Wetlands are essential for global biogeochemical cycles and ecosystem services, with the dynamics of soil organic carbon (SOC) serving as the critical regulatory mechanism for these processes. However, accurately modeling carbon dynamics in wetlands presents challenges due to their complexity. Traditional approaches often fail to capture spatial variations, long-range transport, and periodical flooding dynamics, leading to uncertainties in carbon flux predictions. To tackle these challenges, we introduce a novel extension of the fractional RothC model, integrating temporal fractional-order derivatives into spatial dimensions. This enhancement allows for the creation of a more adaptive tool for analyzing SOC dynamics. Our differential model incorporates Richardson–Richard's equation for moisture fluxes, a diffusion–advection–reaction equation for fractional-order dynamics of SOC compounds, and a temperature transport equation. We examine the influence of diffusive movement and sediment moisture content on model solutions, as well as the impact of including advection terms. Finally, we validated the model on a restored wetland scenario at the Ebro Delta site, aiming to evaluate the effectiveness of flooding strategies in enhancing carbon sequestration and ecosystem resilience.

This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one-dimensional model of a linear microelastic material. We explore different kernel functions, including V-shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one-sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.

In this paper, we introduce new generalized barycentric coordinates (coined as moment coordinates) on convex and nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with n vertices (nodes) in R2, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank n, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.

Numerical simulation of fractional-order partial differential equations is a challenging task and the majority of computing environments does not provide support for these problems. In this paper we describe how to exploit some of the Matlab features (a programming language not supporting fractional calculus in a naive way) to solve partial differential equations with the spectral fractional Laplacian. For shortness we focus on fractional Poisson equations but the proposed approach can be extended, with just some technical difficulties, to more involved problems. This approach cannot be considered as a highly efficient and accurate way to solve fractional partial differential equations, but as an easy-to-use tool for non specialists in numerical computation to obtain solutions without having to produce sophisticated numerical codes.

Deep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the perydinamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.

We study the implementation of a Chebyshev spectral method with forward Euler integrator proposed in Berardi et al.(2023) to investigate a peridynamic nonlocal formulation of Richards’ equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.

In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carathéodory type right-hand side functions. We provide construction of the randomized Euler scheme for DDEs and investigate its error. We also report results of numerical experiments.

Physics-Informed Neural Networks (PINN) are a machine learning tool that can be used to solve direct and inverse problems related to models described by Partial Differential Equations by including in the cost function to minimise during training the residual of the differential operator. This paper proposes an adaptive inverse PINN applied to different transport models, from diffusion to advection–diffusion–reaction, and mobile–immobile transport models for porous materials. Once a suitable PINN is established to solve the forward problem, the transport parameters are added as trainable parameters and the reference data is added to the cost function. We find that, for the inverse problem to converge to the correct solution, the different components of the loss function (data misfit, initial conditions, boundary conditions and residual of the transport equation) need to be weighted adaptively as a function of the training iteration (epoch). Similarly, gradients of trainable parameters are scaled at each epoch accordingly. Several examples are presented for different test cases to support our PINN architecture and its scalability and robustness.

In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic theory is a non-local continuum theory that models the deformation and failure of materials. Richards’ equation describes the unsaturated flow of water through porous media, and it plays an essential role in many applications, such as groundwater management, soil science, and environmental engineering. We develop a peridynamic formulation of Richards’ equation that includes the effect of peridynamic forces and a piecewise smooth initial condition, further introducing a non-standard symmetric influence function to describe such peridynamic interactions, which turns out to provide beneficial effects from a numerical point of view. Moreover, we implement a numerical scheme based on Chebyshev polynomials and symmetric Gauss–Lobatto nodes, providing a powerful spectral method able to capture singularities and critical issues of Richards’ equation with piecewise smooth initial conditions. We also present numerical simulations that illustrate the performance of the proposed approach. In particular, we perform a computational investigation into the spatial order of convergence, showing that, despite the discontinuity in the initial condition, the order of convergence is retained.

The aim of this chapter is to device a computationally effective procedure for numerically solving fractional-time-space differential equations with the spectral fractional Laplacian. A truncated spectral representation of the solution in terms of the eigenfunctions of the usual integer-order Laplacian is considered. Time-dependent coefficients in this representation, which are solutions to some linear fractional differential equations, are evaluated by means of a generalized exponential time-differencing method, with some advantages in terms of accuracy and computational effectiveness. Rigorous a priori error estimates are derived, and they are verified by means of some numerical experiments.

Considering the concept of attainable sets for differential inclusions, we introduce the isochronous manifolds relative to a piecewise smooth dynamical systems in R2 and R3, and study how analytical and topological properties of such manifolds are related to sliding motion and to partially nodal attractivity conditions on the discontinuity manifolds. We also investigate what happens to isochronous manifolds at tangential exit points, where attractivity conditions cease to hold. In particular, we find that isochronous curves in R2, which are closed simple curves under attractivity regime, become open curves at such points.

The problem of modeling water flow in the root zone with plant root absorption is of crucial importance in many environmental and agricultural issues, and is still of interest in the applied mathematics community. In this work we propose a formal justification and a theoretical background of a recently introduced numerical approach, based on the shooting method, for integrating the unsaturated flow equation with a sink term accounting for the root water uptake model. Moreover, we provide various numerical simulations for this method, comparing the results with the numerical solutions obtained by MATLAB pdepe.

In piecewise smooth dynamical systems, a co-dimension 2 discontinuity manifold can be attractive either through partial sliding or by spiraling. In this work we prove that both attractivity regimes can be analyzed by means of the moments solution, a spiraling bifurcation parameter and a novel attractivity parameter, which changes sign when attractivity switches from sliding to spiraling attractivity or vice-versa. We also study what happens at what we call attractivity transition points, showing that the spiraling bifurcation parameter is always zero at those points.

Here some issues are studied, related to the numerical solution of Richards' equation in a one dimensional spatial domain by a technique based on the Transversal Method of Lines (TMoL). The core idea of TMoL approach is to semi-discretize the time derivative of Richards' equation: afterward a system of second order differential equations in the space variable is derived as an initial value problem. The computational framework of this method requires both Dirichlet and Neumann boundary conditions at the top of the column. The practical motivation for choosing such a condition is argued. We will show that, with the choice of the aforementioned initial conditions, our TMoL approach brings to solutions comparable with the ones obtained by the classical Methods of Lines (hereafter referred to as MoL) with corresponding standard boundary conditions: in particular, an appropriate norm is introduced for effectively comparing numerical tests obtained by MoL and TMoL approach and a sensitivity analysis between the two methods is performed by means of a mass balance point of view. A further algorithm is introduced for deducing in a self-sustaining way the gradient boundary condition on top in the TMoL context.

In this work, we consider a special choice of sliding vector field on the intersection of two co-dimension 1 manifolds. The proposed vector field, which belongs to the class of Filippov vector fields, will be called moments vector field and we will call moments trajectory the associated solution trajectory. Our main result is to show that the moments vector field is a well defined, and smoothly varying, Filippov sliding vector field on the intersection Σ of two discontinuity manifolds, under general attractivity conditions of Σ. We also examine the behavior of the moments trajectory at first order exit points, and show that it exits smoothly at these points. Numerical experiments illustrate our results and contrast the present choice with other choices of Filippov sliding vector field.

In this paper, we consider selection of a sliding vector field of Filippov type on a discontinuity manifold Σ of co-dimension 3 (intersection of three co-dimension 1 manifolds). We propose an extension of the moments vector field to this case, and—under the assumption that Σ is nodally attractive—we prove that our extension delivers a uniquely defined Filippov vector field. As it turns out, the justification of our proposed extension requires establishing invertibility of certain sign matrices. Finally, we also propose the extension of the moments vector field to discontinuity manifolds of co-dimension 4 and higher.

In this work, we consider model problems of piecewise smooth systems inR3, for which we propose minimum variation approaches to find a Filippov sliding vector field on the intersection Σ of two discontinuity surfaces. Our idea is to look at the minimum variation solution in theH1-norm, among either all admissible sets of coefficients for a Filippov vector field, or among all Filippov vector fields. We compare the resulting solutions to other possible Filippov sliding vector fields (including the bilinear and moments solutions). We further show that-in the absence of equilibria-also these other techniques select a minimum variation solution, for an appropriately weightedH1-norm, and we relate this weight to the change of time variable giving orbital equivalence among the different vector fields. Finally, we give details of how to build a minimum variation solution for a general piecewise smooth system inR3.

The authors regret that there was a mistake in the statement of Theorem 3.4. The correction follows.

We consider several possibilities on how to select a Filippov sliding vector field on a codimension 2 singularity surface Σ, intersection of two codimension 1 surfaces. We discuss and compare several, old and new, approaches, under the assumption that Σ is nodally attractive. Of specific interest is the selection of a smoothly varying Filippov sliding vector field. As a result of our analysis and experiments, the best candidates of the many possibilities explored are those based on the so-called barycentric coordinates. In the present context, one of these possibilities appear to be new. © 2013 Elsevier B.V. All rights reserved.