Huxley Equation Research Articles

Finite Element Approximation for the Delayed Generalized Burgers–Huxley Equation with Weakly Singular Kernel: Part II Nonconforming and DG Approximation

Finite Element Approximation for the Delayed Generalized Burgers–Huxley Equation with Weakly Singular Kernel: Part II Nonconforming and DG Approximation

Principal Algebra, Invariant Solutions and Representations for Optimal Systems of the Burgers–Huxley Equation

Principal Algebra, Invariant Solutions and Representations for Optimal Systems of the Burgers–Huxley Equation

Numerical Solution to the Time-Fractional Burgers–Huxley Equation Involving the Mittag-Leffler Function

Fractional differential equations play a significant role in various scientific and engineering disciplines, offering a more sophisticated framework for modeling complex behaviors and phenomena that involve multiple independent variables and non-integer-order derivatives. In the current research, an effective cubic B-spline collocation method is used to obtain the numerical solution of the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. It is implemented with the help of a θ-weighted scheme to solve the proposed problem. The spatial derivative is interpolated using cubic B-spline functions, whereas the temporal derivative is discretized by the Atangana–Baleanu operator and finite difference scheme. The proposed approach is stable across each temporal direction as well as second-order convergent. The study investigates the convergence order, error norms, and graphical visualization of the solution for various values of the non-integer parameter. The efficacy of the technique is assessed by implementing it on three test examples and we find that it is more efficient than some existing methods in the literature. To our knowledge, no prior application of this approach has been made for the numerical solution of the given problem, making it a first in this regard.

A third (fourth)-order computational scheme for 2D Burgers-type nonlinear parabolic PDEs on a nonuniformly spaced grid network

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

Exact solutions of the generalized Huxley–Burgers’ equations

This study employs the Extended Kudryashov method to address the fractional-order generalized Burgers–Huxley equation, the fractional-order generalized Huxley equation, the fractional-order generalized Burgers equation, and the fractional-order generalized Fisher equation. Through this approach, exact solutions for these equations are derived. Furthermore, we visualize the obtained results in 3D using Maple [Formula: see text]. The outcomes underscore the effectiveness and reliability of the Extended Kudryashov method as a clear and efficient tool for solving the specified equations.

Uniform large deviations for stochastic Burgers–Huxley equation driven by multiplicative noise

In this paper, we establish a Freidlin–Wentzell type large deviation principle uniformly with respect to initial condition in bounded subsets of an infinite dimensional Banach space for stochastic Burgers–Huxley equation driven by multiplicative small noise. The proof is based on the weak convergence approach obtained by Salins, Budhiraja, Dupuis (Trans. Amer. Math. Soc., 2019).

Analytical and numerical solutions to the Klein–Gordon model with cubic nonlinearity

In this paper, the nonlinear Klein–Gordon equation’s exact solutions are obtained through the application of an appropriate transformation based on He’s semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method’s accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters.

An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the L2 as well as the H1 norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory.

Numerical Solution of Burgers–Huxley Equation Using a Higher Order Collocation Method

In this paper, the collocation method with cubic B-spline as basis function has been successfully applied to numerically solve the Burgers–Huxley equation. This equation illustrates a model for describing the interaction between reaction mechanisms, convection effects, and diffusion transport. Quasi-linearization has been employed to deal with the nonlinearity of equations. The Crank–Nicolson implicit scheme is used for discretization of the equation and the resulting system turned out to be semi-implicit. The stability of the method is discussed using Fourier series analysis (von Neumann method), and it has been concluded that the method is unconditionally stable. Various numerical experiments have been performed to demonstrate the authenticity of the scheme. We have found that the computed numerical solutions are in good agreement with the exact solutions and are competent with those available in the literature. Accuracy and minimal computational efforts are the key features of the proposed method.

A hybrid computational scheme for singularly perturbed Burgers’-Huxley equation

This paper aims to construct and analyze a hybrid computational method for the nonlinear singularly perturbed Burgers’-Huxley equation. The presence of the perturbation parameter and non-linearity in the considered problem makes it difficult to solve the problem analytically and using classical numerical techniques on uniform step sizes as ε goes small. To elucidate such limitations, one can rely on non-classical numerical techniques. In this paper, one such parameter uniform numerical method is designed for the considered problem. The method is constructed:•Firstly, the non-linear terms are linearized via Newton-Raphson-Kantorovich technique. The linearized problem is discretized by the implicit Euler method in the temporal direction.•Secondly, the obtained equation is solved by employing the hybrid computational method comprised of the cubic spline in tension method in the inner layer region and the midpoint upwind method in the outer layer region on a piecewise uniform Shishkin mesh.•Finally, an error analysis of the method is done and observed that the proposed method is parameter uniform convergent with the order of convergence O(τ+N−2ln3N). Three examples are presented and the results are compared to some existing schemes in the literature to demonstrate the reliability of the proposed scheme.

Embedding physical laws into Deep Neural Networks for solving generalized Burgers–Huxley equation

Among the difficult problems in mathematics is the problem of solving partial differential equations (PDEs). To date, there is no technique or method capable of solving all PDEs despite the large number of effective methods proposed. One finds in the literature, numerical methods such as the methods of finite differences, finite elements, finite volumes and their variants, semi-analytical methods such as the Variational Iterative Method, New Iterative Method and others. In recent years, we have witnessed the introduction of neural networks in solving PDEs. In this work, we will propose an adaptation of the method of embedding some physical laws into neural networks for solving Burgers–Huxley equation and revealing the dynamic behavior of the equation directly from spatio-temporal data. We will combine our technique with the Residual-based Adaptive Refinement method to improve its accuracy. We will give a comparison of the proposed method with those obtained by the New Iterative Method.

Analysis of the Burgers–Huxley Equation Using the Nondimensionalisation Technique: Universal Solution for Dirichlet and Symmetry Boundary Conditions

The Burgers–Huxley equation is important because it involves the phenomena of accumulation, drag, diffusion, and the generation or decay of species, which are common in various problems in science and engineering, such as heat transmission, the diffusion of atmospheric contaminants, etc. On the other hand, the mathematical technique of nondimensionalisation has proven to be very useful in the appropriate grouping of the variables involved in a physical–chemical phenomenon and in obtaining universal solutions to different complex engineering problems. Therefore, a deep analysis using this technique of the Burgers–Huxley equation and its possible boundary conditions can facilitate a common understanding of these problems through the appropriate grouping of variables and propose common universal solutions. Thus, in this case, the technique is applied to obtain a universal solution for Dirichlet and symmetric boundary conditions. The validation of the methodology is carried out by comparing different cases, where the coefficients or the value of the boundary condition are varied, with the results obtained through a numerical simulation. Furthermore, one of the cases presented presents a boundary condition that changes at a certain time. Finally, after applying the technique, it is studied which phenomenon is predominant, concluding that from a certain value diffusion predominates, with the rest being practically negligible.

Methodology for Solving Engineering Problems of Burgers–Huxley Coupled with Symmetric Boundary Conditions by Means of the Network Simulation Method

The Burgers–Huxley equation is a partial differential equation which is based on the Burgers equation, involving diffusion, accumulation, drag, and species generation or sink phenomena. This equation is commonly used in fluid mechanics, air pollutant emissions, chloride diffusion in concrete, non-linear acoustics, and other areas. A general methodology is proposed in this work to solve the mentioned equation or coupled systems formed by it using the network simulation method. Additionally, the implementation of the most common possible boundary conditions in different engineering problems is indicated, including the Neumann condition that enables symmetry to be applied to the problem, reducing computation times. The method consists mainly of establishing an analogy between the variables of the differential equations and the electrical voltage at a central node. The methodology is also explained in detail, facilitating its implementation to similar engineering problems, since the equivalence, for example, between the different types of spatial and time derivatives and its correspondence with the electrical device is detailed. As an example, several cases of both the equation and a coupled system are solved by varying the boundary conditions on one side and applying symmetry on the other.

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

Improving Numerical Solutions for the Generalized Huxley Equation: The New Iterative Method (NIM)

Improving Numerical Solutions for the Generalized Huxley Equation: The New Iterative Method (NIM)

Application of the New Iterative Method (NIM) to the Generalized Burgers–Huxley Equation

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, such as Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), and the exact solution, and found that it is efficient and easy to use. NIM has the advantage of quick convergence, easy implementation, and handling of a wide range of initial conditions. The comparison of the present symmetrical results with the existing literature is satisfactory.

An accurate and efficient numerical solution for the generalized Burgers–Huxley equation via Taylor wavelets method: Qualitative analyses and Applications

This paper aims to propose a highly accurate and simple algorithm based on the Taylor wavelet methods for obtaining the approximate solution of the generalized Burgers–Huxley equation. Additionally, various qualitative analyses including positivity-preservation, monotonicity-preservation, boundedness of the obtained solutions as well as convergence analysis of the proposed method have been provided. Furthermore, the applicability and validity of the proposed method are demonstrated on a benchmark equation. By comparing the approximate solutions with the exact solution, it is observed that the proposed method has recorded better results than the other methods in the literature. Although the generalized Burgers–Huxley equation has been studied throughout the study, it is worth emphasizing that the proposed method is a good solver for such nonlinear equations.

The analysis of the solution of the Burgers–Huxley equation using the Galerkin method

AbstractMany scientific methods have been proposed to solve the Burgers–Huxley equation. Among these famous methods, little or nothing has been mentioned in the literature about the coupling of the nonstandard finite difference method in the time and the Galerkin together with the compactness methods in the space variables. We exploit this gap and present a reliable technique aimed to use this coupling technique and show that the solution of the problem under investigation is uniqued in appropriate spaces to be defined. We proceed to show that the numerical solution obtained from the designed scheme is stabled and converges in the as well as the H1‐norms. Furthermore, we use examples together with some numerical experiments to justify the validity of our theoretical results.

Solving the conformable Huxley equation using the complex method

<abstract><p>We apply the complex method to build new exact solutions for the conformable Huxley equation. The results show that abundant new exact solutions were constructed, which extends the results of Cevikel, Bekir and Zahran. Further, we extend the undetermined form for the exponential function solutions in the complex method.</p></abstract>

Natural transform decomposition method for the numerical treatment of the time fractional Burgers–Huxley equation

AbstractIn this article, we studied the time‐fractional Burgers–Huxley equation using the natural transform decomposition method. The fractional operator is treated in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu senses. We employed the natural transform with the Adomian decomposition process on time‐fractional Burgers–Huxley equation to obtain the solution. To establish the uniqueness and convergence of the accomplished solution, the Banach's fixed point theorem is used. The obtained findings are visually shown in two‐ and three‐dimensional graphs for various fractional orders. To illustrate the efficacy of the method under discussion, numerical simulations are provided. The proposed solution captures the behavior of the reported findings for various fractional orders. A comparative study was conducted to ascertain the proposed method's correctness. The findings of this study establish that the technique investigated is both efficient and accurate for solving nonlinear fractional differential equations that arise in science and technology.