The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

Salah, Mohamed; Amer, R. M.; Matbuly, M. S.

doi:10.4236/am.2014.53033

Cited by 9 publications

(15 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Jiwari et al [ 40 ] employed the polynomial differential quadrature method (PDQM) for studying the generalized FN equations with time-dependent coefficients in 1D size. Salah et al [ 41 – 43 ] solved wave propagation, Fisher and FN equations by PDQM with Runge–Kutta fourth order (RK4), perturbation with PDQ block-marching technique and DQM with Implicit Euler.…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 3 , our main contributions include: four schemes based on differential quadrature are presented as polynomial differential quadrature (PDQ) technique, sinc differential quadrature (SDQM) [ 44 – 47 ], discrete singular convolution (DSC) based on delta Lagrange (DLK) and Regularized Shannon kernels (RSK) [ 48 – 57 ], and Runge–Kutta fourth order (RK4) [ 41 – 43 ]. Some numerical examples are presented and discussed in Sect.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

et al. 2022

View full text Add to dashboard Cite

In the present work, polynomial, discrete singular convolution and sinc quadrature techniques are employed as the new techniques to derive accurate and efficient numerical solutions for the reaction–diffusion equations. Three models, Fitzhugh-Nagumo, Newell–Whitehead–Segel, and tumor growth models, were presented. The equations of three models are reduced to nonlinear ordinary differential equations by using different quadrature schemes. Then, Runge–Kutta fourth-order method is employed to solve nonlinear ordinary differential equations. In addition, the MATLAB program is used to solve these problems. Comparisons between the new methods and the existing ones are included, demonstrating the ease of implementation and efficiency. Also, the calculated results are supported by four various statistical errors. It is found that the rate of error reaches ≤ 10–6 in discrete singular convolution depending on regularized Shannon kernel which is better than others. Further, a parametric analysis is presented to discuss the influence of diffusion and reaction parameters on the solution.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

et al. 2022

View full text Add to dashboard Cite

show abstract

“…In recent years, some researchers (Dahiya and Mittal, 2017;Thoudam, 2017;Aswin et al, 2017;Mittal and Rohila, 2017;Salah et al, 2014) have used DQM for the approximate solutions of integer order partial differential equations. To the best of authors' knowledge, DQM has never been implemented for the approximate solutions of any nonlinear fractional partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…Bernstein polynomial-based DQM is used by Mittal and Rohila (2017) for the solution of nonlinear diffusion equations. In Salah et al (2014), implement the DQM along with Runge-Kutta method of order four as well as along with implicit Euler method on the thermal wave propagation model in one and two dimensions.…”

Section: Introductionmentioning

confidence: 99%

Differential quadrature method for nonlinear fractional partial differential equations

Saeed

Rehman

Din³

2018

View full text Add to dashboard Cite

Purpose The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results. Design/methodology/approach The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations. Findings The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method. Originality/value Many engineers can use the presented method for solving their nonlinear fractional models.

show abstract

“…They compared the numerical solution with the results from Runge-Kutta method. They overcome the limitation of stability and one can use large step size [16]. Shu et al early presented block-marching technique with DQ discretization to obtain the solution in the time direction block by block to overcome the above difficulty.…”

Section: Open Access Ammentioning

confidence: 99%

An Efficient Method to Solve Thermal Wave Equation

Salah¹,

Amer²,

Matbuly³

2014

Self Cite

View full text Add to dashboard Cite

In this paper, an efficient technique of differential quadrature method and perturbation method is employed to analyze reaction-diffusion problems. An efficient method is presented to solve thermal wave propagation model in one and two dimensions. The proposed method marches in the time direction block by block and there are several time levels in each block. The global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives. Furthermore, the proposed method is validated by comparing the obtained results with the available analytical ones and also compared with the hybrid technique of differential quadrature method and Runge-Kutta fourth order method.

show abstract

The Differential Quadrature Solution of Reaction-Diffusion Equation Using Explicit and Implicit Numerical Schemes

Cited by 9 publications

References 12 publications

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

Modeling and Solution of Reaction–Diffusion Equations by Using the Quadrature and Singular Convolution Methods

Differential quadrature method for nonlinear fractional partial differential equations

An Efficient Method to Solve Thermal Wave Equation

Contact Info

Product

Resources

About