Diffusion Equation Research Articles

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Removal of the bemacid blue E-TL using granular activated carbon from abundant biomass: adsorption isotherm, thermodynamic and kinetic studies

In this work, we were interested in the adsorption of an acid dye, Bemacid Blue E-TL by granular activated carbon based on olive stones. To shed light on the adsorption process, batch experiments were performed to study the effect of operating parameters on the adsorption process such as equilibrium time, GAC dose, pH, initial dye concentration, and temperature. The experimental results showed that the adsorption of the dye Bemacid Blue E-TL by GAC based on olive stones depends on the equilibrium time at 8 hours, the pH is 2 of the solution and the dose is 5 g.L-1. To explain the adsorption equilibrium, the experimental values were examined by Langmuir, Freundlich, and Temkin models. The equilibrium is perfectly described by the Langmuir model whose correlation coefficient is higher than 0.99. The maximum retained quantity of the Bemacid Blue E-TL is 55.6 mg.g-1. Different kinetic models such as pseudo-first order, pseudo-second order, and the intraparticle diffusion equation were used to evaluate the adsorption kinetics. The results of the kinetic modeling showed that the pseudo-second order fitted the data well, with a high coefficient of determination (R2 > 0.99) and intra-particle diffusion is not the only rate-limiting step. The thermodynamic parameter values enthalpy (ΔH°), entropy (ΔS°), and energy change (ΔG°) show that the adsorption processes are endothermic and spontaneous. The various results obtained are promising and encouraging to consider a more comprehensive study with the objective of showing that the adsorbent chosen for this study is effective and could be used as a low-cost adsorbent for acid dye removal.

Inelastic Scattering of Dark Matter with Heavy Cosmic Rays

We investigate the impact of inelastic collisions between dark matter (DM) and heavy cosmic ray (CR) nuclei on CR propagation. We approximate the fragmentation cross-sections for DM-CR collisions using collider-measured proton-nuclei scattering cross-sections, allowing us to assess how these collisions affect the spectra of CR boron and carbon. We derive new CR spectra from DM-CR collisions by incorporating their cross-sections into the source terms and solving the diffusion equation for the complete network of reactions involved in generating secondary species. In a specific example with a coupling strength of b χ = 0.1 and a DM mass of m χ = 0.1 GeV, considering a simplified scenario where DM interacts exclusively with oxygen, a notable modification in the boron-to-carbon spectrum due to the DM-CR interaction is observed. Particularly, the peak within the spectrum, spanning from 0.1 to 10 GeV, experiences an enhancement of approximately 1.5 times. However, in a more realistic scenario where DM particles interact with all CRs, this peak can be amplified to twice its original value. Utilizing the latest data from AMS-02 and DAMPE on the boron-to-carbon ratio, we estimate a 95% upper limit for the effective inelastic cross-section of DM-proton as a function of DM mass. Our findings reveal that at m χ ≃ 2 MeV, the effective inelastic cross-section between DM and protons must be less than O(10−32)cm2 .

Mathematical modeling of hypoxia and adenosine to explore tumor escape mechanisms in DC-based immunotherapy

Identifying and controlling tumor escape mechanisms is crucial for improving cancer treatment effectiveness. Experimental studies reveal tumor hypoxia and adenosine as significant contributors to such mechanisms. Hypoxia exacerbates adenosine levels in the tumor microenvironment. Combining inhibition of these factors with dendritic cell (DC)-based immunotherapy promises improved clinical outcomes. However, challenges include understanding dynamics, optimal vaccine dosages, and timing. Mathematical models, including agent-based, diffusion, and ordinary differential equations, address these challenges. Here, we employ these models for the first time to elucidate how hypoxia and adenosine facilitate tumor escape in DC-based immunotherapy. After parameter estimation using experimental data, we optimize vaccination protocols to minimize tumor growth. Sensitivity analysis highlights adenosine’s significant impact on immunotherapy efficacy. Its suppressive role impedes treatment success, but inhibiting adenosine could enhance therapy, as suggested by the model. Our findings shed light on hypoxia and adenosine-mediated tumor escape mechanisms, informing future treatment strategies. Additionally, identifiability analysis confirms accurate parameter determination using experimental data.

A hybrid room acoustic modeling approach combining image source, acoustic diffusion equation, and time-domain discontinuous Galerkin methods

In this paper a hybrid model is introduced that constructs a broadband room impulse response using a geometrical (image source method) and a statistical method (acoustic diffusion equation) for the high-frequency range, supported by a wave-based method (time-domain discontinuous Galerkin method) for the low-frequency range. A crucial element concerns the construction of the high-frequency impulse response where a transition from a predominantly specular (image source) to a predominantly diffuse sound-field (diffusion equation) is required. To achieve this transition an analytical envelope is introduced. A key factor is the room-averaged scattering coefficient which accounts for all scattering behavior of the room and determines the speed of transition from a specular to a non-specular sound-field. To evaluate its performance, the model is compared to a broadband wave-based solver for two reference scenarios. The hybrid model shows promising results in terms of reverberation time (T20), center time (Ts) and bass-ratio (BR). Aspects such as the used geometrical complexity, the ‘room-averaged’ scattering coefficients, and other model simplifications and assumptions are discussed.

A numerical validation between the neutron transport and diffusion theories for a spatial kinetics problem

In this paper, a comparative analysis of numerical results of the neutron transport and diffusion theories for steady-state and transient multigroup problems is presented. The neutron transport equation is known as the one that best describes the behavior of the neutron population in a nuclear reactor. However, due to the difficulty of working with its complete form, other models are considered as approximations to this equation. One such approximation is the neutron diffusion equation, which uses the Fick's Law. It is well known, however, that the diffusion model may not work well under specific conditions. The objective of this work is to establish a quantitative comparison of numerical results obtained for the K dominant eigenvalue and for the scalar fluxes from the two theories and to analyze the influence of the model on the results.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

Abstract In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra-Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schl"{a}fli integral of the Volterra-Bessel function. In this respect, we study the asymptotic analysis of the Volterra-Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory.

Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method

Solving Time-fractional Order Radon Diffusion Equation in Water by Finite Difference Method

Layered Patterns in Reaction–Diffusion Models with Perona–Malik Diffusions

In this paper we deal with a reaction–diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona–Malik’s type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria ±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 1$$\\end{document}, described by a parameter θ>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta >1$$\\end{document}. If θ∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta \\in (1,2)$$\\end{document}, we prove existence of steady states oscillating (and touching) ±1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 1$$\\end{document}, called compactons, while in the case θ=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta =2$$\\end{document} we prove the presence of metastable solutions, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for θ>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta >2$$\\end{document}, solutions with an unstable transition layer structure persist only for an algebraically long time.

Discrete Boltzmann model with split collision for nonequilibrium reactive flows

Abstract A multi-relaxation-time discrete Boltzmann model (DBM) with split collision is proposed for both subsonic and supersonic compressible reacting flows, where chemical reactions take place among various components. The physical model is based on a unified set of discrete Boltzmann equations that describes the evolution of each chemical species with adjustable acceleration, specific heat ratio, and Prandtl number. On the righ-hand side of discrete Boltzmann equations, the collision, force, and reaction terms denote the change rates of distribution functions due to self- and cross-collisions, external forces, and chemical reactions, respectively. The source terms can be calculated in three ways, among which the matrix inversion method possesses the highest physical accuracy and computational efficiency. Through Chapman-Enskog analysis, it is proved that the DBM is consistent with the reactive Navier-Stokes equations, Fick's law and Stefan-Maxwell diffusion equation in the hydrodynamic limit. Compared with the one-step-relaxation model, the split collision model offers a detailed and precise description of hydrodynamic, thermodynamic, and chemical nonequilibrium effects. Finally, the model is validated by six benchmarks, including multicomponent diffusion, mixture in the force field, Kelvin-Helmholtz instability, flame at constant pressure, opposing chemical reaction, and steady detonation.

Analysis of pipeline leakage in unsaturated stratum: A new seepage-diffusion model

Accurate prediction of the spread of fluids leaking from underground pipelines is crucial for risk assessment. In this study, we developed a method for predicting the diffusion range of pipeline seepage fluids within an unsaturated stratum. First, we derived a seepage–diffusion equation based on the generalized Darcy’s law and subsequently analyzed the mechanism of fluid diffusion in the unsaturated stratum. We then constructed a model for the seepage–diffusion of a pipeline leakage fluid, incorporating variables such as the saturation, permeability coefficient, diffusion pressure, and diffusion distance of soil microelements across various time intervals, using a stepwise algorithm in tandem with the Green–Ampt model and VG–Mualen permeability coefficient function. We investigated the influence of fluid self-gravity, saturated permeability coefficient, initial saturation, and intra-pipe pressure on the diffusion distance of the pipeline leakage fluid in an unsaturated stratum, focusing on specific cases. The results indicate that an increase in the saturated permeability coefficient, initial saturation, and intra-pipe pressure leads to an increase in the fluid diffusion distance. A simulation test was conducted to validate the proposed seepage–diffusion model. The findings of this study can be employed to predict the diffusion range of pipeline leakage fluids in various formation types, providing a vital foundation for pipeline leakage accident management.

A robust numerical scheme for solving Riesz-tempered fractional reaction–diffusion equations

The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this equation numerically is challenging due to the need to discretize complicated integral operators which increase the computational costs. These complexities are exacerbated by nonlinear source terms, nonsmooth data and irregular domains. In this study, we propose a second order Exponential Time Differencing Finite Element Method (ETD-RDP-FEM) to efficiently solve nonlinear FDE, posed in irregular domains. This approach discretizes matrix exponentials using a rational function with real and distinct poles, resulting in an L-stable scheme that damps spurious oscillations caused by non-smooth initial data. The method is shown to outperform existing second-order methods for FDEs with a higher accuracy and faster computational time.

A simple method of shape transformation using the modified Gray–Scott model

In this paper, based on the original Gray–Scott model, we propose a modified Gray–Scott model by introducing a target term into the reaction–diffusion equations. We apply this modified model in the context of shape transformation problems. To expedite the process from the source shape to the target shape, we utilize the explicit Euler method to solve our proposed modified Gray–Scott model, making our approach simpler and more efficient. To validate the feasibility of our method, we conduct simulation experiments in both two-dimensional (2D) and three-dimensional (3D) spaces. By progressing through experiments of increasing complexity, we demonstrate the natural effectiveness of our simulation method as a viable approach for shape transformation. To demonstrate the efficiency of the method, we provide the runtime consumed by the simulated shape transformation experiment. Additionally, to assess the correspondence between the ground truth values of the target shape and the simulated results, we calculate the corresponding area change rate and volume change rate in 2D and 3D spaces to prove that our proposed method can effectively transform into the target shape.

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

Transport of energetic particles in turbulent space plasmas: pitch-angle scattering, telegraph, and diffusion equations

Introduction: In this article, we revisit the pitch-angle scattering equation describing the propagation of energetic particles through magnetized plasma. In this case, solar energetic particles and cosmic rays interact with magnetohydrodynamic turbulence and experience stochastic changes in the pitch-angle. Since this happens over an extended period of time, a pitch-angle isotropization process occurs, leading to parallel spatial diffusion. This process is described well by the pitch-angle scattering equation. However, the latter equation is difficult to solve analytically even when considering special cases for the scattering coefficient.Methods: In the past, a so-called subspace approximation was proposed, which has important applications in the theory of perpendicular diffusion. Alternatively, an approach based on the telegraph equation (also known as telegrapher’s equation) has been developed. We show that two-dimensional subspace approximation and the description based on the telegraph equation are equivalent. However, it is also shown that the obtained distribution functions contain artifacts and inaccuracies that cannot be found in the numerical solution to the problem. Therefore, an N-dimensional subspace approximation is proposed corresponding to a semi-analytical/semi-numerical approach. This is a useful alternative compared to standard numerical solvers.Results and Discussion: Depending on the application, the N-dimensional subspace approximation can be orders of magnitude faster. Furthermore, the method can easily be modified so that it can be used for any pitch-angle scattering equation.

An α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Robust and Second-Order Accurate Scheme for a Subdiffusion Equation

We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document}. The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank–Nicolson scheme in the classical diffusion case, that is, as α→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 1$$\\end{document}. Using a novel approach, we show that the proposed scheme is α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-robust and second-order accurate in the L2(L2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(L^2)$$\\end{document}-norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end.

Electrically active defects induced by thermal oxidation and post-oxidation annealing of n-type 4H-SiC

In this work, we have performed a detailed study of the defects created in the bulk of 4H-SiC after thermal oxidation and post oxidation annealing using deep level transient spectroscopy and minority carrier transient spectroscopy (MCTS). The study reveals the formation of several shallow and deep level majority carrier traps in the bandgap. The ON1 (EC−0.85 eV), ON2a (EC−1.05 eV), and ON2b (Ec−1.17 eV) levels are the most dominant and are observed across all the samples (EC denotes the conduction band edge). Three shallow levels Ti(k) (EC−0.17 eV), E0.23 (EC−0.23 eV), and C1/2 (EC−0.36/0.39 eV) are observed in the samples. For most of the majority carrier defects, the highest concentration is observed after an NO anneal at 1300°C. This behavior is sustained in the depth profile measurements where the defect concentration after the NO anneal at 1300°C is significantly higher than for the rest of the samples. The origin of most of the majority carrier defects has been attributed to C interstitial injection from the interface during thermal oxidation and annealing. MCTS measurements reveal two prominent minority carrier traps, labeled O0.17 (EV+0.17 eV) and B (EV+0.28 eV), where the concentration of O0.17 is independent of annealing parameters while the concentration of the B level increases after the NO anneal (EV denotes the valence band edge). Furthermore, the depth profiles of the defects are used to evaluate their diffusion parameters by solving the diffusion equation to fit the experimental profiles. The defect concentrations decay exponentially with depth, which evidences that the defects were created at or near the SiO2–SiC interface and migrate toward the bulk during oxidation and post-oxidation annealing.

Effect of radius-dependent diffusion behavior of various gold nanoparticles on photothermal therapy

Among the various anti-cancer treatments, photothermal therapy (PTT) is gaining traction as it is a non-invasive treatment. PTT is a treatment technique involving the use of a laser to raise the temperature of the target tumor until it dies. In this study, the effects of PTT under various conditions of squamous cell carcinoma (SCC) occurring in the skin were numerically analyzed and optimized. Gold nanoparticles (AuNPs) with different radii were injected into the center of the SCC. Subsequently, the diffusion behavior of the AuNPs was analyzed to calculate the distribution area of the AuNPs that changed over time. Furthermore, at each elapsed time point after injection, the temperature distribution in the tissue was calculated, as treatment was performed using varying laser intensities. The diffusion coefficient of AuNPs was calculated using the Stokes–Einstein equation, and diffusion behavior of AuNPs in biological tissues was analyzed using the convection–diffusion equation. Additionally, temperature distribution was analyzed using the Pennes bioheat equation. The effect of PTT under each condition was quantitatively analyzed using apoptotic variables. As a result, As the radius of AuNPs increased, the optimal treatment start time was derived as 2 h, 8 h, 8 h, and 12 h, respectively, and the laser intensity at that time was derived as 0.44 W, 0.46 W, 0.42 W, and 0.42 W, respectively. The study findings will provide reference for the optimization of the efficacy of PTT.

Recovering the temperature distribution for multi-term time-fractional sideways diffusion equations

Recovering the temperature distribution for multi-term time-fractional sideways diffusion equations