Stability analysis and highly accurate numerical approximation of Fisher’s equations using pseudospectral method

Balyan, L. K.; Mittal, A. K.; Kumar, Mukesh; Choube, M.

doi:10.1016/j.matcom.2020.04.012

Cited by 22 publications

(8 citation statements)

References 32 publications

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“…Ma et al [10] applied the ChebyshevGalerkin and Chebyshev collocation methods to solve the two-dimensional Helmholtz model equation. Balyan et al [11] presented a pseudospectral method to solve Fisher's equation in 1D and 2D using Chebyshev-Gauss-Lobatto points and collocation in the spatial and the temporal directions.…”

Section: Introductionmentioning

confidence: 99%

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Laouar

Arar

Abdelhamid

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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This paper aims to develop a numerical approximation for the solution of the advection-diffusion equation with constant and variable coefficients. We propose a numerical solution for the equation associated with Robin's mixed boundary conditions perturbed with a small parameter $\varepsilon$. The approximation is based on a couple of methods: A spectral method of Galerkin type with a basis composed from Legendre-polynomials and a Gauss quadrature of type Gauss-Lobatto applied for integral calculations with a stability and convergence analysis. In addition, a Crank-Nicolson scheme is used for temporal solution as a finite difference method. Several numerical examples are discussed to show the efficiency of the proposed numerical method, specially when $\varepsilon$ tends to zero so that we obtain the exact solution of the classic problem with homogeneous Dirichlet boundary conditions. The numerical convergence is well presented in different examples. Therefore, we build an efficient numerical method for different types of partial differential equations with different boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Laouar

Arar

Abdelhamid

2023

Advances in the Theory of Nonlinear Analysis and Its Application

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“…Moreover Gottlieb and Wang 39 have proposed a fully discrete Fourier collocation spectral method for the numerical solutions, stability analysis, and convergence analysis of viscous burgers equation. Further, rigorous work on pseudospectral methods for partial differential equations was published 40–42 …”

Section: Introductionmentioning

confidence: 99%

“…Further, rigorous work on pseudospectral methods for partial differential equations was published. [40][41][42] The major contributions of this paper are as follows:…”

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confidence: 99%

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Mittal

Balyan

Panda

et al. 2021

Math Methods in App Sciences

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In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

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“…Trofimov and Peskov [40] have also solved the GPE using a conservative finite difference method. Moreover, multidimensional NSE was solved by several other numerical methods, such as riccati expansion method [1], finite difference method [38, 44], finite element method [10] Galerkin finite element method [43], momentum representation method [9], symplectic and multisymplectic methods [2, 39], compact scheme [18], split‐step Fourier scheme [32], compact boundary value method [31], spectral method [3, 28, 29, 30], operational matrix method [23, 27, 37], discrete collocation method based [26], and spline collocation method [22].…”

Section: Introductionmentioning

confidence: 99%

Time–space Jacobi pseudospectral simulation of multidimensional Schrödinger equation

Mittal

Shrivastava

Panda

2020

Numerical Methods Partial

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In this paper, the authors investigate the collision of soliton waves for multidimensional nonlinear Schrödinger equation (NSE) using the time–space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions and to prove the theory of error estimates and stability analysis for the equations. Using the Jacobi derivatives matrices the given problem is reduced to a system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. For numerical experiments, the method is tested on a number of different examples to study the behavior of collision of two and more than two solitons waves and single soliton wave. Moreover, numerical solutions are demonstrated to justify the theoretical results. The rate of convergence of the proposed method is obtained up to six‐order. The proposed method also preserves mass and energy conservation laws. A comparison of the numerical and exact solutions is depicted in the form of figures and tables.

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Stability analysis and highly accurate numerical approximation of Fisher’s equations using pseudospectral method

Cited by 22 publications

References 32 publications

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Efficient spectral Legendre Galerkin approach for the advection diffusion equation with constant and variable coefficients under mixed Robin boundary conditions

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

Time–space Jacobi pseudospectral simulation of multidimensional Schrödinger equation

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