Exact Finite Difference 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.

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

ObjectivesIn this paper, a numerical scheme is designed for solving singularly perturbed Fredholm integro-differential equation. The scheme is constructed via the exact (non-standard) finite difference method to approximate the differential part and the composite Simpson’s 1/3 rule for the integral part of the equation.ResultThe stability and uniform convergence analysis are demonstrated using solution bound and the truncation error bound. For three model examples, the maximum absolute error and the rate of convergence for different values of the perturbation parameter and mesh size are tabulated. The computational result shows, the proposed method is second-order uniformly convergent which is in a right agreement with the theoretical result.

Fitted exact difference method for solutions of a singularly perturbed time delay parabolic PDE

A novel numerical method has been proposed for solving singularly perturbed parabolic time delay PDE by using a fitted mesh exact finite difference method. The traditional numerical methods on uniform mesh fail to accurately approximate the solution of the considered PDE, due to the presence of a boundary layer. Therefore, developing a numerical scheme that can effectively handle the boundary layer behaviour is of great interest. The proposed scheme utilizes Crank–Nicolson method for temporal discretization and fitted mesh exact finite difference method for spatial discretization. The scheme satisfies the discrete maximum principle, stability bound, and uniform convergence with the boundary layer resolving property. Numerical test examples have been conducted to validate the scheme for different values of the perturbation parameter and mesh numbers. The results show that the proposed scheme outperforms existing methods in the literature in terms of accuracy. The proposed approach holds significant potential for solving similar PDEs that exhibit boundary layer behaviour.

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.

Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length.

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.

Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index α=1, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.

Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

We investigate wave-packet dynamics across supercritical barriers for the Klein-Gordon and Dirac equations. Our treatment is based on a multiple scattering expansion (MSE). For spin-0 particles, the MSE diverges, rendering invalid the use of the usual connection formulas for the scattering basis functions. In a time-dependent formulation, the divergent character of the MSE naturally accounts for charge creation at the barrier boundaries. In the Dirac case, the MSE converges and no charge is created. We show that this time-dependent charge behavior dynamics can adequately explain the Klein paradox in a first-quantized setting. We further compare our semianalytical wave-packet approach to exact finite-difference solutions of the relativistic wave equations.

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 discretization of non-commutative space-time

Non-commutative space-time is discussed for discrete case. An exact discretization of the non-commutative space-time is proposed. New discrete operators, which satisfy the same algebraic relations as standard derivatives of integer orders, are used. Exact discretization of the non-commutative space-time is suggested by using generalization of the star-product (the Moyal–Weyl product) based on the recently proposed exact finite differences. The suggested discrete non-commutative space-time corresponds to the standard “continuous” non-commutative space-time without any approximations. The suggested discrete non-commutative space-time can be interpreted as a special physical (crystal) lattice with long-range interaction.

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.

Numerical dynamics of nonstandard finite difference schemes for a Logistics model with feedback control

In this paper, we construct nonstandard finite difference (NSFD) schemes that preserve essential qualitative properties of a Logistics model with feedback control using the Mickens’ methodology. These properties of the model include positivity of solutions and stability of equilibrium points. Dynamical properties of the proposed NSFD schemes are rigorously investigated. The results show that the constructed nonstandard schemes preserve the essential properties of the continuous model for all finite step sizes. Consequently, we obtain NSFD schemes which are dynamically consistent with the continuous model. Especially, the constructed NSFD schemes become exact finite difference ones in some special cases. Besides, some numerical simulations are performed to validate the theoretical results as well as the advantages of the proposed NSFD schemes. The numerical simulations also indicate that the NSFD schemes are effective and appropriate to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the second order and fourth order Runge–Kutta schemes can generate numerical solutions that are completely different from the solutions of the continuous model and therefore, they fail to correctly reflect the correct behavior of the continuous model.

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.

Stochastic consistency and stochastic stability in mean square sense for Cauchy advection problem

This work deals with the construction of finite difference solutions of random advection Cauchy type partial differential equation containing uncertainty through the coefficient of the velocity. Under appropriate hypothesis on the velocity random variable, we establish that the constructed random finite difference solution is mean square consistent and mean square stable over the whole real line. In addition, the main statistical functions, such as the mean, of the approximate solution stochastic process generated by truncation of the exact finite difference solution are given. Finally, we apply the proposed technique to several illustrative examples which show our discussing for the mean square stability.

Exact Finite Differences. The Derivative on Non Uniformly Spaced Partitions

We define a finite-differences derivative operation, on a non uniformly spaced partition, which has the exponential function as an exact eigenvector. We discuss some properties of this operator and we propose a definition for the components of a finite-differences momentum operator. This allows us to perform exact discrete calculations.

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 $\textbf{x}' = A\textbf{x }+ f(\textbf{x})$.

Accelerators in Macroeconomics: Comparison of Discrete and Continuous Approaches

We prove that the standard discrete-time accelerator equation cannot be considered as an exact discrete analog of the continuous-time accelerator equation. This leads to fact that the standard discrete-time macroeconomic models cannot be considered as exact discretization of the corresponding continuous-time models. As a result, the equations of the continuous and standard discrete models have different solutions and can predict the different behavior of the economy. In this paper, we propose a self-consistent discrete-time description of the economic accelerators that is based on the exact finite differences. For discrete-time approach, the model equations with exact differences have the same solutions as the corresponding continuous-time models and these discrete and continuous models describe the same behavior of the economy. Using the Harrod-Domar growth model as an example, we show that equations of the continuous-time model and the suggested exact discrete model have the same solutions and these models predict the same behavior of the economy.