaction was slowed down using the minimization of the interface between two precursor solutions to study the nucleation of DyF3 particles in detail. The time-domain results of the Regularized Inverse Laplace Transform application to the 1H longitudinal nuclear magnetization recovery curves demonstrate three components responsible for solutions of DyCl3, NaF, and DyF3 nanoparticles. The synthesized DyF3 nanoparticles in a 5 mm NMR tube are characterized as DyF3 with an average size of 5 nm using XRD and TEM methods. Thus, a method for monitoring the nanoparticle nucleation using preventing mixing of the precursor solutions by minimizing their interface has been proposed for the first time. The proposed method will allow a detailed study of the chemical reaction course and processes affecting the nanoparticle formation.

Multi-view learning meets state-space model: A dynamical system perspective.

For capturing dynamic information about a filled-cave in the fractured reservoir, a novel Pressure Transient Analysis (PTA) analytical model for a well located at the filled-cave is established. In this new model, we consider the stress-sensitivity of the filled-cave and the inter-porosity flow of fracture. First, Perturbation transformation was used to obtain the pressure distribution in the filled-cave zone. Then, the Warren–Root model was applied to establish the pressure solution in the fractured reservoir. Next, the pressure and its derivative are obtained by the Laplace transformation and Steftest inversion. Lastly, the Bottomhole Pressure (BHP) and Bottomhole Pressure Derivative (BHPD) combined curve reveals the flow regimes of this novel model. The results show the composite model can be used to characterize the fractured reservoir with the filled-caves, and its flow follows the composite flow regimes. The spherical flow has an obvious slope of 0.5 on the BHPD curve, which can identify the size of the filled-caves. The boundary flow can be used to identify stress-sensitivity. Affected by the stress-sensitivity of the filled-cave, the BHPD’s slope of the boundary flow will be greater than 1. This research work provides technical support for capturing cave and fracture parameters in the fractured reservoir.

ABSTRACTA rapid, reagent‐free method for the quantification of chitosan in aqueous solutions was developed using 1H nuclear magnetic resonance (NMR) relaxometry. Transverse relaxation times (T2) of chitosan solutions (5–1000 mg/L) were measured with a Carr–Purcell–Meiboom–Gill (CPMG) sequence on a benchtop NMR relaxometer. The data were processed using three approaches: monoexponential fitting (Bruker software), inverse Laplace transformation (CONTIN), and one‐component curve fitting (Excel Solver). Calibration curves derived from these models demonstrated high linearity and reproducibility, particularly when concentration intervals were segmented. Among the approaches, the most robust correlation was achieved by plotting the absolute area (AA) of the T2 distribution obtained from CONTIN analysis versus chitosan concentration, yielding R2 values of up to 0.9978 for 5–100 mg/L. Comparative analysis at 40°C and 21°C confirmed the method's temperature stability, with improved sensitivity at elevated temperature. Unlike conventional spectrophotometric or chromatographic methods, the proposed protocol requires no chemical derivatization or complex sample preparation. This technique provides a fast, accurate, and non‐destructive alternative for chitosan quantification, especially suitable for material science applications where precise concentration monitoring is critical, such as surface modification of nanomaterials.

For a wide class of problems, the integral Laplace transform leads to a simpler equation with respect to the image of the sought original. The next step is the inversion problem: finding the original by its image. Often, this step cannot be implemented analytically, and approximate inversion methods are required. In this case, the approximate solution is represented as a linear combination of the image and its derivatives at a number of points of the complex half-plane where the image is analytic. However, unlike the image, the original may even have discontinuity points. Finding frameworks and approximate solutions is reduced to solving systems of linear algebraic equations (SLAEs), which, like the original problem, are ill-conditioned. SLAE matrices have different properties, which should be taken into account to reduce their condition number in comparison with existing regularization methods. The results of numerical experiments confirming the efficiency of the proposed inversion algorithms are presented.

A mathematical model of NMR transverse relaxation for pore size distribution estimation in porous media.

This paper presents the design and construction of a cost-effective embeddable nuclear magnetic resonance sensor using 3D printing to improve the construction process. The sensor comprises two 25.4mm diameterx3mm thick neodymium-iron-boron disk magnets and an elliptical radio frequency coil. Magnetic field simulations were employed to determine the optimal separation between magnets, achieving a relatively homogeneous B0 field of 180mT at the center of the array. Custom 3D-printed parts ensured precise magnet alignment and facilitated coil fabrication. The sensor was encased within a Faraday cage constructed from a printed circuit board to mitigate external electromagnetic interference. A remote tuning circuit was developed to tune the coil to 7.66MHz. Initial testing involved using an eraser sample to determine the required 90° and 180° pulse amplitudes and duration. The sensor's performance was further validated under immersion conditions in milk, yogurt, and fresh cement paste, using the Carr-Purcell-Meiboom-Gill technique. The signals obtained were processed by fitting the data to an exponential decay function to obtain the T2 lifetimes and their corresponding signal intensities, and by Inverse Laplace Transformation to obtain the T2 lifetime distribution. Results indicate the sensoŕs capability to detect variations in samples having different compositions.

This study focuses on the dynamic response of porous functionally graded nanomaterials to moving loads. The analysis was performed using two approaches: the Ritz method with the help of the benefits achieved by employing Chebyshev polynomials in the cosine form and the differential quadrature method with further inverse Laplace transformation. Both approaches utilize the formulation of a nano-thin beam considering an improved higher-order beam model and nonlocal strain gradient theory with two characteristic length scales, referred to as nonlocality and strain gradient length scales. Power-law dependencies steer the constituent designs of pore-graded materials toward pore factors that influence pore volume either with a uniform or non-uniform distribution of pores. Moreover, a variable scale modulus was adopted to further improve accuracy by considering the scale effects for graded nano-thin beams. The first part of the study addresses the equation of motion, which is solved by applying the Ritz technique with Chebyshev polynomials. In the second part, the governing equations for nanobeams are discussed where the differential quadrature method is used to discretise them further, and the inverse Laplace transform is used to obtain the dynamic deflections. The results of the present study elucidate the effects of the moving load speed, nonlocal strain gradient factors, porosity, pore number and distribution, and elastic medium on the dynamic deflection of functionally graded nanobeams.

Inverse Laplace transforms (ILTs) are fundamental to a wide range of scientific and engineering applications—from diffusion NMR spectroscopy to medical imaging—yet their numerical inversion remains severely ill-posed, particularly in the presence of noise or sparse data. The primary objective of this study is to develop robust and efficient numerical methods that improve the stability and accuracy of ILT reconstructions under challenging conditions. In this work, we introduce a novel family of Kaczmarz-based ILT solvers that embed advanced regularization directly into the iterative projection framework. We propose three algorithmic variants—Tikhonov–Kaczmarz, total variation (TV)–Kaczmarz, and Wasserstein–Kaczmarz—each incorporating a distinct penalty to stabilize solutions and mitigate noise amplification. The Wasserstein–Kaczmarz method, in particular, leverages optimal transport theory to impose geometric priors, yielding enhanced robustness for multi-modal or highly overlapping distributions. We benchmark these methods against established ILT solvers—including CONTIN, maximum entropy (MaxEnt), TRAIn, ITAMeD, and PALMA—using synthetic single- and multi-modal diffusion distributions contaminated with 1% controlled noise. Quantitative evaluation via mean squared error (MSE), Wasserstein distance, total variation, peak signal-to-noise ratio (PSNR), and runtime demonstrates that Wasserstein–Kaczmarz attains an optimal balance of speed (0.53 s per inversion) and accuracy (MSE = 4.7×10−8), while TRAIn achieves the highest fidelity (MSE = 1.5×10−8) at a modest computational cost. These results elucidate the inherent trade-offs between computational efficiency and reconstruction precision and establish regularized Kaczmarz solvers as versatile, high-performance tools for ill-posed inverse problems.

ABSTRACTThe dispersity (Ð) of a polymer is one of its most important characteristics. While the most established technique for dispersity determination is size exclusion chromatography (SEC), Diffusion‐ordered NMR spectroscopy (DOSY) has emerged as a rival for routine polymer molar mass determination, even if dispersity remained somewhat elusive in DOSY to date. To expand DOSY to reliable dispersity measurement, we synthesize 26 polystyrenes with Ð ranging from 1.1 to 3.7 and number‐average molar mass (Mn) ranging from 3.0 to 22.4 kg∙mol−1 to correlate molar mass distribution (MMD) measurements of SEC to DOSY. We show that the method of inverse Laplace transformation (ILT) fails to represent the shape of the MMD of a polymer when it is not narrowly dispersed. Despite this failure, DOSY does yield the number and weight average molar mass with high accuracy, and thus the dispersity of samples is well accessible from DOSY‐ILT. SEC and DOSY‐ILT are well correlated up to Ð = 2.0. Interestingly, using the standard deviation (σ) and coefficient of variation of the MMD rather than dispersity yields an even better correlation across the entire dispersity and molar mass range covered in this study, underpinning the usefulness of using σ more regularly in polymer characterization.

The present study aimed to determine the strategy for analysis of NMR transverse magnetization decay (the T2 decay) of semicrystalline polymers using two alternative methods, i.e., a least-squares fit of decays and their inverse Laplace transform (ILT). First, these methods are used to analyze the T2 decay of one-phase compounds with physical states ranging from crystals (gypsum) to glassy polymer (polycarbonate, PC) and from amorphous EPDM vulcanizate to melts of PC and high-density polyethylene. Then, physical mixtures of these compounds are analyzed. Finally, the T2 decay of isotactic polypropylene and propylene-ethylene random copolymers with complex physical structures is examined. For the study, a kernel of commonly used ILT software, which is applicable for the analysis of exponential functions, was modified by adding the Gaussian and the Abragamian functions that describe well the decay shape of glassy and crystalline compounds, respectively. To our knowledge, such a composite kernel has not been used before. This method was named the Laplace-like. The Laplace-like analysis offers advantages when the decay rate of different phases does not largely differ, e.g., in materials composed of crystals, glassy phases, and interfaces with restricted molecular mobility when deconvolution of T2 decays with a least-squares method fails. The Laplace-like method can help to identify the number of polymer fractions with restricted chain mobility and distribution of specific (exponential, Gaussian, or Abragamian) T2 relaxation times. However, the ILT method with the exponential kernel cannot provide meaningful data for soft polymeric matters. The reliable analysis requires implementation in the kernel of the Laplace algorithm complex functions describing the shape of NMR decays of soft polymeric matters and polymer melts. The advantages and limitations of a least-squares deconvolution of T2 decays into their components and the Laplace-like method are discussed. The most reliable procedures for extracting information about the phase composition and molecular mobility in different phases of semicrystalline polymers were proposed.

The application of a hybrid spectral collocation method (HSCM) to a class of new multi-term time-fractional viscoelastic non-Newtonian fluid models is studied in this work. The noteworthy addition of this work is that the new model in this study includes a novel time-fractional operator on the spatial derivative, also the considered model includes multi-term time-fractional derivatives with fractional orders ranging from 0 to 2. The proposed HSCM combines the Laplace transform (LT) for temporal discretization and the spatial operators are discretized via the Chebyshev collocation method (CM). It has been proven that the LT is beneficial for solving diffusion-type problems and presents a substitute for the finite difference method (FDM). In FDM, small time step is required to achieve high accuracy and stability, due to such limitations the computational cost is also enhanced. On the other hand, CM utilizes Langrange’s interpolation polynomial, based on Chebyshev collocation points. For smooth problems, the implementation of CM is straightforward and has spectral convergence. The HSCM has three basic steps. Initially, the LT is utilized to reduce the proposed model into a time-independent model; second, the CM is used for spatial discretization; and finally, to get the original solution of the model, inverse LT is used. The Ulam–Hyers stability of the considered model is also discussed. To ensure the stability, accuracy, and computational efficiency of the proposed method, four numerical examples have been examined.

Single and double-selective split-inversion pulse and recovery (SIP-R) sequences for targeted T1 relaxation measurements.

Inverse Laplace transforms (ILTs) are fundamental to a wide range of scientific and engineering applications—from diffusion NMR spectroscopy to medical imaging—yet their numerical inversion remains severely ill-posed, particularly in the presence of noise or sparse data. In this work, we introduce a novel family of Kaczmarz-based ILT solvers that embed advanced regularization directly into the iterative projection framework. We propose three algorithmic variants—Tikhonov-Kaczmarz, Total Variation (TV)-Kaczmarz, and Wasserstein-Kaczmarz—each incorporating a distinct penalty to stabilize solutions and mitigate noise amplification. The Wasserstein-Kaczmarz method, in particular, leverages optimal transport theory to impose geometric priors, yielding enhanced robustness for multi-modal or highly overlapping distributions. We benchmark these methods against established ILT solvers—including CONTIN, Maximum Entropy (MaxEnt), TRAIn, ITAMeD, and PALMA—using synthetic single- and multi-modal diffusion distributions contaminated with 1 % controlled noise. Quantitative evaluation via mean squared error (MSE), Wasserstein distance, total variation, peak signal-to-noise ratio (PSNR), and runtime demonstrates that Wasserstein-Kaczmarz attains an optimal balance of speed (0.53 s per inversion) and accuracy (MSE = 4.7×10−8), while TRAIn achieves the highest fidelity (MSE = 1.5×10−8) at a modest computational cost. These results elucidate the inherent trade-offs between computational efficiency and reconstruction precision and establish regularized Kaczmarz solvers as versatile, high-performance tools for ill-posed inverse problems. To promote reproducibility and further development, all code is freely available at https://github.com/fmarrabal/kaczmarz-ilt-solver.

In the process of gas flooding, the underground miscible law and oil displacement characteristics are complex, and there is no effective evaluation method. As an effective method to invert reservoir parameters and analyze seepage law, well test can be used to analyze the law of underground gas and oil action. This study focuses on the mechanism of gas-crude oil interaction, and establishes a three-zone radial composite well test model including interface skin effect and power-law change of physical properties of transition zone. The model innovatively introduces the interface coefficient to characterize the phase transition effect of the regional boundary, which is solved by dimensionless transformation, Laplace transformation and Stehfest numerical inversion method. Get the fluid phase distribution evaluation chart in the process of gas flooding. The results show that the radius of the crude oil zone (R1) determines the duration of the radial flow, the radius of the gas-oil transition zone (R2) regulates the seepage range, the power law index (θ) controls the derivative curve shape, and the energy storage index (I) has a weak influence on the curve shape. This study establishes a theoretical framework for dynamic monitoring of miscible gas drive wells and nonlinear reservoir parameter inversion.