Nonlinear Reaction-diffusion Model Research Articles

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction–diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction–diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

Nonlocal p-Laplacian involving a nonlinear fractional reaction-diffusion system applied to image restoration

Image restoration has been a longstanding challenge in image processing, involving the enhancement of image content to extract valuable information. Nonlinear models have demonstrated their efficacy in eliminating additive noise, which motivates our goal of investigating a novel nonlocal nonlinear reaction-diffusion model for noise removal. The proposed model integrates a fractional diffusion equation with a nonlinear nonlocal p-Laplace operator, with the fidelity term employing a weak norm to better capture oscillatory patterns and intricate details in textured images. Using the Schauder fixed point theorem, the well-posedness of the proposed model solution is established. The experimental results confirm the effectiveness and efficiency of the proposed model, providing validation for its practical utility.

Wavelets computational modeling of nonlinear coupled reaction–diffusion models arising in chemical processes

This work submits the computational modeling of nonlinear coupled reaction–diffusion (RD) models via uniform scale‐2 Haar wavelets (S2HWs) based numerical algorithm. The nonlinear RD models have many applications in mathematical chemistry, chemical and biological processes, etc. In the design of the algorithm, first of all, semi‐discretization is done in time, and after that, the quasi‐linearization process is used for linearization. Finally, the full discretization is done with the help of S2HWs, and the enlisted linear system is solved by LU decomposition method. Also, the convergence analysis of the algorithm has discussed. In the simulation part, different types of patterns such as Turing, pulse‐splitting, self‐replicating pulse patterns, etc. of different models are captured.

Synchronization of Fractional Partial Difference Equations via Linear Methods

Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a fractional discrete nonlinear bacterial culture reaction-diffusion model using the Caputo h-difference operator and a second-order central difference scheme and an L1 finite difference scheme after deriving the discrete fractional version of the well-known Degn–Harrison system and Lengyel–Epstein system. Using appropriate techniques and the direct Lyapunov method, the conditions for full synchronization are determined.Furthermore, this research shows that the L1 finite difference scheme and the second-order central difference scheme may successfully retain the properties of the related continuous system. The conclusions are proven throughout the paper using two major biological models, and numerical simulations are carried out to demonstrate the practical use of the recommended technique.

Qualitative and quantitative analysis of a nonlinear second-order anisotropic reaction-diffusion model of an epidemic infection spread

<p style='text-indent:20px;'>The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: <inline-formula><tex-math id="M1">\begin{document}$ S_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ E_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ I_0(x) $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M4">\begin{document}$ R_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ D_0(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}</tex-math></inline-formula>, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space <inline-formula><tex-math id="M8">\begin{document}$ W^{1,2}_p(Q) $\end{document}</tex-math></inline-formula>, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.</p><p style='text-indent:20px;'>The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.</p>

On a nonlocal and nonlinear second-order anisotropic reaction-diffusion model with in-homogeneous Neumann boundary conditions

<p style='text-indent:20px;'>In this paper we study the existence and uniqueness of the solution in <inline-formula><tex-math id="M1">\begin{document}$ C((0, T], L^\infty(\Omega)) $\end{document}</tex-math></inline-formula> of a new nonlocal and nonlinear second-order anisotropic reaction-diffusion problem with in-homogeneous Neumann boundary conditions, generalizing other problems in the literature. Then, by using the finite difference method, we propose an explicit in time numerical scheme to approximate the unique solution of our problem. We also present some numerical simulations that come to show the performance of our theoretical model.</p>

Radical Production with Pulsed Beams: Understanding the Transition to FLASH.

Ultra-high dose rate (UHDR) irradiation regimes have the potential to spare normal tissue while keeping equivalent tumoricidal capacity than conventional dose rate radiotherapy (CONV-RT). This has been called the FLASH effect. In this work, we present a new simulation framework aiming to study the production of radical species in water and biological media under different irradiation patterns. The chemical stage (heterogeneous phase) is based on a nonlinear reaction-diffusion model, implemented in GPU. After the first 1 μs, no further radical diffusion is assumed, and radical evolution may be simulated over long periods of hundreds of seconds. Our approach was first validated against previous results in the literature and then employed to assess the influence of different temporal microstructures of dose deposition in the expected biological damage. The variation of the Normal Tissue Complication Probability (NTCP), assuming the model of Labarbe et al., where the integral of the peroxyl radical concentration over time (AUC-ROO) is taken as surrogate for biological damage, is presented for different intra-pulse dose rate and pulse frequency configurations, relevant in the clinical scenario. These simulations yield that overall, mean dose rate and the dose per pulse are the best predictors of biological effects at UHDR.

Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis

We consider a family of exact solutions to a nonlinear reaction–diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher–KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher–KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.

Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent with the theoretical results of other scholars, which verifies the accuracy of the method. We show that stable spatio-temporal patterns can be sustained for a long time; these patterns are different from any previously obtained in numerical studies. Here, we show that behavior patterns can be described well by the fractional FitzHugh–Nagumo and Gray–Scott models, which have unique properties that integer models do not have. Results show that the Fourier spectral method has strong competitiveness, reliability, and solving ability for solving 2D and 3D fractional-in-space nonlinear reaction-diffusion models.

Solution of Nonlinear Reaction-Diffusion Model in Porous Catalysts Arising in Micro-Vessel and Soft Tissue Using a Metaheuristic

The steady-state reactive transport model (RTM) is a generalization of the nonlinear reaction-diffusion model in porous catalysts. The RTM is expressed as a non-linear ordinary differential equation of second-order with boundary conditions. Artificial neural network (ANN), Particle swarm optimization (PSO), and hybrid of PSO-SQP (Sequential Quadratic Programming) are used to obtain accurate, approximate solutions to the non-linear RTM. The proposed technique is applied to three different cases of non-linear RTM. The properties of the nonlinear reactive transport model in porous catalysts are investigated by considering various cases based on variation in the half-saturation concentration “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> ” and the characteristic reaction rate “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> .” The stability, reliability, and exactness of the proposed technique are established through comparison with the outcomes of the standard numerical procedure with the RK4 method and along with the different performance indices, which are Root-Mean-Square Error (RMSE), (TIC), Absolute Error (AE), and Mean Absolute Deviation (MAD).

Weak Universality of the Dynamical ${{\Phi }_{3}^{4}}$ Model on the Whole Space

We prove that the large scale behavior of a class of stochastic weakly nonlinear reaction-diffusion models on \(\mathbb {R}^{3}\) converges to the dynamical \({{\Phi }^{4}_{3}}\) model using paracontrolled distributions on weighted Besov spaces. Our approach depends on the delicate choice of the weight, the localization operator technique and a modified version of the maximum principle from Gubinelli and Hofmanová (Commun. Math. Phys. 368, 1201–1266, 5).

Development of an integrated ultrasonic biofilm detachment model for biofilm thickness control in membrane aerated bioreactors

Controlling biofilm thickness in an aerated membrane biofilm reactor (MBfR) has been recognized as a key for MBfRs to achieve a long-term stable performance. In this context, acoustic cavitation is an effective strategy that can be used for controlling biofilm thickness. However, to maintain the biofilm thickness at an optimum value, it is necessary to understand the effect of acoustic parameters and cavitation bubble distribution on biofilm detachment and establish a link between biofilm detachment and regrowth. The purpose of this study is to provide an integrated mathematical model that describes biofilm development in an aerated membrane biofilm reactor using a nonlinear reaction-diffusion model and its response to mechanical stress generated from acoustic cavitation. The simulation results show that pressure, amplitude and frequency of transducer are two key factors that affect biofilm detachment. Moreover, uniform distribution of cavitation along the biofilm surface is critical to achieve an even biofilm thickness. Furthermore, periodic cavitation detachment with an appropriate resting time in between is important for maintaining the biofilm thickness at a desired value. Therefore, the purpose of this study is to present an integrated modeling approach to optimize acoustic cavitation parameters and methods to achieve effective biofilm thickness control in MBfRs.

Differential dissipativity analysis of reaction–diffusion systems

This note shows how classical tools from linear control theory can be leveraged to provide a global analysis of nonlinear reaction–diffusion models. The approach is differential in nature. It proceeds from classical tools of contraction analysis and recent extensions to differential dissipativity.

A reliable Taylor series solution to the nonlinear reaction-diffusion model representing the steady-state behaviour of a cationic glucose-sensitive membrane

A reliable Taylor series solution to the nonlinear reaction-diffusion model representing the steady-state behaviour of a cationic glucose-sensitive membrane

A Comparative Study of the Numerical Approximations of the Quenching Time for a Nonlinear Reaction-Diffusion Equation

In this paper, we study the numerical methods for solving a nonlinear reaction-diffusion model for the polarization phenomena in ionic conductors. In particular, we propose three types of numerical methods, including the finite difference, cubic B-spline collocation, and local discontinuous Galerkin method, to approximate the quenching time of the model. We prove the conservation properties for all three numerical methods and compare their numerical performance.

Adaptive trainable non‐linear reaction diffusion for Rician noise removal

Rician noise reduction is an essential issue in magnetic resonance imaging (MRI). Recently, learning-based methods have achieved great success in dealing with image restoration problems, which provide fast inference and good performance. One limitation of these methods, however, is that the training procedure is usually noise-level dependent, i.e. the trained models are bound to a specific noise level and lack the ability to automatically adapt to different noise levels. In this study, the authors propose a variational model for Rician noise removal by integrating a noise adaption function into the field of experts image prior, which can adapt to different noise levels. Instead of directly solving the energy minimisation problem, the authors unroll the gradient descent step of the energy functional for several iterations, the time-dependent parameters of which can be learned through a supervised training process. The authors call this methodology as the noise adaptive trainable non-linear reaction–diffusion model. The proposed methodology is robustness against noise level changing and noise distributions. Experimental results over T 1 -, T 2 - and PD-weighted MRI data set demonstrate that the proposed model can achieve superior performance compared with other methods in terms of both the peak signal-to-noise ratio and the structural similarity index.

A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations

This work aims to propose a new analyzing tool, called the fractional iteration algorithm I for finding numerical solutions of nonlinear time fractional-order Cauchy reaction-diffusion model equations. The key property of the suggested technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The proposed approach can be utilized without the use of any transformation, Adomian polynomials, small perturbation, discretization or linearization. The main feature of the fractional iteration algorithm-I is the improvement of an auxiliary parameter that can ensure a rapid convergence. To check the stability, accuracy and speed of the method, obtained results are compared numerically and graphically with the exact solutions and results available in the latest literature. In addition, numerical results are displayed graphically for various cases of the fractional-order α. These results demonstrate the viability of the proposed technique and show that this technique is exceptionally powerful and suitable for solving fractional PDEs.

The leopard never changes its spots

Previous research in pattern formation using reaction-diffusion mostly focused on static domains, either for computational simplicity or mathematical tractability. In this work, we have explored the expressiveness of combining simple mechanisms as a possible explanation for pigmentation pattern formation, where tissue growth plays a crucial role. Our motivation is not only to realistically reproduce natural patterns but also to get insights into the underlying biological processes. Therefore, we present a novel approach to generate realistic animal skin patterns. First, we describe the approximation of tissue growth by a series of discrete matrix expansion operations. Then, we combine it with an adaptation of Turing's non-linear reaction-diffusion model, which enforces upper and lower bounds to the concentrations of the involved chemical reagents. We also propose the addition of a single-reagent continuous autocatalytic reaction, called reinforcement, to provide a mechanism to maintain an already established pattern during growth. By careful adjustment of the parameters and the sequencing of operations, we closely match the appearance of a few real species. In particular, we reproduce in detail the distinctive features of the leopard skin, also providing a hypothesis for the simultaneous productions of the most common melanin types, eumelanin and pheomelanin.

A scaling transformation method and exact solutions of nonlinear reaction–diffusion model

We find a very simple scaling transformation method by which a kind of rank non-homogenous second order nonlinear ODEs is reduced to an integral form. Comparing with the canonical-like transformation method and the first integral method, our method provides a simple mechanism of reduction of such equations. As an application, a reaction–diffusion equation with variable coefficients is integrated, and its exact cnoidal wave solution is obtained. Furthermore, by using the complete discrimination system for polynomial method, more exact solutions are obtained.