Burgers Huxley Equation Research Articles

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.

Numerical treatment of singularly perturbed unsteady Burger-Huxley equation

This article deals with the numerical treatment of a singularly perturbed unsteady Burger-Huxley equation. The equation is linearized using the Newton-Raphson-Kantorovich approximation method. The resulting linear equation is discretized using the implicit Euler method and an exponential spline method for time and space variables, respectively. Richardson's extrapolation technique is employed to increase the accuracy in the temporal direction. The stability and uniform convergence of the proposed scheme are investigated. The scheme is shown uniformly convergent with the order of convergence O(τ + ℓ2) and O(τ2 + ℓ2) before and after Richardson extrapolation, respectively. Several test examples are considered to validate the applicability and efficiency of the scheme. It is observed that the proposed scheme provides more accurate results than the methods available in the literature.

A sixth-order finite difference method for solving the generalized Burgers-Fisher and generalized Burgers-Huxley equations

A sixth-order finite difference method for solving the generalized Burgers-Fisher and generalized Burgers-Huxley equations

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.

Fitted numerical method for singularly perturbed Burger–Huxley equation

This paper deals with the numerical treatment of a singularly perturbed unsteady Burger–Huxley equation. We linearize the problem using the Newton–Raphson–Kantorovich approximation method. We discretize the resulting linear problem using the implicit Euler method and specially fitted finite difference method for time and space variables, respectively. We provide the stability and convergence analysis of the method, which is first-order parameter uniform convergent. We present several model examples to illustrate the efficiency of the proposed method. The numerical results depict that the present method is more convergent than some methods available in the literature.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

The Approximate Solution of Singularly Perturbed Burger-Huxley Equation with RDTM

In this research, the numerical integral method procedure on uniform mesh is used to solve the singularly perturbed problem which has integral boundary value. This method also includes the trapezoid method, the finite difference method, and the Thomas algorithm. The problem is converted to finite difference problem by using finite difference approximations and trapezoid method. Finally, the convergence of the presented method is analyzed through sample application. Thus, the correctness and sufficiency of the method are shown.

Differentiability of the transition semigroup of the stochastic Burgers-Huxley equation and application to optimal control

In this present paper we consider the transition semigroup {Pt}t≥0 related to the stochastic Burgers-Huxley equation which describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports. We are proving that it has a regularizing effect in the Banach space of continuous functions C(0,1), that is, the transition semigroup {Pt}t≥0 possesses strong Feller property. Some estimates on derivative of the transition semigroup related to stochastic Burgers-Huxley equation are established based on the exponential moment estimates of the solution of stochastic Burgers-Huxley equation and its first variation equation. Finally, certain application in Hamilton-Jacobi-Bellman equation arising from stochastic optimal control problem is provided to illustrate our results.

Numerical solution of generalized Burgers-Huxley equations using wavelet based lifting schemes

Numerical solution of generalized Burgers-Huxley equations using wavelet based lifting schemes

Numerical solution of Generalized Burger–Huxley & Huxley’s equation using Deep Galerkin neural network method

In this paper, a deep learning algorithm based on Deep Galerkin method (DGM) is presented for the approximate solution of the generalized Burgers–Huxley equation (gBHE), and generalized Huxley’s equation (gHE). In this method, a deep neural network (DNN) is used for approximating the solution without generating mesh grid, which satisfies the differential operator, boundary and initial conditions. DNN is trained on randomly selected batches of time and space points, thus helping to avoid forming a mesh. Adam optimizer is used for optimizing the parameters of the DNN. Further, the convergence of the cost function and convergence of the neural network to the exact solution is demonstrated. This method shows very encouraging results which have been compared with recent methods such as: A fourth order improved numerical scheme (FDS4), Adomain-decomposition method (ADM), Modified cubic B-spline differential quadrature method (MCB-DQM), Variational iteration method (VIM), and others.

Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation

The present work deals with the global solvability as well as asymptotic analysis of the stochastic generalized Burgers–Huxley (SGBH) equation perturbed by a white-in-time and correlated-in-space noise defined in a bounded interval of $${\mathbb {R}}$$ . We first prove the existence of a unique mild as well as strong solution to the SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of the SGBH equation, that is, irreducibility and the strong Feller property. These two properties guarantee the uniqueness of invariant measures and ergodicity also. Then, under further assumptions on the noise coefficient, we discuss the ergodic behavior of the solution of the SGBH equation by providing a large deviation principle for the occupation measure for large time (Donsker–Varadhan), which describes the exact rate of exponential convergence.

A robust Hermite spline collocation technique to study generalized Burgers-Huxley equation, generalized Burgers-Fisher equation and Modified Burgers’ equation

In this paper, a robust Hermite collocation technique is proposed to find the numerical solution of generalized Burgers-Huxley and Burgers-Fisher equations as well as modified Burgers’ equation. In this technique, Hermite collocation method with fifth order Hermite splines have been used to approximate the solution variable and its spatial derivatives. Crank-Nicolson finite difference scheme is applied on time derivatives. The quasilinearization technique is used to linearize the nonlinear terms in the equation. Von-Neumann method is applied to show stability of the proposed technique. Robustness of proposed technique is shown by solving five test examples of these three equations with different parameters. The computed numerical results are better than the results from other techniques compared in this paper and are also matched well with the exact solutions.

An Implicit Scheme for the Numerical Solution of the Generalized Burgers-Huxley Equation

An Implicit Scheme for the Numerical Solution of the Generalized Burgers-Huxley Equation

Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method

Numerical study of nonlinear generalized Burgers–Huxley equation by multiquadric quasi-interpolation and pseudospectral method

A Numerical Solution of the Generalized Burgers-Huxley Equation

In this study numerical solutions of the generalized Burgers-Huxley equation are obtained utilizing a new approach: The Crank Nicolson logarithmic finite difference method (CN-LFDM). The effectiveness of the suggested method is demonstrated by a numerical example for various parameter cases. Presented tables demonstrate that the obtained results are in excellent agreement with the exact solutions and better than numerical results acquired by other methods in the literature. The method was analyzed with the von-Neumann stability analysis method and it was shown that the method was unconditionally stable.

1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as  Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and  Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

Data-driven Discovery of Modified Kortewegde Vries Equation, Kdv–Burger Equation and Huxley Equation by Deep Learning

Data-driven Discovery of Modified Kortewegde Vries Equation, Kdv–Burger Equation and Huxley Equation by Deep Learning

Isolated periodic wave trains in a generalized Burgers–Huxley equation

We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.

Uniformly convergent computational method for singularly perturbed unsteady burger-huxley equation

This paper deals with the numerical treatment of a singularly perturbed unsteady non-linear Burger-Huxley problem. Due to the simultaneous presence of a singular perturbation parameter and non-linearity in the problem applying classical numerical methods to solve this problem on a uniform mesh are unable to provide oscillation-free results unless they are applied with very fine meshes inside the region. Thus, to resolve this issue, a uniformly convergent computational scheme is proposed. The scheme is formulated:•First, the non-linear singularly perturbed problem is linearized using the Newton-Raphson-Kantorovich quasilinearization technique.•The resulting linear singularly perturbed problem is semi-discretized in time using the implicit Euler method to yield a system of singularly perturbed ordinary differential equations in space.•Finally, the system of singularly perturbed ordinary differential equations are solved using fitted exponential cubic spline method.The stability and uniform convergence of the proposed scheme are investigated. The scheme is stable and ε−uniformly convergent with first order in time and second order in space directions. To validate the applicability of the proposed scheme several test examples are considered. The obtained numerical results depict that the proposed scheme provides more accurate results than some methods available in the literature.