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Research Papers

Numerical Solution of Nonlinear Reaction–Advection–Diffusion Equation

[+] Author and Article Information
Anup Singh

Department of Mathematical Sciences,
Indian Institute of Technology (BHU),
Varanasi 221005, India

S. Das

Department of Mathematical Sciences,
Indian Institute of Technology (BHU),
Varanasi 221005, India
e-mail: sdas.apm@iitbhu.ac.in

S. H. Ong

Institute of Mathematical Sciences,
University of Malaya,
Kuala Lumpur 50603, Malaysia;
Department of Actuarial Science
and Applied Statistics,
UCSI University,
Kuala Lumpur 56000, Malaysia

H. Jafari

Department of Mathematical Sciences,
University of South Africa (UNISA),
Pretoria 0003, South Africa

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 17, 2018; final manuscript received January 16, 2019; published online February 15, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(4), 041003 (Feb 15, 2019) (6 pages) Paper No: CND-18-1173; doi: 10.1115/1.4042687 History: Received April 17, 2018; Revised January 16, 2019

In the present article, the advection–diffusion equation (ADE) having a nonlinear type source/sink term with initial and boundary conditions is solved using finite difference method (FDM). The solution of solute concentration is calculated numerically and also presented graphically for conservative and nonconservative cases. The emphasis is given for the stability analysis, which is an important aspect of the proposed mathematical model. The accuracy and efficiency of the proposed method are validated by comparing the results obtained with existing analytical solutions for a conservative system. The novelty of the article is to show the damping nature of the solution profile due to the presence of the nonlinear reaction term for different particular cases in less computational time by using the reliable and efficient finite difference method.

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Figures

Grahic Jump Location
Fig. 1

A partition of the (x,t)-plane into uniform cells of size h×k

Grahic Jump Location
Fig. 2

Comparison between analytical and numerical results at t =2.5, 10 when λ=0

Grahic Jump Location
Fig. 3

(a) Plots of u(x,t)/u0 versus x at t=2.5, 5, 10, 15, 20, D =0.6 and V =0.6 when λ=0 and (b) plots of u(x,t)/u0 versus x at t=2.5, 5, 10, 15, 20, D =0.6 and V =0.6 when λ≠0

Grahic Jump Location
Fig. 4

(a) Plots of u(x,t)/u0 versus x at t=1, 2.5, 5, 10,D =0.6 and V =0 when λ=0 and (b) plots of u(x,t)/u0 versus x at t=1,  2.5, 5, 10,D =0.6 and V =0 when λ≠0

Grahic Jump Location
Fig. 5

(a) Plots of u(x,t)/u0 versus x at t=2.5,  5,  10,  15,  20, D =1 and V =1 when λ=0 and (b) plots of u(x,t)/u0 versus x at t=2.5,  5,  10,  15,  20,D =1 and V =1 when λ≠0

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