Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method

Mittal, R. C.; Rohila, Rajni

doi:10.1016/j.chaos.2016.09.007

Cited by 44 publications

(29 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Furthermore, it is demonstrated that the nonuniform grids improve the accuracy of the solution considerably without any extra computational cost (see Table , Figure a,b). The present scheme offered better results than for the linear model, and similar patterns to for the Gray–Scott and Brusselator models.…”

Section: Resultssupporting

confidence: 71%

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Yadav

Jiwari

2018

Numerical Methods Partial

View full text Add to dashboard Cite

In this article, we establish the existence and uniqueness of solutions to the coupled reaction-diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank-Nicolson and the predictor-corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second-order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme. KEYWORDS a priori error estimate, Brusselator, coupled reaction-diffusion models, Crank-Nicolson method, existence and uniqueness, Galerkin method, Gray-Scott, predictor-corrector method INTRODUCTIONThe coupled reaction-diffusion models frequently arise in the field of chemistry [1, 2], biology [3,4], sociology [5], physics, geology, ecology, and so forth. These models are naturally applied in chemistry, for example, the Brusselator model. However, such models can also describe dynamical processes of nonchemical nature, for instance, the predator-prey model. In 1952, in his pioneering work on morphogenesis, Turing [6] first proposed that the reaction-diffusion systems may be used to study the replicating patterns such as stripes, spots, and dappling, seen on the skin of many animals such as zebras, lions, and cats. In [6], Turing explained that the involvement of diffusion, Numer Methods Partial Differential Eq. 2019;35:830-850. wileyonlinelibrary.com/journal/num

show abstract

Section: Resultssupporting

confidence: 71%

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Yadav

Jiwari

2018

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…With the help of the proposed method, this problem is solved for two sets of parameters: ε = 0.001, ε 1 = 0.1, ε 2 = 0.01 (reaction dominated case) and ε = 1, ε 1 = 0.1, ε 2 = 0.01 (diffusion dominated case), when α = 1 and M = 20, t = 0.01, 0.001. In Tables 11 and 12, the computed solutions are matched with the exact solution as well as with those reported in Mittal and Jiwari (2016). For comparison purpose, we have set (s, t) ∈ [0, π/2] × [0, 0.1].…”

Section: Computational Resultsmentioning

confidence: 99%

“…For comparison purpose, we have set (s, t) ∈ [0, π/2] × [0, 0.1]. From these tables, one can clearly see that the current approach produces more accurate solutions than (Mittal and Jiwari 2016) with less number of nodal points and time steps. Moreover, in Table 13, error norms are reported for the reaction and diffusion dominated cases at t max = 0.1 when α = 0.9 and s ∈ [0, π].…”

Section: Computational Resultsmentioning

confidence: 99%

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

et al. 2019

View full text Add to dashboard Cite

In this article, we compute numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete E 2 , E ∞ and E rms error norms. Varying the number of nodal points M and time step-size t, convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.

show abstract

“…Dag and Saka [21] applied B-spline collocation method to study equal width equation. Mittal [22] used B-splines to get numerical solutions of coupled reaction diffusion systems. Wave equations have been studied by Mittal and Bhatia [23] using modified cubic B-spline functions.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method

Rohila

Mittal

2018

Math Sci

Self Cite

View full text Add to dashboard Cite

Fourth-order B-spline collocation method has been applied for numerical study of Fisher's equation which represents several important phenomena such as biological invasions, reaction diffusion in chemical processes and neutron multiplication in nuclear reactions, etc. Results are found to be better than second-order B-spline collocation method. It is observed that when time becomes sufficiently large, local initial disturbance propagates with constant limiting speed. Proposed method is satisfactorily efficient in terms of accuracy and stability.

show abstract

Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method

Cited by 44 publications

References 38 publications

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method

Contact Info

Product

Resources

About