Exact Finite Difference Schemes Research Articles

Some novel exact and non-standard finite difference schemes for a class of diffusion–advection–reaction equation

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The L 2 , L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes.

An exact finite difference scheme for a three-layer beam model

ABSTRACT In this work, we propose an efficient computational technique for a recognized three-layer beam model which is described by a third-order linear ordinary differential equation. By using the Mickens' methodology, an exact finite difference (EFD) scheme for this model is constructed. The EFD scheme provides an explicit recurrence formula to compute the solution of the beam model, but it is more efficient than the formula of the exact solution. Moreover, the EFD scheme is also more efficient than well-known standard finite difference schemes in both computational time and accuracy. Finally, two numerical experiments are performed to illustrate the theoretical results as well as to show advantages of the EFD scheme.

On the convergence of NSFD schemes for a new class of advection–diffusion–reaction equations

This paper deals with the two classes of non-linear advection–diffusion–reaction (ADR) equations subject to certain initial and boundary conditions. We derive exact finite difference (FD) schemes with the help of solitary wave solutions of ADR equations. Furthermore, we construct non-standard FD schemes for both the equations. The positivity and boundedness properties of the equations are satisfied by the proposed non-standard FD schemes. We also prove that these schemes are stable under certain conditions. Moreover, they are shown to be first order consistent in both time and space. We compute solutions of the ADR equations by applying proposed non-standard FD schemes under given initial and boundary conditions. The obtained results demonstrate that the proposed methods are simple and powerful tool in finding the numerical solutions of ADR equations. The schemes give good accuracy even for few spatial divisions.

Exact and nonstandard finite difference schemes for the Burgers equationB(2,2)

In this paper, we consider the Burgers equation B(2,2). Exact and nonstandard finite difference schemes (NSFD) for the Burgers equation B(2,2) are designed. First, two exact finite difference schemes for the Burgers equation B(2,2) are proposed using traveling wave solution. Then, two NSFD schemes are represented for this equation. These two NSFD schemes are compared with a standard finite difference (SFD) scheme. Numerical results show that the NSFD schemes are accurate and efficient in the numerical simulation of the kink-wave solution of the B(2,2) equation. We see that although the SFD scheme yields numerical instability for large step sizes, NSFD schemes provide reliable results for long time integration. Local truncation errors show that the NSFD schemes are consistent with the B(2,2) equation.

An efficient Mickens' type NSFD scheme for the generalized Burgers Huxley equation

We consider the generalized Burgers–Huxley (GBH) equation subject to certain initial and boundary conditions (BCs). Using a solitary wave solution, we derive an exact finite difference (EFD) scheme for the GBH equation. Furthermore, we propose a non-standard finite difference (NSFD) scheme which operates for all The qualitative properties, i.e. positivity and boundedness, are satisfied by the proposed NSFD scheme. Moreover, the stability and consistency of the NSFD scheme are also discussed. Our scheme is stable under certain conditions with the first-order accuracy in both time and space. We compute solutions of the GBH equation for various values of at a different time using the NSFD scheme and calculate their respective maximum errors. The maximum error of NSFD solutions is compared with the maximum error of several other methods to depict the supremacy of the proposed method. We also compute CPU time for all the computations which reveal that our scheme gives an accurate result within few seconds which saves our time. Our scheme gives precise results with only a few spatial divisions.

Exact and nonstandard finite difference schemes for the generalized KdV\u2013Burgers equation

We consider the generalized KdV–Burgers operatorname{KdVB}(p,m,q) equation. We have designed exact and consistent nonstandard finite difference schemes (NSFD) for the numerical solution of the operatorname{KdVB}(2,1,2) equation. In particular, we have proposed three explicit and three fully implicit exact finite difference schemes. The proposed NSFD scheme is linearly implicit. The chosen numerical experiment consists of tanh function. The NSFD scheme is compared with a standard finite difference(SFD) scheme. Numerical results show that the NSFD scheme is accurate and efficient in the numerical simulation of the kink-wave solution of the operatorname{KdVB}(2,1,2) equation. We see that while the SFD scheme yields numerical instability for large step sizes, the NSFD scheme provides reliable results for long time integration. Local truncation error reveals that the NSFD scheme is consistent with the operatorname{KdVB}(2,1,2) equation.

On the stability of Micken's type NSFD schemes for generalized Burgers Fisher equation

ABSTRACTWe propose exact finite difference scheme (EFD) for Generalized Burgers Fisher (GBF) Equation using solitary wave solution. Moreover a non-standard finite difference (NSFD) scheme is also proposed. The proposed EFD and NSFD scheme works for all This scheme preserves the positivity and boundedness properties which is also discussed here. The method is shown to be stable, consistent and first-order accurate in both space and time. Approximate solutions of the GBF equation are obtained using NSFD scheme and in order to show the accuracy, the computed solutions are compared with the exact solution. Comparisons are shown with the other available methods to indicate that our method gives better result. These methods give accurate results even for relatively bigger values of step size.

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.

Exact Finite Difference Schemes for Three-Dimensional Linear Systems with Constant Coefficients

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for problems with periodic solutions on very large time interval demonstrate the efficiency and exactness of the EDS compared with high-order numerical methods. This result can be extended for constructing EDS for general systems of $n$ linear differential equations with constant coefficients and nonstandard finite difference (NSFD) schemes preserving stability properties for quasi-linear system of equations $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.

Numerical solutions of FitzHugh–Nagumo equation by exact finite-difference and NSFD schemes

In this paper, numerical solution of the FitzHugh–Nagumo (FN) equation is presented based on the nonstandard finite-difference (NSFD) scheme. At first, two exact finite-difference schemes for FN equation are obtained. Moreover, two NSFD schemes are presented for this equation. The positivity, boundedness, and local truncation error of the schemes are discussed. The numerical results obtained by the NSFD schemes are compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD schemes.

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.

Exact finite-difference scheme and nonstandard finite-difference scheme for coupled Burgers equation

Abstract This work develops exact finite-difference schemes for the two-dimensional nonlinear coupled viscous Burgers equation using the analytic solution. We extend the explicit nonstandard finite-difference schemes on the basis of the exact finite-difference schemes to solve the coupled Burgers equation. Numerical examples are presented to verify the efficiency and accuracy of the methods. MSC:39A10, 65L12, 74H15.

Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

Exact finite difference schemes for three-dimensional advection-reaction equations

Exact finite difference schemes for three-dimensional advection-reaction equations

Exact finite difference scheme for linear differential equation with constant coefficients

An exact finite difference equation for the n-th order linear differential equation with real, constant coefficients is constructed. The exact finite difference scheme is expressed differently but equivalent to that given by Potts [3].

A note on exact finite difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions

We construct the exact finite difference equation discretizations for the nonlinear differential equations whose solutions are the Jacobi cosine and sine functions. Our derivations clarify and extend previous work done on this topic.

Exact Finite-Difference Schemes ford-Dimensional Linear Stochastic Systems with Constant Coefficients

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

Exact finite difference and non-standard finite difference schemes for

Exact finite difference schemes and non-standard finite difference schemes are constructed for the first-order differential equation , for and . In particular, we show that the central finite difference scheme is an exact scheme for the differential equation .