Analytical model of reactive transport processes with spatially variable coefficients

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).

In this paper a general integral formula for the fundamental solution of a linear partial difference equation with constant coefficients is investigated. For equations in twodimensional space, especially for the discretization of the Helmholtz equation, special integral formulae are given and the convergence of corresponding integrals is proved. The limits of the fundamental solution at infinity are calculated in several cases.

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

The locations of a significant number of industrial facilities, landfills, and almost all mineral ore bodies are characterized by high in situ stresses and fractures, which act as flow paths for fluids underground. Toxic chemical transport is one of the main reasons for loss in groundwater quality and reactive transport modelling is an important tool necessary for predicting and controlling their fate. Modelling reactive transport has largely been achieved using the hydrodynamic transport equation, with an additional reaction and/or decay term. To date, however, most approaches to modelling reactive transport often consider an already existing mixture (multi-phase and/or multi-species) undergoing such transport in homogeneous media. The motivation of this research is to use the transport solutions of two single-phase species in homogeneous media to describe the reactive transport of two species before and after mixing, due to formation heterogeneity. This practical scenario can be observed where regional-scale fracture systems transporting pollution from spatially isolated source locations cause the mixing of chemical pollutants from different source origins due to fracture–fracture flux across two or more intersecting fractures. Here, the authors present a framework for determining the exact solution of the reactive groundwater transport model by the coupling of the homogeneous and non-homogeneous model sub-components. The solution of the homogeneous sub-component is obtained using the Laplace transform. The exact solution of the proposed non-homogeneous equation is obtained analytically using Green’s function method. Further, the two equations are numerically discretized using the Crank–Nicolson scheme and their stability conditions are also established. Instead of presenting the specific solution of a particular case study scenario, a general model to approximate the contribution of reactive processes in the system is presented. It is found that the fate and contribution of each species within the system is captured by the model for which the reaction kinetics and equilibrium partitioning between liquid and solid phases are all assumed to be linear first order.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

Understanding the complex, interacting processes that determine the fate and transport of contaminants in groundwater is a major challenge for evaluating and predicting risks to clean water, human and ecological receptors and for designing effective remediation plans. Different physical and biogeochemical processes including advection, hydrodynamic dispersion, dissolution, sorption and biodegradation affect the migration of contaminants in saturated porous media like a groundwater system. In this chapter an overview of these processes is presented together with the basic theory on contaminant transport modeling, which represents an essential tool for a quantitative description of contaminant migration in the subsurface. Numerical simulations of typical contamination scenarios are presented, with the main goal of identifying the influence of different parameters on contaminant fate and transport such as transverse dispersivity, thickness and strength of the contamination source, recharge, biodegradation rates and mixing enhancement through flow focusing in high permeability zones. These numerical simulations are complemented by two examples, i.e. the reactive transport of toluene from a LNAPL source and a field study, where ammonium is continuously released from a leaking landfill to the underlying aquifer. The principal processes at the landfill site have been quantitatively integrated into the framework of a two-dimensional reactive transport model.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

Partial differential equations (PDEs) are core tools for characterizing physical laws and are widely used in fluid mechanics, power systems, additive manufacturing, and other fields. However, traditional numerical methods are limited by mesh partitioning, leading to a trade-off between accuracy and efficiency in complex geometric domains and high-dimensional problems. While Physically Informed Neural Networks (PINNs) achieve meshless PDE solutions by embedding prior physical knowledge, they still face challenges such as insufficient accuracy, low training efficiency, and poor stability. This study aims to address the core bottlenecks of PINN in solving PDEs by proposing a systematic optimization strategy to improve its numerical solution accuracy, efficiency, and stability. Methodologically, firstly, a composite loss function containing PDE residuals and boundary initial condition constraints is constructed based on automatic differentiation techniques to ensure that the network satisfies physical laws. Secondly, the PINN training system is optimized, including using Neural Architecture Search (NAS-PINN) to automatically match the optimal network structure, designing an adaptive data sampling strategy to improve data utilization, and introducing parallel computing and hardware acceleration techniques to reduce training time. Finally, the effectiveness of the strategy is verified through classical PDEs and engineering problems. The results show that the optimized PINN achieves high-precision numerical approximation in classical PDE solutions and can efficiently solve forward design and backward parameter inversion problems in engineering scenarios. Compared with traditional methods, it exhibits stronger adaptability in complex geometric domains and high-dimensional problems, while improving training efficiency by more than 30% and significantly enhancing stability. This research not only enriches the optimization theory of PINN and provides an efficient new path for PDE numerical solutions, but also promotes the practical application of PINN in engineering fields such as fluid mechanics, additive manufacturing, and power systems, possessing significant theoretical value and application significance.

We introduce a discrete-time microscopic single-particle model for kinetic transport. The kinetics are modeled by a two-state Markov chain, and the transport is modeled by deterministic advection plus a random space step. The position of the particle afterntime steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to 0 linearly, we obtain a random variableS(t), which is closely connected to a well-known (deterministic) partial differential equation (PDE), reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed parts of the solute at timetdo exist, and satisfy the PDEs.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

Loss landscape is a useful tool for characterizing and comparing neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly nonconvex in very high-dimensional space. In this paper, we develop the “roughness” concept for understanding such landscapes in high dimensions and apply this technique to study two neural network models arising from solving differential equations. Our main innovation is the proposal of a well-defined and easy-to-compute roughness index (RI) which is based on the mean and variance of the (normalized) total variation for one-dimensional functions projected on randomly sampled directions. A large RI at the local minimizer indicates an oscillatory landscape profile and indicates a severe challenge for the first-order optimization method. Particularly, we observe the increasing-then-decreasing pattern for RI along the gradient descent path in most models. We apply our method to two types of loss functions used to solve partial differential equations (PDEs) when the solution of PDE is parametrized by neural networks. Our empirical results on these PDE problems reveal important and consistent observations that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method.

The purpose of this article is to describe the transcendental entire solutions of quadratic trinomial partial differential equations (PDDEs) with constant coefficients. We establish theorems on the forms of finite order transcendental entire solutions for such PDDEs, which generalize and improve previous theorems. Some examples confirm the existence and the forms of transcendental entire solutions with finite order of such equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/37/abstr.html

Multisummability of formal power series solutions of partial differential equations with constant coefficients

The fractional partial differential equation is an equation with non-integer partial derivative which is now widely used in solving various problems so that in recent years, many researchers are interested to study fractional partial differential equation. The equation which is used in this paper is the non-homogeneous fractional partial differential equation. The aim of this study is to determine general solution of the partial differential equation through three different order equations (α; 1), (β; 2) and (α; β), where 0 < α ≤ 2 and 0 < β ≤ 1 by using Homotopy Analysis Method (HAM). Furthermore, by taking a convergent sequence, it will be analyzed that the convergence of the sequence of differential equations results in the sequence of solutions of partial fractional non-homogeneous partial differential equations converging to the solution function of the fractional partial differential equation.