Trigonometric cubic B-spline collocation algorithm for numerical solutions of reaction–diffusion equation systems

Onarcan, Aysun Tok; Adar, Nihat; Dağ, İdris

doi:10.1007/s40314-018-0713-4

Cited by 15 publications

(7 citation statements)

References 15 publications

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“…In modal DG technique, the preference of orthogonal polynomial functions considerably facilitates the beneficence of the high-order moments of the approximate solutions to the reaction terms associated vector Θ in Eq. (30). Due to the orthogonal property of the polynomial functions and a simultaneous correlation ψ 1 = 1, when the scaled Legendre polynomial functions, ψ N , are multiplied by the mapping Jacobian…”

Section: Novel Reaction Terms Treatmentmentioning

confidence: 99%

“…Tamsir and co-authors [28,29] proposed a hybrid numerical method of cubic trigonometric B-spline base functions and differential quadrature algorithm for solving Fisher type NCRD systems. Onarcan et al [30] presented a trigonometric cubic B-splines based collocation scheme for obtaining the spatiotemporal dynamics of the NCRD systems. So far, some of the most well-known computational algorithms for developing the biological based spatiotemporal patterns are also carried out in the literature, for example, finite difference [31], positive finite volume [32], finite volume spectral element [33], and meshless local Petrov-Galerkin [34] methods.…”

Section: Introductionmentioning

confidence: 99%

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Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

Singh¹,

Battiato²,

Kumar³

2022

Preprint

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The nonlinear coupled reaction-diffusion (NCRD) systems are important in the formation of spatiotemporal patterns in many scientific and engineering fields, including physical and chemical processes, biology, electrochemical processes, fractals, viscoelastic materials, porous media, and many others. In this study, a mixed-type modal discontinuous Galerkin approach is developed for one-and two-dimensional NCRD systems, including linear, Gray-Scott, Brusselator, isothermal chemical, and Schnakenberg models to yield the spatiotemporal patterns. These models essentially represent a variety of complicated natural spatiotemporal patterns such as spots, spot replication, stripes, hexagons, and so on. In this approach, a mixed-type formulation is presented to address the second-order derivatives emerging in the diffusion terms. For spatial discretization, hierarchical modal basis functions premised on the orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge-Kutta algorithm. The spatiotemporal patterns generated with the present approach are very comparable to those found in the literature. This approach can readily be expanded to

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Section: Novel Reaction Terms Treatmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

Singh¹,

Battiato²,

Kumar³

2022

Preprint

View full text Add to dashboard Cite

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“…We can linearize the nonlinear terms with aid of the study given in previous studies 1,60,61 as

{(u_{i}^{j + 1})}^{2} = 2 u_{i}^{j} u_{i}^{j + 1} - {(u_{i}^{j})}^{2}, {(v_{i}^{j + 1})}^{2} = 2 v_{i}^{j} v_{i}^{j + 1} - {(v_{i}^{j})}^{2}, u_{i}^{j + 1} v_{i}^{j + 1} = u_{i}^{j + 1} v_{i}^{j} + u_{i}^{j} v_{i}^{j + 1} - u_{i}^{j} v_{i}^{j}, {(u_{i}^{j + 1})}^{2} v_{i}^{j + 1} = 2 u_{i}^{j + 1} u_{i}^{j} v_{i}^{j} + {(u_{i}^{j})}^{2} v_{i}^{j + 1} - 2 {(u_{i}^{j})}^{2} v_{i}^{j}, {(v_{i}^{j + 1})}^{2} u_{i}^{j + 1} = 2 u_{i}^{j} u_{i}^{j + 1} v_{i}^{j} + {(v_{i}^{j})}^{2} u_{i}^{j + 1} - 2 {(v_{i}^{j})}^{2} u_{i}^{j} .

Then, we have

{f_{1}}_{i}^{j} + {f_{1}}_{i}^{j + 1} = {G_{1}}_{i}^{j} u_{i}^{j + 1} + R_{1}

…”

Section: Numerical Approximationmentioning

confidence: 99%

“…Nonlinear reaction–diffusion equations (NRDEs) are a mathematical tool that describes many phenomena in engineering and science. Due to the existence of the reaction and diffusion terms, NRDSs can tackle complicated behaviors such as the Gray–Scott model, Belousov–Zhabotinskii reaction systems, Gierer–Meinhardt model, Lengyel–Epstein system, wave optics, and spread of infectious diseases; see Onarcan et al 1 …”

Section: Introductionmentioning

confidence: 99%

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

Zahra

Hikal

Băleanu

2021

Math Methods in App Sciences

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In this article, two nonstandard high‐order schemes on a uniform and nonuniform time stepping combined with the multi‐parameter exponential fitting technique (MPEF) have been developed to solve the time‐fractional nonlinear reaction–diffusion system. The first method based on the MPEF combined with the 3‐weighted shifted‐Grünwald–Letnikov approximation with uniform time stepping, this scheme leads to a numerical solution that suffers from the singularity near t = 0. In order to frustrate this singularity, a nonstandard higher‐order L1‐approximation for a nonuniform time‐stepping scheme is developed. The developed scheme's convergence and unconditionally stability analysis have been verified. Numerical results effectively validate the theoretical aspects.

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“…Hale [20] attained explicit nontrivial still patterns in the unidimensional Gray-Scott Reaction-diffusion model for cubic autocatalysis. Mittal and Rohila [21] solve the Gray-Scott model numerically by applying modified cubic B-spline, Jiwari et al [22] captured patterns of Gray-Scott model by accomplishing differential quadrature algorithm, Onarcan [23] introduced an algorithm for GSM reaction model.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution to the Gray-Scott Reaction-Diffusion equation using Hyperbolic B-spline

Kaur

Joshi

2022

J. Phys.: Conf. Ser.

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In the present paper, the one-dimensional Gray-Scott Reaction-diffusion equation is solved numerically with the usage of Hyperbolic B-spline along with the differential quadrature method. The hyperbolic B-spline is used to discretize the partial derivatives, by which the ordinary differential equations will be obtained which further are solved with the SSP-RK43 scheme. The efficiency and accuracy of the method are to be checked by using L ∞ and L 2 errors. The obtained numerical results are shown with help of 2D and 3D figures. As a deduction, it is concluded that the method is an efficient and effective technique for elucidating the Gray-Scott Reaction-diffusion equation and likewise for the variety of partial differential equations.

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Trigonometric cubic B-spline collocation algorithm for numerical solutions of reaction–diffusion equation systems

Cited by 15 publications

References 15 publications

Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

Spatiotemporal pattern formation in nonlinear coupled reaction-diffusion systems with a mixed-type modal discontinuous Galerkin approach

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

Numerical solution to the Gray-Scott Reaction-Diffusion equation using Hyperbolic B-spline

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