Space–time localized radial basis function collocation method for solving parabolic and hyperbolic equations

Hamaidi, Mohammed; Naji, Ahmed; Charafi, A.

doi:10.1016/j.enganabound.2016.03.009

Cited by 39 publications

(22 citation statements)

References 15 publications

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“…To recall the space-time localized RBF method defined in [13], let Ω ⊂ ℝ d be a bounded domain with a sufficiently regular boundary ∂Ω and consider the following time-dependent boundary value problem…”

Section: Space-time Localized Rbf Methods Formulationmentioning

confidence: 99%

“…Although few works were published concerning the space-time meshless method for solving PDEs with independent variable coefficients [13][14][15][16][17][18], up to date and to our best knowledge, there is no investigation on the application of space-time meshless method for solving PDEs with either variable or time-dependent variable coefficients. In this paper, we investigate the application of space-time localized meshless collocation radial basis functions developed in [13] to a general second-order parabolic and hyperbolic problems, with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

Hamaidi

Naji

Taik

2021

Modelling and Simulation in Engineering

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In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.

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Section: Space-time Localized Rbf Methods Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

Hamaidi

Naji

Taik

2021

Modelling and Simulation in Engineering

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“…Some other efficient space-time methods developed can be found in [23,14,22,1,30]. A recent work [8,11] was carried out to develop time-space methods for numerically approximating the time-dependent partial differential equations. In the present work, a time-space numerical technique is constructed which is based on time-space radial kernels for solving the generalized Black-Scholes equation (1).…”

Section: Marjan Uddin and Hazrat Alimentioning

confidence: 99%

Space-time kernel based numerical method for generalized Black-Scholes equation

Uddin¹,

Ali²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

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“…Hence, using the Hopf-Cole transformation given by Equation (21), the (exact) Fourier solution to the problem given by Equations (18)- (20) is obtained as…”

Section: Examplementioning

confidence: 99%

“…Various types of space-time formulations can be found in [14][15][16][17][18]. Recently, the authors in the work [19,20] developed efficient space-time methods for solving time-dependent PDEs. In this work, a space-time numerical scheme is constructed using anisotropic kernels to solve nonlinear Burgers' equations.…”

Section: Introductionmentioning

confidence: 99%

The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

Uddin

Ali

2018

Mathematics

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It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations.

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Space–time localized radial basis function collocation method for solving parabolic and hyperbolic equations

Cited by 39 publications

References 15 publications

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

Space-time kernel based numerical method for generalized Black-Scholes equation

The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

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