Nonstandard finite‐difference scheme to approximate the generalized Burgers‐Fisher equation

Namjoo, Mehran; Zeinadini, M.; Zibaei, Sadegh

doi:10.1002/mma.5283

Cited by 14 publications

(5 citation statements)

References 35 publications

(77 reference statements)

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“…The advantages of the hybrid methods are seen especially from the steep behavior of the produced results. We show that our methods stabilize the solutions much earlier than the methods suggested by some literature [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]26]. The present works of literature are aimed at obtaining numerical solutions accurately.…”

Section: Introductionmentioning

confidence: 79%

“…Recently, spline functions with some numerical schemes have been used in acquiring numerical solutions of the Burgers' equation such as cubic and quadratic B-spline collocation method [14], modified cubic B-spline collocation method [15], B-spline Galerkin method and B-spline collocation method [16], collocation method based on Hermite formula and cubic B-splines [17], a cubic B-spline Galerkin method with higher order splitting approaches [18], cubic B-spline and fourth-order compact finite difference method [19], and cubic B-spline and differential quadrature method [20]. Also, implicit fractional step θ-scheme and conforming finite element method [21], radial basis functions (RBF) meshless method [22], nonstandard finite difference method [23], and a sixth-order compact finite difference scheme for space integration and Crank-Nicolson scheme for time discretization were used in [24].…”

Section: Introductionmentioning

confidence: 99%

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A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

Cakay,

Selim

2024

Journal of Applied Mathematics

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This paper presents a study on the numerical solutions of the Burgers’ equation with forcing effects. The article proposes three hybrid methods that combine two-point, three-point, and four-point discretization in time with the Galerkin finite element method in space (TDFEM2, TDFEM3, and TDFEM4). These methods use backward finite difference in time and the finite element method in space to solve the Burgers’ equation. The resulting system of the nonlinear ordinary differential equations is then solved using MATLAB computer codes at each time step. To check the efficiency and accuracy, a comparison between the three methods is carried out by considering the three Burgers’ problems. The accuracy of the methods is expressed in terms of the error norms. The combined methods are advantageous for small viscosity and can produce highly accurate solutions in a shorter time compared to existing numerical schemes in the literature. In contrast to many existing numerical schemes in the literature developed to solve Burgers’ equation, the methods can exhibit the correct physical behavior for very small values of viscosity. It has been demonstrated that the TDFEM2, TDFEM3, and TDFEM4 can be competitive numerical methods for addressing Burgers-type parabolic partial differential equations arising in various fields of science and engineering.

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

Cakay,

Selim

2024

Journal of Applied Mathematics

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“…Nowadays, NSFD schemes have been widely used as a powerful and efficient class of numerical methods for solving differential equations arising in real-world situations. We refer the readers to [32][33][34][35][36][37][38][39] and [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] for good reviews and some recent notable works related to NSFD schemes, respectively. Recently, we have successfully developed the Mickens' methodology to construct NSFD schemes for differential equation models arising in real-world applications [55][56][57][58][59][60].…”

Section: Introductionmentioning

confidence: 99%

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

Hoang

Ngo²,

Truong

2023

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this work, we introduce a simple method for investigating the asymptotic stability of discrete dynamical systems, which can be considered as an extension of the classical Lyapunov's indirect method. This method is constructed based on the classical Lyapunov's indirect method and the idea proposed by Ghaffari and Lasemi in a recent work. The new method can be applicable even when equilibia of dynamical systems are non-hyperbolic. Hence, in many cases, the classical Lyapunov's indirect method fails but the new one can be used simply. In addition, by combining the new stability method with the Mickens' methodology, we formulate some nonstandard finite difference (NSFD) methods which are able to preserve the asymptotic stability of some classes of differential equation models even when they have non-hyperbolic equilibrium points. As an important consequence, some well-known results on stability-preserving NSFD schemes for autonomous dynamical systems are improved and extended. Finally, a set of numerical examples are performed to illustrate and support the theoretical findings.

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“…e general alternating difference schemes with variable time steps are constructed and proved to be unconditionally stable. Namjoo et al [22] presented the numerical solution of the generalized Burgers-Fisher equation on the basis of the nonstandard finite-difference (NSFD) scheme. e positivity, consistency, and boundedness of the scheme are discussed.…”

Section: Introductionmentioning

confidence: 99%

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

Pan

Yang

2020

Mathematical Problems in Engineering

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This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

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Nonstandard finite‐difference scheme to approximate the generalized Burgers‐Fisher equation

Cited by 14 publications

References 35 publications

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation

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