Solution Of Diffusion Equation Research Articles

High-precision meshfree methods for solving 3D radiation diffusion problem in irregular spaces

Radiation diffusion is a significant phenomenon in inertial confinement fusion, geology and so on. Solving nonlinear radiation diffusion equation in irregular spaces is challenging. In this paper, the direct linearisation algorithm and the successive permutation iteration algorithm are constructed for solving 3D radiation diffusion equation, which are full-implicit discretization on time. The high-precision meshfree methods are provided by the Kansa's non-symmetric collocation method with the compactly supported radial basis function. Ultimately, the accuracy and efficiency of the presented algorithms are verified through the comparison of the fixed point, Newton and SOR methods in 3D numerical experiments. These novel meshfree methods not only avoid the complexity of grid generation, but also provide high-precision numerical solution for nonlinear radiation diffusion equation.

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.

Fractional diffusion equations interpolate between damping and waves

Abstract The behaviour of the solutions of the time-fractional diffusion equation,
based on the Caputo derivative, is studied and its dependence on the fractional
exponent is analysed. The time-fractional convection-diffusion equation is also solved
and an application to Pennes bioheat model is presented. Generically, a wave-like
transport at short times passes over to a diffusion-like behaviour at later times.

Exact solution of the nonlinear boson diffusion equation for gluon scattering

An exact analytical solution of the nonlinear boson diffusion equation is presented. It accounts for the time evolution toward the Bose–Einstein equilibrium distribution through inelastic and elastic collisions in the case of constant transport coefficients. As a currently interesting application, gluon scattering in relativistic heavy-ion collisions is investigated. An estimate of the time-dependent gluon-condensate formation in overoccupied systems through number-conserving elastic scatterings in Pb–Pb collisions at relativistic energies is given.

Thermal Spreading Resistance of Near Surface Localized Heating-Induced Size Effects in Semiconductors

Abstract Localized heating is encountered in various scenarios, including the operation of transistors, light-emitting diodes, and some thermal spectroscopy techniques. When localized heating occurs on a scale comparable to the mean free path of the dominant energy carriers, additional thermal resistance is observed due to ballistic effects. The main objective of this study is to find a relation between this resistance, problem geometry, and material thermal properties in situations involving localized heating. Models based on the solution of the Fourier heat diffusion equation and the gray phonon Boltzmann Transport Equation are solved simultaneously to calculate the additional thermal resistances that arise from localized heating. Subsequently, the results are analyzed to derive the desired relationship. It is noted that in the context of localized heating resistance, the effects of geometrical variables are non-linear and substantial, particularly when the Knudsen numbers for the boundary and heat source exceed certain thresholds. Specifically, when the Knudsen number for the heat source becomes comparable to 1 localized heating resistance is observed. However, when the Knudsen number based on heat source height and width surpasses 8 and 20, respectively the heat source behaves akin to a point source, no longer significantly affecting the localized heating resistance. At this juncture, the maximum resistance limit is reached.

Analytical and numerical study of diffusion propelled surface growth phenomena

To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

Quasi-static bending analysis of composite laminated cylindrical panels under hygrothermal conditions

The combined effects of temperature and humidity, termed hygrothermal, are included in the deformation analysis of laminated composite cylindrical panels based on first-order shear deformation theory (FSDT). The panel is under transverse mechanical load, and both steady-state and transient hygrothermal loads are considered. The problem is solved using the minimum potential energy theorem for different boundary conditions. The moisture profile through the thickness is obtained by analytical solution of Fick diffusion equation. The dependency of moisture diffusion coefficient on temperature is considered and the material temperature is assumed to be the same as the environmental temperature. Using a semi-analytical approach, displacement field of the panel subjected to evaluated profile of moisture in a specific time and thermo-mechanical loads is obtained. The accuracy and validation of the present solution are demonstrated by solving numerical examples and comparing them with the results available in the literature. Furthermore, new results are presented and effects of various important parameters such as panel geometry, edge boundary conditions, lamination layups and imposed moisture conditions on the deformation of the composite cylindrical panel are studied. Solved examples demonstrate that higher length-to-arc and radius-to-thickness ratios result in increased deflection and amplify the influence of mechanical load compared to hygrothermal effects on the overall deformation of the panel. In most case studies, the hygroscopic deformation observed at one-tenth of the saturation time exceeded half of the deformation observed after the saturation time, which shows the importance of transient hygroscopic analysis.

Spacetime symmetries and geometric diffusion

We examine relativistic diffusion through the frame and observer bundles associated with a Lorentzian manifold (M, g). Our focus is on spacetimes with a non-trivial isometry group, and we detail the conditions required to find symmetric solutions of the relativistic diffusion equation. Additionally, we analyze the conservation laws associated with the presence of Killing vector fields on (M, g) and their implications for the expressions of the geodesic spray and the vertical Laplacian on both the frame and the observer bundles. Finally, we present several relevant examples of symmetric spacetimes.

Existence of Mild Solutions to Delay Diffusion Equations with Hilfer Fractional Derivative

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that limt→∞⁡ln⁡‖u(t,⋅)‖pg(t)=0 in the almost surely sense where p∈R and the function g:R+→R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals.

Fundamental solution of the time-space bi-fractional diffusion equation with a kinetic source term for anomalous transport

The purpose of this paper is to study the fundamental solution of the time-space bi-fractional diffusion equation incorporating an additional kinetic source term in semi-infinite space. The equation is a generalization of the integer-order model ∂tρ(x,t)=∂x2ρ(x,t)-ρ(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\partial _{{t}} {\\rho (x,t)}} = {\\partial ^2_{{x}} {\\rho (x,t)}} - { \\rho (x,t)}$$\\end{document} (also known as the Debye–Falkenhagen equation) by replacing the first-order time derivative with the Caputo fractional derivative of order 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\alpha < 1$$\\end{document}, and the second-order space derivative with the Riesz-Feller fractional derivative of order 0<β<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\beta <2$$\\end{document}. Using the Laplace-Fourier transforms method, it is shown that the parametric solutions are expressed in terms of the Fox’s H-function that we evaluate for different values of 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} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}.

Time-fractional fabric to quantify non-Fickian diffusion in porous media: New vision from previous studies

The diffusion kinetics of different substances in porous media is a priori described by the second Fick's law of diffusion. Herein, we demonstrate that such a description sometimes yields erratic results. In contrast, the time-fractional diffusion equation is found to reflect the diffusion kinetics in the case of Fick's law failure. The analytic solutions of the time-fractional diffusion equation are based on the Mittag–Leffler function. At large times, it is shown that the utilization of the approximation of the Mittag–Leffler function for the large values of its argument for the description of the experimental data gives almost the same results as those obtained using the Mittag–Leffler function itself. The time-fractional diffusion is applied on a phenomenological basis. Nevertheless, we speculate that the deviations from Fick's law may be governed by strong adsorption of the diffusing species inside the small micropores of a porous material.

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

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 Schläfli 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.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

AbstractThe aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

A one-dimensional diffusive transport model to evaluate H2S generation in salt caverns hydrogen storage sites

Salt caverns are considered a potential hydrogen storage site given the ductility, tightness, and chemical inertness of halite. In addition, unlike porous media, salt caverns require less cushion gas and allow more cycles of injection and production within a year. However, hydrogen is an electron donor for many hydrogenotrophic microorganisms including halophilic sulfate-reducing microbes (SRM) and hydrogen sulfide (H2S) can be generated in a cavern once sulfate and hydrogen become available in the aqueous phase. As salt deposits are often associated with large amounts of impurities such as anhydrite and gypsum, their dissolution during the leaching phase can release the required sulfate ions for SRMs to produce H2S once hydrogen diffuses and dissolves in the brine at the bottom of the cavern. In this study, a one-dimensional diffusive transport model was developed using PHREEQC to simulate sulfate reduction and H2S generation in a cavern at the gas-brine interface. The solution of Fick's diffusion equation was used to calculate the hydrogen concentration at each level of the 1D model while the kinetic rates of reactions were derived from reported experimental and field site observations of brine samples taken from salt caverns. The results obtained indicated that hydrogen consumption at the gas-brine interface can trigger H2S and methane formations. This is followed by a reduction in sulfate and carbonate concentration and a slight increase in pH due to the reduction of protons in the solution. It was also observed that the presence of Fe2+ ions can reduce the amount of the hydrogen sulfide production while Ca2+ ions slightly favour sulfate reduction due to the production of methane. Although the kinetic reaction rate of the reactions in the model is subject to a certain degree of uncertainty, the workflow and results this study can serve as a guide for the future storage of hydrogen in salt caverns.

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitably good exponent range. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves as an admissible solution of the corresponding equation everywhere away from the point singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. Here we work in the whole Euclidean space.In this setting we show how the diffusion process distributes mass from the initial infinite singularity with different rates along the different space directions. Indeed, and up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences.If we consider the family of fundamental solutions for different masses, we prove that they all share the same universal tail behaviour (i.e., for large |x|) as the VSS. Namely, their tail is asymptotically convergent to the unique VSS tail. This means that the VSS partition formula holds also for the fundamental solutions at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.

Travelling wave solution of fourth order reaction diffusion equation using hybrid quintic hermite splines collocation technique

AbstractFourth order extended Fisher Kolmogorov reaction diffusion equation has been solved numerically using a hybrid technique. The temporal direction has been discretized using Crank Nicolson technique. The space direction has been split into second order equation using twice continuously differentiable function. The space splitting results into a system of equations with linear heat equation and non linear reaction diffusion equation. Quintic Hermite interpolating polynomials have been implemented to discretize the space direction which gives a system of collocation equations to be solved numerically. The hybrid technique ensures the fourth order convergence in space and second order in time direction. Unconditional stability has been obtained by plotting the eigen values of the matrix of iterations. Travelling wave behaviour of dependent variable has been obtained and the computed numerical values are shown by surfaces and curves for analyzing the behaviour of the numerical solution in both space and time directions.

Unique recovery of the initial state of distributed order time fractional diffusion equation

We use Phragm\'en-Lindel\"of-Liouville argument to prove the uniqueness for the determining the initial state of solution for the time fractional diffusion equation with distributed order derivative. Several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Effects of pore-fluid pressure on the motion of debris flows

ABSTRACT Pore-fluid pressure mediates the shear strength between soil particles and significantly controls the movement of debris flows. This study analyzes the effects by proposing a numerical model to solve the pore-fluid diffusion equation coupled to changes in surge motion of debris flows. The model was established by combining a Lagrangian method solving the depth-averaged one-dimensional equations for the debris-flow motion and a Fourier-series solution for the pore-fluid diffusion equation. Owing to the source term related to the change in surge depth, the Fourier-series solution shows that the excess pore-fluid pressure near the bed dissipates faster and exhibits a non-monotonic profile. The results of numerical calculations reveal that the diffusivity of excess pore-fluid pressure and the liquefaction condition have significant impacts on the depth profiles and velocities of debris-flow surges.

Application of The Homotopy Perturbation Method to the Neutron Diffusion Equation

The Homotopy Perturbation Method (HPM) has been shown to be effective in solving both linear and nonlinear differential equations in mathematics, making it useful in a wide range of applications in the fields of physics and engineering. In this study, the Homotopy Perturbation Method was applied to the neutron diffusion equation for a one-dimensional time-independent approach. The Laplace operator of the neutron diffusion equation was considered for Cartesian, spherical and cylindrical coordinates. The critical radius values obtained for three different systems were calculated for all possible values of the relevant material parameter B. The results show that the solution of the neutron diffusion equation is agree with the literature.