Diffusion Equation Research Articles

Numerical discretization of a Darcy–Forchheimer flow with variable density and heat transfer

In this paper, we study a heat transfer scenario in Darcy–Forchheimer porous media with variable density. The block-centered finite difference method is applied to discretize the non-isothermal flow equations governing the system. Specifically, the pressure field is modeled using the nonlinear Darcy–Forchheimer formulation, while the density and temperature are described by convection-dominated diffusion equations, which are treated via the characteristic method. Theoretical analyses are rigorously developed for pressure, velocity, density, temperature, and auxiliary flux across non-uniform grids. Several numerical experiments are carried out to illustrate the merits of our method by comparing numerical results to analytical solutions.

A high-accuracy compact finite difference scheme for time-fractional diffusion equations

A high-accuracy compact finite difference scheme for time-fractional diffusion equations

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Depth detection limit of a fluorescent object in tissue-like medium with background emission in continuous-wave measurements: a phantom study.

Although the depth detection limit of fluorescence objects in tissue has been studied, reports with a model including noise statistics for designing the optimum measurement configuration are missing. We demonstrate a variance analysis of the depth detection limit toward clinical applications such as noninvasively assessing the risk of aspiration. It is essential to analyze how the depth detection limit of the fluorescence object in a strong scattering medium depends on the measurement configuration to optimize the configuration. We aim to evaluate the depth detection limit from theoretical analysis and phantom experiments and discuss the source-detector distance that maximizes this limit. Experiments for detecting a fluorescent object in a biological tissue-mimicking phantom of ground beef with background emission were conducted using continuous wave fluorescence measurements with a point source-detector scheme. The results were analyzed using a model based on the photon diffusion equations. Then, variance analysis of the signal fluctuation was introduced. The model explained the measured fluorescence intensities and their fluctuations well. The variance analysis showed that the depth detection limit in the presence of ambient light increased with the decrease in the source-detector distance, and the optimum distance was in the range of 10 to 15mm. The depth detection limit was found to be with this optimum distance for the phantom. The presented analysis provides a guide for the optimum design of the measurement configuration for detecting fluorescence objects in clinical applications.

Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

Switching Response in Organic Electrochemical Transistors by Ionic Diffusion and Electronic Transport.

The switching response in organic electrochemical transistors (OECT) is a basic effect in which a transient current occurs in response to a voltage perturbation. This phenomenon has an important impact on different aspects of the application of OECT, such as the equilibration times, the hysteresis dependence on scan rates, and the synaptic properties for neuromorphic applications. Here we establish a model that unites vertical ion diffusion and horizontal electronic transport for the analysis of the time-dependent current response of OECTs. We use a combination of tools consisting of a physical analytical model; advanced 2D drift-diffusion simulation; and the experimental measurement of a poly(3-hexylthiophene) (P3HT) OECT. We show the reduction of the general model to simple time-dependent equations for the average ionic/hole concentration inside the organic film, which produces a Bernards-Malliaras conservation equation coupled with a diffusion equation. We provide a basic classification of the transient response to a voltage pulse, and the correspondent hysteresis effects of the transfer curves. The shape of transients is basically related to the main control phenomenon, either the vertical diffusion of ions during doping and dedoping, or the equilibration of electronic current along the channel length.

Maximum-Principle-Preserving High-Order Conservative Difference Schemes for Convection-Dominated Diffusion Equations

Maximum-Principle-Preserving High-Order Conservative Difference Schemes for Convection-Dominated Diffusion Equations

A non-autonomous framework for climate change and extreme weather events increase in a stochastic energy balance model

We develop a three-timescale framework for modeling climate change and introduce a space-heterogeneous one-dimensional energy balance model. This model, addressing temperature fluctuations from rising carbon dioxide levels and the super-greenhouse effect in tropical regions, fits within the setting of stochastic reaction–diffusion equations. Our results show how both mean and variance of temperature increase, without the system going through a bifurcation point. This study aims to advance the conceptual understanding of the extreme weather events frequency increase due to climate change.

Graphene Oxide Decorated with Nickel Cobaltite Nanoparticles as an Adsorbent for Cationic Methyl Green Dye: Kinetic, Isotherm, and Thermodynamic Studies

‎تم في هذه الدراسة ، تزيين ‏رقائق أكسيد الجرافين (‏GO‏) بجسيمات ‏كوبلتيت النيكل النانوية ‏NiCo2O4‎‏(‏NC‏) عن طريق الترسيب في ‏الموقع ، وتم استخدام المتراكب ‏المحضر (‏NC: GO‏) كسطح ماز لإزالة ‏صبغة الميثيل الخضراء ( ‏MG‏) من ‏المحاليل المائية. تم التحقق من ‏التغطية الناجحة لأوكسيد الجرافين ‏بجزيئات كوبلتيت النيكل النانوية ‏‏(‏NC‏) باستخدام دراسات ‏FT-IR‏ وحيود ‏الأشعة السينية (‏XRD‏). كانت أحجام ‏الجسيمات البلورية لل ‏NC‏ و ‏ لل ‏GO‏ المزين ب NCهي ‏‎10.53‎‏ نانوميتر ‏‏9.30 نانومتر على التوالي. تم دراسة ‏تأثير العديد من العوامل التجريبية ‏، بما في ذلك زمن التلامس ، وكمية ‏السطح الماز ، ودرجة الحرارة. كان ‏وقت التلامس الأمثل وكمية السطح 120 ‏دقيقة و 3 مجم / لتر على التوالي. ‏تتلائم بيانات الامتزاز بشكل أفضل مع ‏ايزوثيرم ‏Freundlich‏. تم استخدام ‏أربعة نماذج حركية لتتبع عملية ‏الامتزاز: معادلة زائفة من الدرجة ‏الأولى ، ومعادلة زائفة من الدرجة ‏الثانية ، ومعادلة الانتشار داخل ‏الجسيمات ، ومعادلة بويد. أظهرت ‏نمذجة البيانات التجريبية أن حركية ‏الامتزاز كانت ممثلة بشكل جيد ‏بنموذج الرتبة الثانية الزائفة (‏R2 ‎‎= 0.9945‎‏) مع معدل ثابت سرعة يبلغ 3.2 ‏‏× 10 - 3 (جم / مجم. دقيقة). يتم ‏امتصاص صبغة ‏MG‏ تدريجيًا بواسطة ‏الجسيمات النانوية ‏NC‏ من خلال ‏الانتشار داخل الجسيمات ويتم ‏الاحتفاظ بها بعد ذلك في مسام أصغر. ‏أظهرت قيم التحليل الديناميكي ‏الحراري أن عملية امتزاز صبغة ‏MG‏ ‏كانت ماصة للحرارة بطبيعتها ، و ‏تلقائية وعملية الامتزاز فيزيائية. الكلمات المفتاحية: الانتشار داخل الجسيمات ، المسافات البينية، النموذج الزائف من الدرجة الأولى ، معادلة بويد

Spectral methods utilizing generalized Bernstein‐like basis functions for time‐fractional advection–diffusion equations

This paper presents two methods for solving two‐dimensional linear and nonlinear time‐fractional advection–diffusion equations with Caputo fractional derivatives. To effectively manage endpoint singularities, we propose an advanced space‐time Galerkin technique and a collocation spectral method, both employing generalized Bernstein‐like basis functions (GBFs). The properties and behaviors of these functions are examined, highlighting their practical applications. The space‐time spectral methods incorporate GBFs in the temporal domain and classical Bernstein polynomials in the spatial domain. Fractional equations frequently produce irregular solutions despite smooth input data due to their singular kernel. To address this, GBFs are applied to the time derivative and classical Bernstein polynomials to the spatial derivative. A thorough error analysis confirms the technique's accuracy and convergence, offering a robust theoretical basis. Numerical experiments validate the method, demonstrating its effectiveness in solving both linear and nonlinear time‐fractional advection–diffusion equations.

Exploiting locality in sparse polynomial approximation of parametric elliptic PDEs and application to parameterized domains

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a highly nonlinear way on a parameter y ∈ [−1, 1]N. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove ℓp-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

Image Enhancement Using Bidimensional Empirical Mode Decomposition and Morphological Operations for Brain Tumor Detection and Classification.

The three steps of brain image processing – preprocessing, segmentation, and classification are becoming increasingly important in patient care. The aim of this article is to present a proposed method in the mentioned three-steps, with emphasis on the preprocessing step, which includes noise removal and contrast enhancement. The fast and adaptive bidimensional empirical mode decomposition and the anisotropic diffusion equation as well as the modified combination of top-hat and bottom-hat transforms are used for noise reduction and contrast enhancement. Fast C-means clustering with enhanced image is used to detect tumors and the tumor cluster corresponds to the maximum centroid. Finally, Ensemble learning is used for classification. The Figshare brain tumor dataset contains magnetic resonance images used for data selection. The optimal parameters for both noise reduction and contrast enhancement are investigated using a tumor contaminated with Gaussian noise. The results are evaluated against state-of-the-art results and qualitative performance metrics to demonstrate the dominance of the proposed approach. The fast C-means algorithm is applied to detect tumors using twelve enhanced images. The detected tumors were compared to the ground truth and showed an accuracy and specificity of 99% each, and a sensitivity and precision of 90% each. Six statistical features are retrieved from 150 enhanced images using wavelet packet coefficients at level 4 of the Daubechies 4 wavelet function. These features are used to develop the classifier model using ensemble learning to create a model with training and testing accuracy of 96.7% and 76.7%, respectively. When this model is applied to classify twelve detected tumor images, the accuracy is 75%; there are three misclassified images, all of which belong to the pituitary disease group. Based on the research, it appears that the proposed approach could lead to the development of computer-aided diagnosis (CADx) software that physicians can use as a reference for the treatment of rain tumor.

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

Entropy solutions to the fully nonlocal diffusion equations

AbstractWe consider the fully nonlocal diffusion equations with nonnegative ‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations.

Complete synchronization of two spirals by a messenger wave in a reaction diffusion system

Synchronization phenomena are ubiquitous in nature. They can be observed in physical, chemical, and biological systems. In the present study, we examine synchronization phenomena in chemical reaction–diffusion systems in the experimental Belousov–Zhabotinsky reaction. We study how two counter-rotating spirals pinned to unexcitable heterogeneities separated by a wall interact with a third free spiral of higher frequency. We found that the latter, which we call the messenger wave, synchronizes the two non-interacting pinned spirals. We also carried out numerical simulations of a model system with Barkley’s reaction–diffusion equations and found corroborating results.

Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.

Wet ball milling activated oyster shells for multifunctional slow-release compound fertilizer production

Slow-release fertilizers have been known to boost crop yields, but their high cost and slow degradation make them less practical for use. In this study, oyster shell (OS) was modified by mechanical ball milling with monosodium glutamate (MSG) wastewater as an activator to prepare activated oyster shell (AOS) with an average particle size of 32.7 nm. Compared to natural oyster shell (NOS), the AOS increased water-soluble Ca by 22 times, expanded the average desorption pore size by 155 %, enhanced the specific surface area by 9 times, and improved the total pore volume by 1587 %. AOS was mixed with traditional fertilizers and extruded to develop a new type of activated oyster-based slow-release compound fertilizer (AOF). Physical effects mainly controlled the slow-release ability of AOF. AOS formed vaterite with a large peak shape, high strength, and well-defined crystal structure, providing more binding sites for NH4+, HPO42−, K+, Ca2+ and Mg2+. In addition, AOS was rich in Ca-OH and Mg-OH, which increased the potential for ligand exchange with fertilizer ions, which further improved the chemical slow-release properties of AOF. The AOF continuously releases N, P, K, Ca, and Mg nutrients over a period exceeding 70 days. The nutrient release behavior in AOF was accurately described by the logistic equation, the Elovich equation, and the parabolic diffusion equation. AOF increased the dry weight of green plum by 190 % and the yield by 61 %. The AOF prepared in this study proved to be an innovative and environmentally sustainable slow-release fertilizer.

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.

3-D Modelling and Simulation of a Reservoir for Surfactant-Polymer Flooding Using Eclipse Software

Surfactant-polymer flooding is a tertiary enhanced oil recovery method used to recover oil that remained in the reservoir after the primary and secondary oil recovery mechanisms. Predicting the pressure in the reservoir is important for oil production as pressure changes with time. A suitable approach to achieve this task is to derive fluid flow equation based on the reservoir characteristics and solve them numerically which provide the solution to the mathematical fluid flow model (diffusivity equation). In this study, 3-D reservoir was modelled using Eclipse software. The fluid flow equations in a porous media were derived based on the simulated model and the reservoir conditions. Numerical solution using implicit formulation to solve the mathematical fluid flow model (diffusivity equation) was investigated by developing Python codes using Jupyter library to ascertain the pressure distribution for the reservoir and imported into Eclipse simulator. Simulation was carried out using surfactant-polymer and reservoir properties to determine the oil recovery. The results of the study showed that pressure increases with time as oil production continued, and water saturation decreased for the grid-cells of the reservoir. Waterflooding had oil recovery of 38.0% and water-cut of 59.0%, while surfactant flooding had oil recoveries of 42.0%, 46.5%, 49.0% and water-cut of 57.0%, 51.0%, 46.3%. In addition, polymer flooding had oil recoveries of 44.3%, 48.4%, 54.0% and water-cut of 50.0%, 45.0% and 33.0% respectively at different concentrations of 0.3%wt. 0.4%wt. and 0.5%wt.

Weak long‐range diffusion

AbstractWe study the spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.