An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

Oruç, Ömer

doi:10.1007/s00366-019-00734-z

Cited by 38 publications

(17 citation statements)

References 58 publications

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“…Therefore, it is very important to devise reliable and efficient numerical techniques. There are plenty of numerical techniques that have been devised for acquiring numerical solution of PDEs so Linear barycentric interpolation far, such as finite element, finite difference, meshless, wavelet and spectral methods (Li et al, 2020;Chen and Chen, 2020;Zhang, 2020a, 2020b;Wang et al, 2019;Oruç et al, 2016;Oruç, 2019aOruç, , 2020aDehghan and Taleei, 2012).…”

Section: A Brief Review Of Numerical Methodsmentioning

confidence: 99%

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Oruç¹

2020

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Purpose The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations. Design/methodology/approach The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times. Findings Numerical simulations corroborate the efficiency and accuracy of the proposed method. Originality/value Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.

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Section: A Brief Review Of Numerical Methodsmentioning

confidence: 99%

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Oruç¹

2020

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“…Luther was the first researcher who found the wave speed of this model [19]. Many researchers have investigated the traveling wave solutions of this model, and they have also improved it into (2+1)-dimensional form, that is given by [20][21][22][23]…”

Section: Introductionmentioning

confidence: 99%

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

Khater

Alabdali

2021

Mathematics

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The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique’s conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

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“…Various types of algorithms are considered in the literature, for instance, finite-difference schemes (Garvie, 2007;Ruuth, 1995), finite volume spectral element method (Shakeri and Dehghan, 2011), positive finite volume methods (Gerisch and Chaplain, 2006), discontinuous Galerkin methods (Zhang et al, 2014;Zhu et al, 2009), differential quadrature (DQ) methods (Mittal and Rohila, 2016;Jiwari et al, 2017), variational multiscale element-free Galerkin and local discontinuous Galerkin methods (Dehghan and Abbaszadeh, 2016), the method of fundamental solutions (Wang et al, 2020), the singular boundary method (Qiu et al, 2019), meshfree local Petrov-Galerkin methods (Ilati and Dehghan, 2017), operator splitting and approximate matrix factorization time-stepping schemes (Gerisch and Verwer, 2002) and implicit-explicit time-stepping schemes (Madzvamuse, 2006). Recently, some novel algorithms such as Lin et al (2018Lin et al ( , 2020a; Lin and Reutskiy (2020); Lin et al (2020b); Oruç (2020b); Oruç et al (2019); Oruç (2020aOruç ( , 2020c were developed for convection-diffusion-reaction problems.…”

Section: Introductionmentioning

confidence: 99%

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

Jiwari

Gerisch

2021

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Purpose This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time. Design/methodology/approach In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight. Findings The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems. Originality/value The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

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An efficient wavelet collocation method for nonlinear two-space dimensional Fisher–Kolmogorov–Petrovsky–Piscounov equation and two-space dimensional extended Fisher–Kolmogorov equation

Cited by 38 publications

References 58 publications

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Multiple Novels and Accurate Traveling Wave and Numerical Solutions of the (2+1) Dimensional Fisher-Kolmogorov- Petrovskii-Piskunov Equation

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

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