Solution of nonlinear time fractional order reaction advection diffusion equation using neural network approach

Correction to: A nonuniform and fast two-grid L1-FEM algorithm for nonlinear multi-term time-fractional diffusion equation and its robust error analysis

We formulate and prove new Aronson-Bénilan and Li-Yau type gradient estimates for positive solutions to nonlinear slow diffusion equations. The framework is that of a smooth metric measure space (i.e., a weighted manifold) and the estimates make use of a range of Harnack quantities with suitable time-variable coefficients. The proofs exploit the intricate relation between geometry, nonlinearity and dynamics of the equation and the results extend, unify and improve various earlier estimates on slow diffusion equations. A number of important corollaries and implications, notably, to parabolic Harnack inequalities and global bounds are presented and discussed.

On the explicit solution for some nonlinear diffusion equations in heterogeneous porous media

Releasing capsules are widely employed in biomedical applications as smart carriers of therapeutic agents, including drugs and bioactive compounds. Such delivery vehicles typically consist of a loaded core, enclosed by one or multiple concentric coating strata. In this work, we extended existing mechanistic models to account for such multi-strata structures, including possible concurrent erosion of the capsule itself, and we characterized the release kinetics of the active substance into the surrounding medium. We presented a computational study of drug release from a spherical microcapsule, modeled through a non-linear diffusion equation incorporating radial asymmetric diffusion and space- and time-discontinuous coefficients, as suggested by the experimental data specifically collected for this study. The problem was solved numerically using a finite volume scheme on a grid with adaptive spatial and temporal resolution. Analytical expressions for concentration and cumulative release were derived for all strata, enabling the exploration of parameter sensitivity-such as coating permeability and internal diffusivity-on the overall release profile. The resulting release curves provide mechanistic insight into the transport processes and offer design criteria for achieving controlled release. Model predictions were benchmarked against in vitro experimental data obtained under physiologically relevant conditions, showing good agreement and validating the key features of the model. The proposed model thus serves as a practical tool for predicting the behavior of composite coated particles, supporting performance evaluation and the rational design of next-generation drug delivery systems with reduced experimental effort.

This paper explores new analogues of the Leibniz rule for Hadamard and Caputo–Hadamard fractional derivatives. Unlike classical derivatives, fractional ones have a strong nonlocal character, meaning that the value of the derivative at a given point depends on the entire history of the function. Because of this nonlocality, the standard product rule cannot be directly applied. The study develops refined formulas for differentiating the product of two functions, which include additional integral terms representing memory effects inherent to fractional calculus. The paper also establishes a series of inequalities that make it possible to estimate the fractional derivatives of nonlinear expressions, such as powers of a function, through the derivative of the function itself. In particular, it is shown that a specific inequality holds for positive functions that relates the fractional derivative of the function power to the function product and its fractional derivative. These theoretical results are of great importance for the study of linear and nonlinear fractional diffusion equations. They provide useful tools for proving the existence, uniqueness, and stability of their solutions and for deriving a priori estimates that describe the qualitative behavior of such systems.

A nonuniform and fast two-grid L1-FEM algorithm for nonlinear multi-term time-fractional diffusion equation and its robust error analysis

We investigate thermalization and time-dependent Bose-Einstein condensate formation in ultracold 164Dy using a nonlinear boson diffusion equation. As compared to alkali atoms such as 39K or 87Rb, the strong magnetic dipole interaction modifies the scattering-length dependence of the transport coefficients that govern thermalization and condensate formation. A prediction for the time-dependent condensate fraction in 164Dy is made.

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable v=−ε2Δu+f(u), the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only 1/60 of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods.

An initial-boundary value problem for a nonlinear diffusion equation with a Volterra-type memory term in time is investigated. The model describes diffusion processes in media with hereditary (long-memory) effects and covers, as particular cases, a range of applications in heat and mass transfer. For its numerical solution we construct an implicit two-level difference scheme with a quadrature approximation of the memory integral. A discrete energy identity is derived, which yields energy stability of the scheme under natural constraints on the time step. The nonlinearity is treated by an inner iterative linearization procedure; at each iteration a tridiagonal system of linear algebraic equations arises and is solved efficiently by the sweep (Thomas) method. The convergence of the scheme is rigorously justified and its convergence rate is estimated in the corresponding energy norm. Numerical experiments are presented that confirm the theoretical error bounds and demonstrate the accurate reproduction of long-memory effects.

In this paper, a fast two-grid algorithm is constructed for solving the nonlinear time fractional diffusion equations by using the lowest order Raviart-Thomas mixed finite element (RTMFE) space and the temporal graded mesh. In the algorithm, the Caputo time fractional derivative is discretized by the well-known L1 formula on the graded mesh, the spatial domain is divided into coarse and fine grids, then a fast two steps algorithm is proposed by using two-grid computing method. The existence, uniqueness and unconditional stability for the proposed algorithm on the temporal graded mesh are derived in detail. In addition, when the analytical solution satisfies different regularity assumptions, the asymptotically optimal a priori error estimates in spatial direction are obtained on both the temporal uniform and graded meshes, which show that when spatial coarse and fine grid parameters satisfy $H=\mathcal{O}(h^{1/2})$, the fast algorithm can obtain the same accuracy as the RTMFE algorithm. Finally, two numerical examples with different regularity conditions are provided to demonstrate the theoretical results.

ABSTRACT In this paper, we study analytically an abstract fractional diffusion equation in a Hilbert space with locally Lipschitz sources perturbed by a multiplicative ‐regular space‐time white noise. We first point out the equivalence between classical solutions and mild solutions. Next, we prove the existence, and uniqueness of maximal solutions of the problem and show continuous dependence of solutions on the initial values and associated parameters. Notably, in certain cases, we demonstrate that the solution of the problem is blow‐up in a finite time.

A nonlinear diffusion equation with two spatial variables and several time delay variables is considered. The problem is discretized. Constructions of the alternating directions method with piecewise linear interpolation and extrapolation by continuation are presented. This method has the second order of smallness with respect to the time discretization step $\Delta$ and the space discretization step $h$. As a result, the method is reduced to solving two tridiagonal systems of linear algebraic equations at each time step, which have diagonal dominance. These systems are efficiently solved using the sweep method. The order of the residual without interpolation of the method is studied. Under certain assumptions, the convergence of the method with the order $O(\Delta^2+h^2)$ is justified. The results of numerical modeling for a diffusion equation with two delay variables are presented. The computable orders of convergence for each discretization step in the examples turned out to be close to the theoretically obtained orders of convergence for the corresponding discretization steps.

In this article, we study the complicated asymptotic behavior of doubly nonlinear diffusion equations in unbounded spaces. We find a workspace in which is unbounded and can exhibit complicated asymptotic behavior of the solution to the Cauchy problem. To overcome the difficulties caused by the nonlinearity of the equations and the unbounded solutions, we establish propagation estimates, growth estimates, and Weighted \(L^1\)-\(L^\infty\) estimates for the solutions. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2025/108/abstr.html

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modeling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.

Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space

For 3D strongly nonlinear equilibrium radiation diffusion equation, a kind of new meshfree iteration methods based on Richtmyer, factorization, and Newton linearization techniques in complex domains is provided. The new methods avoid the difficulty of mesh generation while ensuring high accuracy. Through specific numerical experiments, it is shown that the new methods have good approximation performance for solving 3D strongly nonlinear problems in complex domains, and all achieve good precision with iteration counts of 2 nearly under the given parameters. The comparison between finite element method (FEM) with the presented methods shows that our new methods match with FEM when the time step is relatively small, but avoid mesh generating.

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: An entropy formulation

When water is present in a medium with pore sizes in a range of approximately 10 nm, the corresponding freezing-point depression will cause long-range broadening of a melting front. Describing the freezing-point depression by the Gibbs–Thomson equation and the pore-size distribution by a power law, we derive a nonlinear diffusion equation for the fraction of melted water. This equation yields superdiffusive spreading of the melting front with a diffusion exponent, which is given by the spatial dimension and the exponent describing the pore size distribution. We derive this solution analytically from energy conservation in the limit where all the energy is consumed by the melting and explore the validity of this approximation numerically. Finally, we explore a geological application of the theory to the case of one-dimensional subsurface melting fronts in granular or soil systems. These fronts, which are produced by heating of the surface, spread at a superdiffusive rate and affect the subsurface to significantly larger depths than a system without the effects of freezing-point depression.

ABSTRACT In this paper, we construct a linearized finite difference scheme for a class of multi‐term time‐fractional nonlinear diffusion equations (MTFNDEs) with initial boundary value conditions. Due to the initial singularity of the solutions of MTFNDEs, it is well known that many existing numerical methods for MTFNDEs often suffer from the phenomenon of order reduction. In order to solve the above problem, based on the technique of variable transformation , we propose an improved Euler method to solve MTFNDEs. Then, we prove that the scheme has order accuracy in time for . Meanwhile, the unconditional stability and convergence of the scheme are proved through the energy method. Finally, numerical experiments have been conducted to support the theoretical results and efficiency of our proposed scheme.