Search

This paper deals with a class of fractional 1-Laplacian diffusion equations with variable orders, proposed as a model for removing multiplicative noise in images. The well-posedness of weak solutions to the proposed model is proved. To overcome the essential difficulties encountered in the approximation process, we place particular emphasis on studying the density properties of the variable-order fractional Sobolev spaces. Numerical experiments demonstrate that our model exhibits favorable performance across the entire image. [ABSTRACT FROM AUTHOR]

We study a nonlocal abstract Ginzburg–Landau type equation. The equation includes variable coefficients with convolution terms and an abstract linear operator function in a Fourier-type Banach space . For sufficiently smooth initial data, assuming growth conditions for the operator and the coefficient , the existence and uniqueness of the solution and the -regularity properties are established. We obtain the existence and uniqueness of the solution, and the regularity of different classes of nonlocal Ginzburg–Landau-type equations by choosing the space and operator that occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]

Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.

The objective of the current article is to use the Banach fixed point theorem to present proofs for the existence and uniqueness theorems for a nonlinear time-fractional differential equation, namely, the linear and nonlinear, homogeneous and nonhomogeneous time-fractional diffusion equations. In addition, we explore exact solutions to nonlinear time fractional partial differential equations using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). To demonstrate the effectiveness of the suggested scheme, three examples are provided along with graphs and numerical tables to support our work. The outcomes demonstrate how effortless and effective the ANDM is for addressing fractional partial differential equations in multi-dimensional spaces, some of which we examined in this study as special cases.AMS Classifications 34A08, 65P99, 49J15, 35R11, 26A33, 74G10. [ABSTRACT FROM AUTHOR]

Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics. [ABSTRACT FROM AUTHOR]

The boundary conditions for describing the mass transfer of carbon from oxygen-free low-pressure atmospheres into steel during vacuum carburizing are analyzed. The effect of the content of carbide-forming alloying elements on the values of the carbon potential of low-pressure acetylene atmospheres is determined depending on the temperature of the process of vacuum carburizing. Temperature dependences of the carbon mass transfer coefficient are derived. [ABSTRACT FROM AUTHOR]

Abstract This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.

A nonlinear vertex-centered finite volume scheme that preserves the discrete maximum principle (DMP) is proposed for anisotropic diffusion equations on polygonal meshes. The new scheme is constructed from a vertex-centered linearity-preserving scheme combined with a nonlinear correction technique. The key ingredient is the introduction of the correction coefficient and the limiter. Particularly, the choice of the limiter is discussed and two new and effective limiters are proposed. Different from most existing nonlinear DMP-preserving schemes, the new nonlinear scheme does not rely on the nonlinear combination of linear flux approximations and requires no interpolation of the auxiliary unknowns. More important is that, it can be proved theoretically that the new scheme is symmetric and coercive on general polygonal meshes with star-shaped cells. Moreover, other theoretical results such as the DMP property and stability of the correction scheme are also presented. Numerical experiments demonstrate the accuracy, efficiency, and DMP property of the new scheme on general polygonal meshes with heterogeneous and anisotropic diffusion tensors. [ABSTRACT FROM AUTHOR]

Copyright of Bulletin of Ablai Khan KazUIRandWL: Series 'International Relations & Regional Studies' is the property of Kazakh Ablai Khan University of International Relations & World Languages and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a novel spectral scheme for the numerical solution of a two-dimensional time-fractional diffusion equation. The proposed scheme approximates the unknown function and its derivatives in space by finite Lucas and Fibonacci polynomial expansion, and time derivative/variable by L1 formula with θ-weighted difference scheme. Error estimates are derived and convergence is proved. Six test problems are then solved to verify the theory numerically. The obtained results confirm the low computational cost and better accuracy than known results in the literature.

In this paper, two well-known analytic approaches for solving diffusion equations are implemented. We propose a modified version of the homotopy perturbation method and Adomian decomposition methods utilizing the ZZ transform. In this sense, the fractional derivative of the Atangana–Baleanu derivative is provided. In addition, concrete examples are as well provided to demonstrate the precision of the proposed methodologies. The proposed solution is observed to have the desired rate of convergence towards the exact answer. The key advantage of the proposed method is the small amount of calculations. To demonstrate the validity of the suggested methods, we give graphical representation of analytical and exact solutions that are in close contact. [ABSTRACT FROM AUTHOR]

In this paper, a new class of spectral volume (SV) methods are proposed, analyzed, and implemented for diffusion equations, with the viscous flux taken as an interior penalty or direct discontinuous Galerkin formulation. The control volumes are constructed by using four kinds of special points (including Legendre–Gauss, Legendre–Gauss–Lobatto, right Legendre–Gauss–Radau and left Legendre–Gauss–Radau points) in subintervals of the underlying meshes, which leads to four different SV schemes. A framework for the stability analysis and error estimates of the four SV schemes is established. In particular, the influence of the choice of the parameters in the numerical fluxes on the convergence rate and the optimal choices of coefficients for each SV scheme are discussed and provided. Numerical experiments are presented to demonstrate the stability and accuracy of the four SV schemes for both linear and nonlinear diffusion equations. [ABSTRACT FROM AUTHOR]

In this work, we find analytical solutions to the Chavy-Waddy–Kolokolnikov equation, a continuum approximation for modeling aggregate formation in bacteria moving toward the light, also known as phototaxis. We used three methods to obtain the solutions, the generalized Kudryashov method, the e − R (ξ) -expansion, and exponential function methods, all of them being very efficient for finding traveling wave-like solutions. Findings can be classified into the case where the nonlinear term can be considered a small perturbation of the linear case and the regime of instability and pattern formation. Standing waves and traveling fronts were also found among the physically interesting cases, in addition to recovering stationary spike-like solutions. [ABSTRACT FROM AUTHOR]

In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag–Leffler function. [ABSTRACT FROM AUTHOR]

This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the fragmentation of the habitat on biodiversity. A simple neutral model for the evolution of abundances in a vegetal metacommunity is introduced. Migration between the communities is explicitely modelized in a deterministic way, while the reproduction process is dealt with using Wright-Fisher models, independently within each community. The large population limit of the model is considered. The hydrodynamic limit of this split-step method is proved to be the solution of a partial differential equation with a deterministic part coming from the migration process and a diffusion part due to the Wright-Fisher process. Finally, the diversity of the metacommunity is adressed through one of its indicators, the mean extinction time of a species. At the limit, using classical comparison principles, the exchange process between the communities is proved to slow down extinction. This shows that the existence of corridors seems to be good for the biodiversity.

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper we prove existence (and summability properties) of solutions for the following elliptic system$ \left\{ \begin{array}{cl} -{\rm{div}}(A(x)\,{\nabla} u) + u^{{\lambda}} = -{\rm{div}}(u^{{\lambda}} \, M(x)\,{\nabla}\psi) + f(x)\,, & {\rm{in }}\; \Omega , \\ -{\rm{div}}(M(x)\,{\nabla}\psi) = u^{\rho}\,, & {\rm{in }}\; \Omega , \\ u = 0 = \psi & \;{\rm{on}}\; \partial\Omega , \end{array} \right. $under some assumptions on $ {\lambda} > 0 $, $ \rho > 0 $ and $ f(x) $ in $ L^{{m}}(\Omega) $, $ m \geq 1 $. 'Return of the Patriarca' (see [3]) [ABSTRACT FROM AUTHOR]

The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest's method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs. [ABSTRACT FROM AUTHOR]

Investigation of floods as one of the most common natural destructive phenomena is very important. Due to the problem of deficiency and defect of hydraulic data in most hydrometric stations in the country, flood hydrograph simulation by hydrological and hydraulic models is helpful. There are several hydrological and hydraulic models to simulate flood hydrographs. The present study has investigated the performance of Advection-Diffusion relationships of 36 flood phenomena in 5 study reaches of Kashkan river and its tributary. Physical properties related to the studied periods and hydrometery data related to each flood phenomenon, including discharge and scale of the input and output station, were used to determine the Manning roughness coefficient and flood simulation. Evaluation of obtained hydrographs indicates the acceptable accuracy of the model in term of simulating values of flood trends and discharge. In order to evaluate the accuracy of the model results, different coefficients such as: Nash-Sutcliffe, RMSE, Volumetric Efficiency, Kling-Gupta and R correlation coefficient were calculated. A value above 0.5 for the Nash-Sutcliffe coefficient in 29 flood phenomena out of 36 studied phenomena, and an acceptable range of other coefficients for all studied flood phenomena, indicates the verifiable results of flood hydrograph simulation using simplified propagation methods at reach scale for studied watershed. Therefore, Advection-Diffusion relationships can be used in similar watersheds to the study area to simulate floods.

In this paper, we propose a new nonlinear conservative and maximum-principle-preserving finite volume element method for heterogeneous anisotropic diffusion equations on triangular or strictly convex quadrilateral meshes. First, we decompose the numerical flux of the standard finite volume element method on each dual edge into two parts: main part having a two-point-flux structure, and residual part owning a multi-point-flux structure. Then, according to the sign of the residual part and the distribution of the local extremum, the residual part is modified to a two-point-flux structure with a nonlinear term. The resulting nonlinear scheme not only inherits conservation and accuracy of original scheme, but also possesses a maximum-preserving structure without extra restrictions on the meshes and diffusion coefficients. Several numerical experiments verify that our nonlinear scheme has an optimal convergence order and satisfies the discrete maximum principle. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space R n , where n ≥ 2 , assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as | x | → ∞ . Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference w between the solution and the self-similar profile, but also an antiderivative W obtained by solving a linear elliptic problem which involves w as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero. [ABSTRACT FROM AUTHOR]

In this work we investigate the well-posed for diffusion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators and the corresponding quasi-geostrophic model with drift D are investigated. Examples on SU (2) for diffusion problems with fractional diffusion are analysed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fourth‐order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one‐dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one‐dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves. [ABSTRACT FROM AUTHOR]

In this paper we consider a numerical scheme for the treatment of an integro-differential equation. The latter represents the formulation of a nonlocal diffusion type equation. The discretization procedure relies on the application of the line method. However, quadrature formulae are needed for the evaluation of the integral operator. They are based on generalized Bernstein polynomials. Numerical evidence shows that the proposed method is a suitable and reliable approach for the problem. [ABSTRACT FROM AUTHOR]

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotopy analysis generalized transform method is suggested to obtain the solutions of fractional order diffusion equations. The results of this research are demonstrated in graphical and tabular descriptions. This study shows that the applied computational technique is very effective, accurate, and beneficial for managing such kind of fractional order nonlinear models occurring in oil pollution. [ABSTRACT FROM AUTHOR]

In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with a computational approach based on the Jacobi wavelets for linear and nonlinear time-fractional diffusion equations of distributed order. We derive the Jacobi wavelet operational vector for the Riemann–Liouville fractional integral operator. By applying this operational vector via the Gauss–Legendre quadrature formula and collocation method in our approach, the problems can be reduced to systems of linear or nonlinear algebraic equations which can be solved by the Newton method. The convergence and some error bounds of the expressed method are theoretically investigated. In addition, the presented method is implemented for six test problems. Comparisons between the obtained numerical results and other methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We generalize the concept of diffusion equations on weighted graphs, which is also known as ω-diffusion equations, to study fractional order diffusion equations on weighted graphs. More precisely, we replace the ordinary first order derivative in time by a fractional derivative of order α in the sense of Riemann-Liouville and Caputo fractional derivatives. We prove the existence of solutions of fractional order diffusion equations on graphs using the concept of α-exponential matrix and illustrate the solutions through numerical simulation in various examples. [ABSTRACT FROM AUTHOR]

During the recent years all crop species achieved the best possible field distribution so a high yield is to be expected. In this paper the solutions of two different diffusion equations are determined, which describe the optimal distribution of cereal grains over a field. Therefore, there are two different partial differential equations of cereal seed distribution-distinction is made between the longitudinal spacing (seeds in a row), and transverse distance (between two rows), as well as the sowing depth. In particular, closed forms of solutions are derived in each case. Although the result of the diffusion equation with respect to the distribution of the lateral seed distance of two adjacent rows is already known, a new solving method is presented in this paper. By this method, the partial differential equation is reduced to an ordinary one, which is easier to solve. In this paper it is shown that the distribution of lateral resp. longitudinal and in-depth wheat seed distances is achieved by a normal Gauss function resp. a log-normal function. Furthermore, it is demonstrated that the fitting functions of the best experimental results of wheat seeding distributions are particular solutions of the individual differential equations. Normal Gauss function describes lateral distribution with R2 = 0.9325; RSME = 1.2450, and log-normal function describes longitudinal distribution with R2 = 0.9380; RSME = 1.4696 as well as depth distribution with R2 = 0.9225; RSME = 2.0187.

In this paper we shall develop a face-based local tangential lifting method to handle the problem of discrete conservation laws for diffusion equations and solve Cahn-Hilliard equations over curved surfaces. To do so, we need to give a new discrete approximation of the Laplace-Beltrami operators on functions over curved surfaces. Our method is a simple effective numerical method for solving diffusion equations on curved surfaces. The key idea of our face-based local tangential lifting method is to approximate locally the underlying surface and functions with respect to faces in the discrete model of the surface by using the centroid method. Indeed, our algorithm is a generalized finite difference method and an intrinsic geometry method. It gives a natural way to approximate directly the Laplace-Beltrami operators on a triangular mesh. Discrete conservation laws for diffusion equations on triangular meshes are also discussed. [ABSTRACT FROM AUTHOR]

The main purpose of this manuscript is to present a new approximation method to find the numerical solution of partial differential equations with Caputo fractional derivatives. We have also included the stability analysis of the proposed technique. To illustrate the effectiveness and reliability of the method, we have examined the method with several numerical examples. The high computational efficiency of the proposed method can be seen by observing the accurate approximate solutions obtained from the process. The suggested procedure is a potent tool and it can be used to solve other similar problems in modeling real-world problems.

The paper considers an approach to formulating and finding solutions to scalar diffusion equations taking into account the fuzzy perception of the timeflow in the processes of propagation of physical substances and information flows. The description of an unconventional method of time accounting is based on the use of fuzzy structured numerical sets, which is based on the principle of forming a fuzzy original with its subsequent replication on the numerical axis. The formalization of the fuzzy original is to define two functions parametrically set on the interval [0, 1] that determine the rate of the subjective perception of a time unit. The diffusion equation describing the information dissemination in the social environment is proposed and analyzed. A solution is obtained that determines the state of the propagation process taking into account the "fast" and "slow" time flows. The proposed methodology allows us to formalize the problem of fuzzy description taking into account the subjective perception of time accounting when solving various dynamics problems. [ABSTRACT FROM AUTHOR]

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the corresponding macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain a macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis on the finite-difference scheme and the mesoscopic MRT-LB model, and find that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis. • A general fourth-order MRT-LB model is developed for the two-dimensional diffusion equations. • The macroscopic finite-difference scheme of the MRT-LB model is obtained theoretically. • The accuracy and stability of the MRT-LB model and finite-difference scheme are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, fractional optimal control problem for two-dimensional coupled diffusion system with final observation is investigated. The fractional time derivative is considered in Atangana–Baleanu sense. Constraints on controls are imposed. First, by means of the classical control theory, the existence and uniqueness of the state for these systems is proved. Then, the necessary and sufficient optimality conditions for the fractional Dirichlet problems with the quadratic performance functional are derived. Finally we give some examples to illustrate the applicability of our results. The optimization problem presented in this paper constitutes a generalization of the optimal control problem of diffusion equations with Dirichlet boundary conditions considered in recent papers to coupled systems with Atangana–Baleanu time derivatives.

We present a backward diffusion flow (i.e., a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a later time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given, including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch-Tung-Striebel smoothing equations in the linear setting. [ABSTRACT FROM AUTHOR]