Solution Of Diffusion Equation Research Articles

Feynman-Kac formula for tempered fractional general diffusion equations driven by TFBM

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u ( t , x ) = L u ( t , x ) + u ( t , x ) B ˙ H , λ ( t ) , ( t , x ) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ ( 0 , 1 ) and tempered parameter η > 0, 𝔏 is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by P t f(x): = 𝔼 x [f(X t )], and { B H , λ ( t ) } t ≥ 0 is a TFBM with Hurst index H ∈ ( 1 2 , 1 ) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator D β, η (t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM B H, λ (t) and the inverse tempered β-stable subordinator D β, η (t), we show that the Feynman-Kac representation u ( t , x ) = E D [ P D β , η ( t ) f ( x ) e ∫ 0 t ∫ 0 t δ ( s − r ) d D β , η ( s ) d B H , λ ( r ) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator D β, η (t) and the novel estimate of stochastic integrals with respect to B H, λ (t) and D β, η (t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution.

Comparison of formulas derived from Fourier series solution of diffusion equation

The first Fourier series solution satisfying the boundary condition of zero concentration at the diffusion distance was derived by assuming that the solution consists of a steady-state solution and a transient solution. The first total amount which has passed through the first layer surface was derived from the first Fourier series solution. When the diffusion distance is large, the exponential term in the first total amount cannot be zero. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method, the plot of the first total amount against time was in good agreement with the quadratic regression curve. The second Fourier series solution satisfying the boundary condition of constant concentration at the diffusion distance was derived by assuming that the diffusion distance is the thickness of the first layer. The second total amount which has passed through the second layer surface was derived from the second Fourier series solution and is a linear function of time because the exponential term is zero due to the very small diffusion distance, the thickness of the first layer. The plot of the second total amount against time was in good agreement with the linear regression line. The time-lag formula was derived from the second total amount. The diffusion coefficients determined from the second total amount are much larger than those of the other three types. All four types showed excellent linearity in Arrhenius plot, but the activation energy for the time-lag formula is somewhat greater than the rest.

Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion

It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).

Hybrid finite element and laplace transform method for efficient numerical solutions of fractional PDEs on graphics processing units

Fractional Partial Differential equations (FPDEs) are essential for modeling complex systems across various scientific and engineering areas. However, efficiently solving FPDEs presents significant computational challenges due to their inherent memory effects, often leading to increased execution times for numerical solutions. This study proposes a highly parallelizable hybrid computational approach that combines the Finite Element Method (FEM) for spatial discretization with Numerical Inversion of the Laplace Transform (NILT) for time-domain solutions, optimized for execution on Graphics Processing Units (GPUs). The NILT method’s high parallelizability, stemming from the independence of its series terms, combined with the robust spatial discretization provided by FEM, enables the efficient and accurate solution of FPDEs on GPUs, demonstrating substantial performance improvements over traditional CPU-based implementations. We observe a generalized pattern in execution time behavior that accounts for both the number of nodes and the number of NILT terms. Specifically, execution time scales quadratically with the number of nodes, while showing only a logarithmic marginal increase with the number of NILT terms These behaviors not only enables consistent performance assessment but also highlights potential areas for algorithm optimization. Validation against exact solutions of fractional diffusion and wave equations, employing Caputo, modified Caputo-Fabrizio, and modified Atangana-Baleanu derivatives, demonstrates the accuracy and convergence of the hybrid FEM-NILT method. Notably, the exact solutions of wave equation are novel in literature. The results highlight the method’s potential for enabling high-precision, large-scale simulations in fractional calculus applications, thereby advancing computational capabilities and efficiency in the field.

On multidimensional exact solutions of the nonlinear diffusion equation of the pantograph type with variable delay

We consider a multidimensional pantograph-type nonlinear diffusion equation with a linearly increasing time delay and scaling with respect to spatial variables in the source (sink). It is proposed to construct exact solutions by the reduction method using two ansatzes with a quadratic dependence on spatial variables. The dependence of the solution on spatial variables is found from a system of algebraic equations, and the dependence on time is found from a system of ordinary differential equations with a linearly increasing delay of the argument. A number of examples of exact solutions are given, both radially symmetric and anisotropic with respect to spatial variables.

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.

A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation

A class of partial differential equations with fractional derivatives in the spatial variables are called space-fractional diffusion equations. They can be applied to simulate anomalous diffusion, in which the classical diffusion equation does not accurately describe how a plume of particles disperses. Analytically solving fractional diffusion equations can be problematic due to the typically complex structures of fractional derivative models. Hence, this study proposes the utilisation of a satisfier function in combination with the Ritz method to effectively address fractional diffusion equations in the Caputo sense. By employing this approach, the equations are transformed into an algebraic system, so facilitating their solution and providing a numerical result. This method can achieve a high level of accuracy in solving the Caputo fractional diffusion equations by utilising only a small number of terms from the shifted Legendre polynomials in two variables. The precision and effectiveness of our approach may be evaluated, as it yielded dependable approximations of the solutions.

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

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.

Analytical Solution of Fractional Order Diffusion Equations Using Iterative Laplace Transform Method

In this present article, by using the Iterative Laplace Transform Method (ILTM), the diffusion equation of fractional order is solved. The ILTM, which works as a combination of two methods, the iterative method and the other is the Laplace transform method, is applied to several diffusion equations to obtain analytical solutions. The proposed method gives the closed-form of series solutions in terms of the Mittag-Leffler function, which is a queen of functions in fractional calculus. The main aim of this work is to present a simple but reliable algorithm for the solution of diffusion equations of the multi-dimensional type, which clearly describes the materials of density dynamics in the diffusion process. The results obtained by using the ILTM approach indicate that this approach is attractive computationally and implemented easily. Due to its straightforward approach and comfortable way of solving problems, the ILTM can be utilized to solve nonlinear fractional problems in various applied and engineering sciences.

Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.

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.

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.

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.

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.

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}.