Nonlinear Reaction-diffusion Equations Research Articles

Mathematical Model of Synaptic Long-Term Potentiation as a Bistability in a Chain of Biochemical Reactions with a Positive Feedback.

Nitric oxide (NO) is involved in synaptic long-term potentiation (LTP) by multiple signaling pathways. Here, we show that LTP of synaptic transmission can be explained as a feature of signal transduction-bistable behavior in a chain of biochemical reactions with positive feedback, formed by diffusion of NO to the presynaptic site and facilitating the release of glutamate (Glu). The dynamics of Glu, calcium (Ca2+) and NO is described by a system of nonlinear reaction-diffusion equations with modified Michaelis-Menten (MM) kinetics. Numerical investigation reveals that the chain of biochemical reactions analyzed can exhibit a bistable behavior under physiological conditions when production of Glu is described by MM kinetics and decay of NO is modeled by means of two enzymatic pathways with different kinetic properties. Our finding extends understanding of the role of NO in LTP: a short high-intensity stimulus is "memorized" as a long-lasting elevation of NO concentration. The conclusions obtained by analysis of the chain of biochemical reactions describing LTP can be generalized to other chains of interactions or for creating the logical elements for biological computers.

Numerical investigations and simulation of calcium distribution in the alpha-cell

The α-cells are a part of islets of Langerhans located in the pancreas and are responsible for glucagon secretion. Calcium signaling is crucial for the regulation of the functions and structure of these α-cells and the same is still not well understood. Here a mathematical model is framed to obtain more insights into calcium signaling in α-cells. The non-linear reaction-diffusion equation for calcium signaling along with boundary conditions is employed to propose the model for a one-dimensional steady-state case. The numerical solutions were obtained using the Newton-Raphson method and the cubic spline method. The combination of Newton-Raphson and cubic spline has proved to be quite effective in numerical simulations and in generating deeper insights into calcium regulation in an α-cell under various conditions. The results provide information about changes in source influx, buffers, ER leak, and SERCA pump leading to disturbances in calcium homeostasis, which can be responsible for the development of diabetes and other metabolic disorders.

Numerical approximation of the stochastic equation driven by the fractional noise

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT-θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests.

Finite volume simulation of calcium distribution in a cholangiocyte cell

Cholangiocytes are the cells of the liver having a major role in the conditioning of bile used in digestion. Other functions of cholangiocytes are in apoptosis and bicarbonate secretion. The Calcium in the intracellular environment of various cells including cholangiocytes regulates a large number of functions. This regulating mechanism in cholangiocytes has been poorly understood to date. In order to analyze the calcium regulation in cholangiocyte cells, a mathematical model for a one-dimensional steady-state case is constructed in this study. This involves a non-linear reaction-diffusion equation with appropriate boundary conditions. The influx from IP$_{3}$ receptor, ryanodine receptor (RYR), and plasma membrane as well as the efflux of calcium from SERCA pump and plasma membrane have been employed in the model. The finite volume method and Newton-Raphson method have been used to solve the problem. Numerical findings have been used to examine the effects of parameters like diffusion coefficient, rate of SERCA pump efflux, buffer, and influx from plasma membrane on calcium concentration in cholangiocyte cells. The information generated from the model can be useful for understanding the mechanism of cholestatic disorders which can be further useful in the diagnosis and treatment of these disorders.

Modeling glucose isomerization in a packed- bed reactor: A complete theoretical and numerical approach

Immobilized enzymes are used in many applications in the food industry presently. Some researchers employ apparent kinetic parameters in immobilized enzyme reactor systems, which only apply to these specific case studies. To increase productivity in a packed bed reactor, it is vital to establish a thorough understanding of all the factors affecting the process, including axial dispersion, convective mass transport, intra-particle diffusive mass transfer, and kinetics. This paper discusses the packed-bed reactor's steady-state model of immobilized glucose isomerase. This model is based on a nonlinear reaction-diffusion equation containing nonlinear terms associated with the enzymatic reaction of Michaelis–Menten kinetics. Akbari-Ganji's method (AGM) and Taylors series with Pade's approximation method are used to calculate the species concentration and normalized current for all possible kinetic parameters and Thiele modulus. The saturated and unsaturated kinetics also have been investigated analytically. A significant agreement between the simulation results and the analytical expression of concentration is observed.

A matrix–free high–order solver for the numerical solution of cardiac electrophysiology

We propose a matrix–free solver for the numerical solution of the cardiac electrophysiology model consisting of the monodomain nonlinear reaction–diffusion equation coupled with a system of ordinary differential equations for the ionic species. Our numerical approximation is based on the high–order Spectral Element Method (SEM) to achieve accurate numerical discretization while employing a much smaller number of Degrees of Freedom than first–order Finite Elements. We combine vectorization with sum–factorization, thus allowing for a very efficient use of high–order polynomials in a high performance computing framework. We validate the effectiveness of our matrix–free solver in a variety of applications and perform different electrophysiological simulations ranging from a simple slab of cardiac tissue to a realistic four–chamber heart geometry. We compare SEM to SEM with Numerical Integration (SEM–NI), showing that they provide comparable results in terms of accuracy and efficiency. In both cases, increasing the local polynomial degree p leads to better numerical results and smaller computational times than reducing the mesh size h. We also implement a matrix–free Geometric Multigrid preconditioner that results in a comparable number of linear solver iterations with respect to a state–of–the–art matrix–based Algebraic Multigrid preconditioner. As a matter of fact, the matrix–free solver proposed here yields up to 45× speed–up with respect to a conventional matrix–based solver.

Exploration of new solitons solutions for the Fitzhugh–Nagumo-type equations with conformable derivatives

The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics, the Fitzhugh–Nagumo equation is also frequently used in circuit theory. In this study, we gave solutions to the fractional Fitzhugh–Nagumo (FN) equation, the fractional Newell–Whitehead–Segel (NWS) equation, and the fractional Zeldovich equation. We have obtained exact solutions within time fractional conformable derivative for these equations.

Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction-diffusion equation

In this paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction-diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction-diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy.

Fibonacci Wavelet Method for the Numerical Solution of Nonlinear Reaction-Diffusion Equations of Fisher-Type

This article aims to propose an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation of Fisher-type. The underlying numerical scheme starts by formulating operational matrices of integration corresponding to the Fibonacci wavelets. Besides, we study the error analysis and convergence theorem of the proposed technique. Subsequently, a set of algebraic equations are formed corresponding to the given problem, which could be handled via any conventional method, for instance, the Newton iteration technique. To demonstrate the efficiency of the proposed wavelet-based numerical method, we compare the obtained absolute,L∞,L2, and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometricB-spline (CTB) differential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more effective and yields precise outcomes in comparison to the existing ones.

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

SELF-DIFFUSIVE SPATIOTEMPORAL PATTERNS IN A FOOD CHAIN MODEL THROUGH WEAKLY NONLINEAR ANALYSIS

Density distributions of populations with self-diffusion and interaction in a spatial domain are dynamically visualized with coupled nonlinear reaction–diffusion equations. Incorporating self-diffusion terms creates a more pragmatic modeling paradigm and provides meaningful descriptions of influences on spatiotemporal pattern formation phenomena. This paper examines the effect of self-diffusion in a food chain system with a Holling type-IV functional response and the type of spatial structures forms on a geographical scale due to the random movement of species. We discussed the existence and uniqueness of a positive equilibrium solution and obtained the Turing instability conditions for the self-diffusive food chain model. Moreover, weakly nonlinear analysis close to the Turing bifurcation boundary is used to derive the amplitude equations. The stability of the amplitude equations and sufficient conditions for the emanation of spatiotemporal patterns (such as spots, stripes, and blended patterns) are investigated. The analytical results are verified with numerical simulations. The results are applicable to all environments and can be used to understand the effects of self-diffusion in other food chain models both qualitatively and quantitatively.

Computing the area-minimizing surface by the Allen-Cahn equation with the fixed boundary

<abstract><p>The Allen-Cahn equation is a famous nonlinear reaction-diffusion equation used to study geometric motion and minimal hypersurfaces. This link has been scrutinized to construct minimal surfaces for many years. The shape of soap film is very interesting, and it can stimulate mathematical inspirations since it explains curvatures and equilibrium shapes in nature. There are many interesting ways to create area-minimizing surfaces with the boundaries, called frame boundaries. However, dealing with surface's ends (boundaries) numerically is not easy for constructing surfaces. This paper presents a mathematical formulation and numerical construction of area-minimizing surfaces, also known as minimal surfaces. We use differential geometry knowledge for numerical verification. The proposed numerical scheme involves fixed frame boundary conditions in the Laplacian operator. We treat the Laplacian with the constraint implicitly and explicitly solve the nonlinear free energy term. This approach ensures stable and efficient construction of area-minimizing surfaces with frame boundaries. In the numerical aspect, we suggest the construction of minimal surfaces by illustrating two classical examples, which are Scherk's minimal surface and catenoid. Both examples have the frame boundaries. Scherk's first surface is a doubly periodic, complete and properly embedded one with parallel ends. The catenoid is formed between two coaxial circular rings and is classified mathematically as the only properly embedded minimal surface with two ends and finite curvature. To be specific, we deal with two different frame boundaries, right angle frame and round frame boundaries, via two examples, Scherk's surface and catenoid.</p></abstract>

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

<abstract><p>In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.</p></abstract>

Application of the newly \(\varphi^{6}\)-model expansion approach to the nonlinear reaction-diffusion equation

In this paper, we use the \(\varphi ^{6}\)-model expansion method to construct the traveling wave solutions for the reaction-diffusion equation. The method of \(\varphi ^{6}\)-model expansion enables the explicit retrieval of a wide variety of solution types, such as bright, singular, periodic, and combined singular soliton solutions. Kink-type solitons, also known as topological solitons in the context of water waves, are another type of solution that can be explicitly retrieved. This study’s results might enhance the equation’s nonlinear dynamical properties. The method proposes a practical and efficient method for solving a sizable class of nonlinear partial differential equations. The dynamical features of the data are explained and highlighted using exciting graphs.

Analysis of dead-core formation in catalytic reaction and diffusion processes with generalized diffusion flux

Dead-core and non-dead-core solutions to the nonlinear diffusion–reaction equation based on the generalized diffusion flux with gradient-dependent diffusivity and the power-law reaction kinetics in catalyst slabs are established. The formation of dead zones where the reactant concentration vanishes is characterized by the critical Thiele modulus that is derived as a function of reaction order and diffusion exponent in the generalized diffusion flux. The effects of reaction order and diffusion exponent on the reactant concentration distribution in the slab and dead-zone length are analyzed. It is particularly demonstrated that by contrast to the model based on the standard Fick’s diffusion, dead-core solutions exist in the case of first-order reactions. Also, the relationship between critical Thiele moduli for models based on the generalized and standard Fick’s diffusion fluxes is established.

On Reaction-Diffusion Equation with Generalized Composite Fractional Derivative

In this paper, we obtain the Sumudu transform of generalized composite fractional derivative and some lemmas related to inverse Sumudu transform. Further, we find solution of nonlinear reaction diffusion equation with generalized composite fractional derivative by applying the Sumudu and Fourier transforms.

Gap functions and global error bounds for differential variational–hemivariational inequalities

This paper concerns with the study of a differential variational–hemivariational inequality (DVHVI, for short) in infinite-dimensional Banach spaces. We first introduce the new concept of gap functions for the variational control system of (DVHVI). Then, we consider two kinds of gap functions which are regularized gap function and Moreau–Yosida regularized gap function, respectively, and examine the relevant properties of the gap functions. Moreover, two global error bounds which depend implicitly on the regularized gap function and the Moreau–Yosida regularized gap function, accordingly, are obtained. Finally, in order to illustrate the applicability of the theoretical results, we investigate a coupled dynamic system which is formulated by a nonlinear reaction–diffusion equation described by a time-dependent nonsmooth semipermeability problem.

Optimal Boundary Control of a Nonlinear Reaction Diffusion Equation via Completing the Square and Al’brekht’s Method

The two contributions of this article are as follows. The first is the solution of an infinite dimensional, boundary controlled linear quadratic regulator by the simple and constructive method of completing the square. The second contribution is the extension of Al’brekht’s method to the optimal stabilization of a boundary controlled, nonlinear reaction diffusion system.

Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

AbstractNew classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

Optimal control problems of a reaction–diffusion ecological model with a protection zone

In this paper the authors discuss optimal control problems of a new ecological model governed by nonlinear reaction–diffusion equations. There are four major ingredients. The first is the proof of the well posedness of the solution to the state equations. The second is to prove the existence of the optimal control by using functional analysis techniques. The third ingredient is the establishment of first order necessary optimality conditions satisfied by the optimal control. The fourth ingredient is an example of optimal control problem and its numerical results that confirm the theoretical results obtained in this paper.