Collocation method for Convection-Reaction-Diffusion equation

Shokofeh Sharifi; Jalil Rashidinia

doi:10.1016/j.jksus.2018.10.004

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Collocation method for Convection-Reaction-Diffusion equation

Shokofeh Sharifi, Jalil Rashidinia^⁎

Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

⁎Corresponding author. rashidinia@iust.ac.ir (Jalil Rashidinia)

Received: 2018-7-16, Accepted: 2018-10-8, Published: 2018-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

A collocation method based on B-spline is developed for solving Convection-Reaction-Diffusion equation subjected to Dirichlet’s boundary conditions. The method comprises an explicit finite difference associated with extended cubic B-spline collocation method. We analyze the convergence and stability of the presented method. The proposed method is applied to various test examples and results are reported in Tables. The obtained numerical results have been compared with the results of the existing methods. Numerical results verify the efficiency and applicability of the scheme.

Keywords

65M06

65M12

65M15

65M70

Convection-Reaction-Diffusion equation

Redefined extended cubic B-spline

Stability and convergence analysis

1 Introduction

In this work, we consider the following one-dimensional parabolic Convection-Reaction-Diffusion equation:

(1)

Hu = g (u),

where H is defined as

Hu = (D_{t} + D_{x}) u,

and

D_{t} u (x, t) = (\frac{\partial}{\partial t}) u (x, t),

D_{x} u (x, t) = (a (x, t) \frac{\partial}{\partial x} - b (x, t) \frac{\partial^{2}}{\partial x^{2}}) u (x, t), (x, t) \in [c, d] \times [t_{0}, T],

with the initial condition

(2)

u (x, t_{0}) = f (x), c ⩽ x ⩽ d,

and Dirichlet boundary conditions

(3)

u (c, t) = Q_{0} (t), u (d, t) = Q_{1} (t), t_{0} ⩽ t ⩽ T

where

u (x, t)

is the unknown quantity being investigated.

a (x, t)

is the velocity of the medium which is called convection velocity,

b (x, t)

is the diffusion coefficient and

g (u)

is reaction source (Makungu et al., 2012).

The application of Convection-Reaction-Diffusion equation (CRD) can be classified into three processes: The first process is Convection which is due to the movement of materials from one region to another, the second process is Reaction which is due to decay, adsorption and reaction of substances with other components and the last process is Diffusion which is due to the movement of materials from high concentration to the low region. Qualitatively all the three processes from CRD model that describe how to disturbed quantity being studied in the medium changes (Makungu et al., 2012).

The Convection- Reaction- Diffusion partial differential equation provides very useful mathematical models in applied sciences such as physics, biology, hydrology and, engineering (Morton, 1996). The descriptions of Eq. (1) includes the transport chemistry in the atmosphere (McRea et al., 1982), heat transfer in a draining film (Isenberg and Gutfinge, 1972), finance (Wilmott et al., 1993), fluid flow (Cengel and Cimbala, 2006; Roache, 1972), hydrology (Cunningham et al., 1991), modelling of biological systems and population dynamics (Murray, 1989; Witelski, 1997).

During past years, many researchers paid attention to analysis, improvement, and implementation of the stable method for numerical solutions of nonlinear Convection-Reaction-Diffusion, see for example (Kaya, 2015).

A new two-level implicit difference method of order O( $k^{2}$ + ${kh}^{2}$ + $h^{4}$ ) based on three spatial grid points and arithmetic average discretization was applied by Mohanty et al. (1990), Mohanty et al. (2007) to determine the solution of one-dimensional quasi-linear and nonlinear parabolic partial differential equations, respectively.

In Duan et al. (2012), Hsieh and Yang (2016), Theeraek et al. (2011) authors proposed several finite element methods for approximation the solution of CRD equation. The Discontinuous Galerkin method has been applied for solving nonlinear CRD equation in Uzunca et al. (2014), Yucel et al. (2013). Monotone iterative methods for finding solution of CRD equation and improving the rate of convergence is suggested in Wang (2011). In Cui (2015) High order Compact exponential method has been used to solve fractional CRD equation with variable coefficients. Nonstandard finite difference combined with tension spline to determine the solution of CRD equation is used in Rashidinia and Shekarabi (2016). In Rashidinia and Sharifi (2016), Raslan et al. (2010) redefined extended B-spline collocation and quintic B-spline finite element method has been applied for solution of the partial differential equation.

In this paper, we develop the collocation method using a combination of finite difference and extended B-spline function for the solution of one-dimensional CRD equation. This article is arranged as follows: In Section 2, the description of the method is explained. Section 3, is devoted to prove the convergence and stability of the method. In Section 4, numerical illustrations are included to verify the accuracy and effectiveness of the presented approach. The paper ends with a brief conclusion.

2 Description of the method

2.1

2.1 Temporal discretization

In this section, we denote $Δ$ be a uniform partition of the domain $[c, d] \times [t_{0}, T]$ , where $x_{j} = c + jh, j = 0, \dots, N$ and $t_{n} = t_{0} + nk, n = 0, 1, 2, \dots$ .

Considering the finite difference operator

(4)

u_{t}^{n} ≅ \frac{δ_{t}}{k (1 - γ δ_{t})} u^{n}, γ \neq 1,

by discretizing time variable and substituting the operator (4) into Eq. (1), we obtain:

\frac{δ_{t}}{k (1 - γ δ_{t})} u^{n} + a (x, t_{n}) u_{x}^{n} - b (x, t_{n}) u_{xx}^{n} = g (u^{n}),

so we have

(5)

δ_{t} u^{n} + a (x, t_{n}) k (1 - γ δ_{t}) u_{x}^{n} - b (x, t_{n}) k (1 - γ δ_{t}) u_{xx}^{n} = k (1 - γ δ_{t}) g (u^{n}) .

By using some algebraic manipulations, we can obtain the following differential equation

(6)

b {\hat{u}}_{xx} - a {\hat{u}}_{x} + \frac{1}{γ k} \hat{u} + g (\hat{u}) = \hat{q} (x),

associated with boundary conditions,

(7)

\hat{u} (c) = Q_{0} (t), \hat{u} (d) = Q_{1} (t),

where,

\{\begin{matrix} \hat{u} \equiv u^{n + 1}, {\hat{u}}_{x} \equiv {(u_{x})}^{n + 1}, {\hat{u}}_{xx} \equiv {(u_{xx})}^{n + 1}, \\ g (\hat{u}) = g (x, t_{n}, u^{n + 1}), b (x, t_{n}) = b, a (x, t_{n}) = a, \\ \hat{q} (x) = b \frac{1 + γ}{γ} u_{xx}^{n} - a \frac{1 + γ}{γ} u_{x}^{n} + \frac{1}{γ k} u^{n} + \frac{1 + γ}{γ} g (u^{n}), \end{matrix}

g (\hat{u})

can be linear or nonlinear function.

For starting the computations we must use the initial condition $u^{0} = u (x, 0) = f (x)$ by Taylor’s series expansions of u at $t = t_{0} + k$ we have:

(8)

u^{1} = u^{0} + {ku}_{t}^{0} + \frac{k^{2}}{2!} u_{tt}^{0} + O (k^{3}) .

Via differentiating the initial condition with respect to x, we can achieve

u_{x}, u_{xx}, u_{xxx}, \dots

t = t_{0}

, by considering Eq. (1) we have,

(9)

u_{t}^{0} = {[{bu}_{xx} - {au}_{x} + g (u)]}^{0} .

By differentiating (9) with respect to x we obtain,

(10)

u_{xt}^{0} = {[{bu}_{xxx} - {au}_{xx} + g_{x} + g_{u} u_{x}]}^{0}, u_{xt}^{0} = {[{bu}_{xxx} - {au}_{xx} + g_{x}]}^{0},

similarly differentiating (9) with respect to t and using the Eq.(10) we have,

(11)

u_{tt}^{0} = {[{bu}_{xxt} - {au}_{xt} + g_{t} + g_{u} u_{t}]}^{0}, u_{tt}^{0} = {[{bu}_{xxt} - {au}_{xt} + g_{t}]}^{0},

substituting (2), (9),(11) into (8) we can compute

u^{1}

in the following forms:

(12)

u^{1} = f (x) + k {[{bu}_{xx} - {au}_{x} + g (u)]}^{0} + \frac{k^{2}}{2!} {[{bu}_{xxt} - {au}_{xt} + g_{t}]}^{0} + O (k^{3}),

and

(13)

u^{1} = f (x) + k {[{bu}_{xx} - {au}_{x} + g (u)]}^{0} + \frac{k^{2}}{2!} {[{bu}_{xxt} - {au}_{xt} + g_{t} + g_{u} u_{t}]}^{0} + O (k^{3}) .

Now by using Eqs. (12) and (13) for linear function

g (\hat{u})

and nonlinear function

g (\hat{u})

respectively, we can start the processes of our scheme (6).

Theorem 2.1

The time discretization steps (4)–(6) give the second order convergence.

Proof

Suppose $\hat{u} (t_{i})$ be the exact and ${\hat{u}}^{i}$ be the approximate solution of the Eq. (1) at ith time level and also assume that $e_{i}$ be the local truncation error in (6) as $e_{i} = {\hat{u}}^{i} - \hat{u} (t_{i})$ , by considering the operator $(1 - γ δ_{t})$ in Eq. (4) and applying Taylor’s series expansion we have:

(14)

| e_{i} | ⩽ ν_{i} k^{3},

for

γ = \frac{- 1}{2}

, where

ν_{i}

is finite constant independent of k.

Let $E_{n + 1}$ be the global error then $E_{n + 1} = \sum_{i = 1}^{n} e_{i}, (k ⩽ \frac{T}{n})$ , therefore by applying (14) we obtain: $| E_{n + 1} | = |\sum_{i = 1}^{n} e_{i}| ⩽ \sum_{i = 1}^{n} | e_{i} | ⩽ \sum_{i = 1}^{n} ν_{i} k^{3} ⩽ n . ν^{'} k^{3} ⩽ ν^{'} n (\frac{T}{n}) k^{2} = ϑ k^{2},$ where $ϑ = ν^{'} T$ , which concludes the proof of Theorem 2.1. □

2.2

2.2 Redefined extended cubic B-spline

In this section, we apply redefined extended B-splines collocation method to approximate solution of Eqs. (6) and (7). We consider $Δ = {c = x_{0} < \dots < x_{N} = d}$ be a uniform partition of the interval $[c, d]$ by the knots $x_{j}$ with $h = \frac{c - d}{N}$ .

The extended cubic B-spline $B_{j} (x, λ)$ is defined as:

(15)

B_{j} (x, λ) = \frac{1}{24 h^{4}} \{\begin{matrix} 4 h (1 - λ) {(x - x_{j - 2})}^{3} + 3 λ {(x - x_{j - 2})}^{4}, x_{j - 2} < x ⩽ x_{j - 1}, \\ (4 - λ) h^{4} + 12 h^{3} (x - x_{j - 1}) + 6 h^{2} (2 + λ) {(x - x_{j - 1})}^{2} \\ - 12 h {(x - x_{j - 1})}^{3} - 3 λ {(x - x_{j - 1})}^{4}, x_{j - 1} < x ⩽ x_{j}, \\ (4 - λ) h^{4} + 12 h^{3} (x_{j + 1} - x) + 6 h^{2} (2 + λ) {(x_{j + 1} - x)}^{2} \\ - 12 h {(x_{j + 1} - x)}^{3} - 3 λ {(x_{j + 1} - x)}^{4}, x_{j} < x ⩽ x_{j + 1}, \\ 4 h (1 - λ) {(x_{j + 2} - x)}^{3} + 3 λ {(x_{j + 2} - x)}^{4}, x_{j + 1} < x ⩽ x_{j + 2}, \\ 0, otherwise . \end{matrix}

The extended B-spline function has one arbitrary parameter

λ

, when

λ

tends to zero the extended B-spline reduced to convectional B-spline function. The extended B-spline has the properties such as: local support, non-negativity, partition of unity and

C^{2}

continuity, the parameter

λ

control the tension of the solution curve (Xu and Wang, 2008).

We focus on developing the collocation method to approximate the solution of Eqs. (1)–(3). Let $\hat{U} (x, t)$ and $\hat{u} (x, t)$ are the approximate solution and exact solution of the problems (1)–(3) respectively. We describe the approximate solution at the boundary points in the following form: Rashidinia and Sharifi (2016)

(16)

\hat{U} (x, t) = \sum_{j = - 1}^{N + 1} {\hat{c}}_{j} B_{j} (x, λ) = \frac{B_{- 1} (x, λ)}{B_{- 1} (x_{0}, λ)} Q_{0} (t) + \frac{B_{N + 1} (x, λ)}{B_{N + 1} (x_{N}, λ)} Q_{1} (t) + \sum_{j = 0}^{N} {\hat{c}}_{j} {\hat{B}}_{j} (x, λ),

where

\frac{B_{- 1} (x, λ)}{B_{- 1} (x_{0}, λ)} Q_{0} (t) + \frac{B_{N + 1} (x, λ)}{B_{N + 1} (x_{N}, λ)} Q_{1} (t) = W (x, t)

is weight function that takes care of the given boundary conditions, also we define the functions

{\hat{B}}_{j} (x, λ)

as follows:

${\hat{B}}_{j} (x, λ) = B_{j} (x, λ) - \frac{B_{j} (x_{0}, λ)}{B_{- 1} (x_{0}, λ)} B_{- 1} (x, λ), j = 0, 1,$

${\hat{B}}_{j} (x, λ) = B_{j} (x, λ), j = 2, \dots, N - 2,$

${\hat{B}}_{j} (x, λ) = B_{j} (x, λ) - \frac{B_{j} (x_{N}, λ)}{B_{N + 1} (x_{N}, λ)} B_{N + 1} (x, λ), j = N - 1, N .$

${\hat{B}}_{j} (x, λ), j = 0, \dots, N$ is as the new set of redefined extended cubic B-spline functions which vanish on the Dirichlet’s boundary conditions. Suppose $\hat{U} (x, t)$ satisfies the Eq. (6) associated with boundary conditions (7), so we obtain

(17)

H \hat{U} (x_{j}, t) = \hat{q} (x_{j}), 0 ⩽ j ⩽ N,

(18)

\hat{U} (x_{0}, t) = Q_{0} (t), \hat{U} (x_{N}, t) = Q_{1} (t),

where

H \hat{u} = b {\hat{u}}_{xx} - a {\hat{u}}_{x} + \frac{1}{γ k} \hat{u} + g (\hat{u})

. By applying Eq. (16) in Eqs. (17) and (18) we proceed as follows:

(19)

b (\sum_{j = 0}^{N} {\hat{c}}_{j} {\hat{B}}_{j}^{″} (x, λ) + W_{xx} (x, t)) - a (\sum_{j = 0}^{N} {\hat{c}}_{j} {\hat{B}}_{j}^{'} (x, λ) + W_{x} (x, t)) + \frac{1}{γ k} (\sum_{j = 0}^{N} {\hat{c}}_{j} {\hat{B}}_{j} (x, λ) + W (x, t)) - g (x_{j}, \sum_{j = 0}^{N} {\hat{c}}_{j} {\hat{B}}_{j} (x, λ) + W (x, t)) = \hat{q} (x_{j}), 0 ⩽ j ⩽ N .

By using the properties of extended cubic B-spline function in Eq. (19), we can obtain the following linear or nonlinear system of equations,

(20)

\begin{matrix} A \hat{C} + h^{2} \hat{G} = h^{2} \hat{ϕ}, \\ A = (\begin{matrix} z_{0} & 0 & 0 & 0 & \dots & 0 \\ x & y & z & 0 & \dots & 0 \\ 0 & x & y & z & \dots & 0 \\ ⋮ & ⋮ & ⋮ & \dots & 0 \\ ⋮ & ⋮ & ⋮ & \dots & 0 \\ 0 & \dots & 0 & x & y & z \\ 0 & \dots & 0 & 0 & 0 & z_{1} \end{matrix}), \end{matrix}

where,

z_{0} = - 12 b (\frac{2 + λ}{4 - λ}) - ah (\frac{8 + λ}{4 - λ}), z_{1} = - 12 b (\frac{2 + λ}{4 - λ}) + ah (\frac{8 + λ}{4 - λ}),

x = \frac{b (2 + λ)}{2} + \frac{ah}{2} + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}), y = - b (2 + λ) + \frac{h^{2}}{γ k} (\frac{8 + λ}{12}),

z = \frac{b (2 + λ)}{2} - \frac{ah}{2} + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}),

and

\hat{G}

can be as linear or nonlinear term

\hat{G} = (\begin{matrix} {\hat{g}}_{0} \\ {\hat{g}}_{1} \\ ⋮ \\ {\hat{g}}_{N} \end{matrix}),

where,

{\hat{g}}_{i} = \hat{g} (x_{i}, \frac{4 - λ}{24} {\hat{c}}_{i - 1} + \frac{8 + λ}{12} {\hat{c}}_{i} + \frac{4 - λ}{24} {\hat{c}}_{i + 1}),

\hat{C} = (\begin{matrix} {\hat{c}}_{0} \\ {\hat{c}}_{1} \\ ⋮ \\ {\hat{c}}_{N} \end{matrix}), h^{2} \hat{ϕ} = (\begin{matrix} h^{2} {\hat{q}}_{0} - (12 b (\frac{2 + λ}{4 - λ}) - 12 ah (\frac{1}{4 - λ}) + \frac{h^{2}}{γ k}) Q_{0} (t) \\ h^{2} {\hat{q}}_{1} \\ ⋮ \\ h^{2} {\hat{q}}_{N - 1} \\ h^{2} {\hat{q}}_{N} - (12 b (\frac{2 + λ}{4 - λ}) - 12 ah (\frac{1}{4 - λ}) + \frac{h^{2}}{γ k}) Q_{1} (t) \end{matrix}) .

3 Stability and convergence analysis

3.1

3.1 Stability analysis

In this section, we will prove the stability and convergence analysis of our scheme for solution of problem (1) associated with boundary conditions (2) and (3).

We prove the stability analysis by using the Von-Neumann stability method. The homogeneous part of the Eq. (6) is as follows

(21)

\frac{1}{γ k} u_{j}^{n + 1} - a {(u_{x})}_{j}^{n + 1} + b {(u_{xx})}_{j}^{n + 1} = \frac{1}{γ k} u_{j}^{n} - a (\frac{1 + γ}{γ}) {(u_{x})}_{j}^{n} + b (\frac{1 + γ}{γ}) {(u_{xx})}_{j}^{n} .

In Eq. (21)

u, u_{x}, u_{xx}

are replaced by the values of extended cubic B-spline functions, so we obtain

(22)

x {(\hat{C})}_{j - 1}^{n + 1} + y {(\hat{C})}_{j}^{n + 1} + z {(\hat{C})}_{j + 1}^{n + 1} = p {(\hat{C})}_{j - 1}^{n} + q {(\hat{C})}_{j}^{n} + r {(\hat{C})}_{j + 1}^{n},

where,

x = \frac{b (2 + λ)}{2} + \frac{ah}{2} + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}), y = - b (2 + λ) + \frac{h^{2}}{γ k} (\frac{8 + λ}{12}),

z = \frac{b (2 + λ)}{2} - \frac{ah}{2} + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}), p = \frac{b (2 + λ)}{2} (\frac{1 + γ}{γ}) + \frac{ah}{2} (\frac{1 + γ}{γ}) + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}),

q = - b (2 + λ) (\frac{1 + γ}{γ}) + \frac{h^{2}}{γ k} (\frac{8 + λ}{12}), r = \frac{b (2 + λ)}{2} (\frac{1 + γ}{γ}) - \frac{ah}{2} (\frac{1 + γ}{γ}) + \frac{h^{2}}{γ k} (\frac{4 - λ}{24}) .

Now we consider the trial solution at the mesh point $(x_{j}, t_{n})$ as: ${(\hat{C})}_{j}^{n} = ξ^{n} \exp (i θ jh),$ here $i = \sqrt{- 1}$ and $θ$ is real.

By using ${(\hat{C})}_{j}^{n} = ξ^{n} \exp (i θ jh)$ in Eq. (22) and simplifying we have

(23)

ξ = \frac{p \exp (- i θ h) + q + r \exp (i θ h)}{x \exp (- i θ h) + y + z \exp (i θ h)},

by substituting the values of

x, y, z, p, q, r

in (23), we get

(24)

ξ = \frac{- b (\frac{1 + γ}{γ}) (2 + λ) (1 - \cos β) + \frac{h^{2}}{12 γ k} (2 λ \sin^{2} \frac{β}{2} + 4 (1 + 2 \cos^{2} \frac{β}{2})) - iah (\frac{1 + γ}{γ}) \sin β}{- b (2 + λ) (1 - \cos β) + \frac{h^{2}}{12 γ k} (2 λ \sin^{2} \frac{β}{2} + 4 (1 + 2 \cos^{2} \frac{β}{2})) - iah \sin β},

here we have taken,

θ h = β

The simplified form of Eq. (24) can be written as:

(25)

ξ = (\frac{1 + γ}{γ}) \frac{X_{1} + (\frac{γ}{1 + γ}) X_{2} - {iY}_{1}}{X_{1} + X_{2} - {iY}_{1}},

where,

X_{1} = - b (2 + λ) (1 - \cos β), X_{2} = \frac{h^{2}}{12 γ k} (2 λ \sin^{2} \frac{β}{2} + 4 (1 + 2 \cos^{2} \frac{β}{2})), Y_{1} = ah \sin β .

By substituting

(\frac{1 + γ}{γ}) = α

and

\cos β = μ

in Eq. (24), we have:

(26)

\begin{matrix} ξ = α \frac{- b (2 + λ) (1 - μ) + \frac{h^{2}}{12 γ k α} (λ (1 - μ) + 4 (2 + μ)) - iah \sqrt{1 - μ^{2}}}{- b (2 + λ) (1 - μ) + \frac{h^{2}}{12 γ k} (λ (1 - μ) + 4 (2 + μ)) - iah \sqrt{1 - μ^{2}}}, \\ | ξ |^{2} = α^{2} \frac{{(- b (2 + λ) (1 - μ) + \frac{h^{2}}{12 γ k α} (λ (1 - μ) + 4 (2 + μ))}^{2} + a^{2} h^{2} (1 - μ^{2})}{{(- b (2 + λ) (1 - μ) + \frac{h^{2}}{12 γ k} (λ (1 - μ) + 4 (2 + μ))}^{2} + a^{2} h^{2} (1 - μ^{2})}, \end{matrix}

for all

θ

and

λ ⩾ - 2

, the numerator in Eq. (26) is less than denominator, therefore

| ξ | ⩽ 1

, thus the proposed method is unconditionally stable.

3.2

3.2 Convergence analysis

By following Henrici (1962), we want to prove the convergence of our method for the nonlinear function $g (u)$ . For this purpose, the following Lemma and Theorem are required.

Lemma 3.1

Let ${B_{- 1} (x, λ), \dots, B_{N + 1} (x, λ)}$ be the set of extended B-splines, which the $B_{i} (x, λ)$ satisfy in the following inequality:

(27)

|\sum_{j = - 1}^{N + 1} B_{j} (x, λ)| ⩽ \frac{7}{4}, x \in [0, 1] .

Proof

For the proof see Rashidinia and Sharifi (2016). □

Theorem 3.1

Let $\hat{u} (x) \in C^{4} [c, d], Δ$ be the partition of $[c, d]$ and $U^{*} (x)$ be the B-spline interpolation function $\hat{u} (x)$ , we have:

(28)

‖ D^{i} (\hat{u} - U^{*}) ‖_{\infty} ⩽ φ_{i} h^{4 - i}, i = 0, \dots, 3 .

Proof

For the proof see Hall (1968). □

Theorem 3.2

Suppose $\hat{u} (x)$ be the exact solution of (6) and $\hat{U} (x)$ be the extended B-spline approximate solution of $\hat{u} (x)$ then the error bound in the solution is:

$‖ \hat{u} (x) - \hat{U} (x) ‖ ⩽ β h^{2}, β = φ_{0} h^{2} + \frac{7}{4} R_{2}$ .

Proof

Let $\hat{U} (x)$ be the unique extended B-spline approximate solution of $\hat{u} (x)$ , suppose $U (x)$ be the computed extended B-spline approximate solution of $\hat{U} (x)$ as follows: $\hat{U} (x) = \sum_{j = - 1}^{N + 1} {\hat{c}}_{j} B_{j} (x, λ),$ $U (x) = \sum_{j = - 1}^{N + 1} c_{j} B_{j} (x, λ),$ by substituting $U (x)$ into (20) we get

(29)

AC + h^{2} G = h^{2} ϕ,

where,

G = (\begin{matrix} g_{0} \\ g_{1} \\ ⋮ \\ g_{N} \end{matrix}), C = (\begin{matrix} c_{0} \\ c_{1} \\ ⋮ \\ c_{N} \end{matrix}), h^{2} ϕ = (\begin{matrix} h^{2} q_{0} - (12 b (\frac{2 + λ}{4 - λ}) + 12 ah (\frac{1}{4 - λ}) + \frac{h^{2}}{γ k}) Q_{0} (t) \\ h^{2} q_{1} \\ ⋮ \\ h^{2} q_{N - 1} \\ h^{2} q_{N} - (12 b (\frac{2 + λ}{4 - λ}) - 12 ah (\frac{1}{4 - λ}) + \frac{h^{2}}{γ k}) Q_{1 (t)} \end{matrix}) .

By subtracting (20) from (29) we have,

(30)

A (C - \hat{C}) + h^{2} (G - \hat{G}) = h^{2} (ϕ - \hat{ϕ}),

the bound on

‖ ϕ - \hat{ϕ} ‖

can be obtained by using Theorem 3.1 By applying (17) we can obtain:

|q (x_{j}) - \hat{q} (x_{j}) | = | HU (x_{j}) - H \hat{U} (x_{j}) | = | b (U^{″} (x_{j}) - {\hat{U}}^{″} (x_{j})) + a (U^{'} (x_{j}) - {\hat{U}}^{'} (x_{j})) + \frac{1}{γ k} (U (x_{j}) - \hat{U} (x_{j})) + g (x_{j}, U (x_{j})) - \hat{g} (x_{j}, \hat{U} (x_{j}))|,

therefore we have,

(31)

|q (x_{j}) - \hat{q} (x_{j})| ⩽ |b (U^{″} (x_{j}) - {\hat{U}}^{″} (x_{j}))| + |a (U^{'} (x_{j}) - {\hat{U}}^{'} (x_{j}))| + |\frac{1}{γ k} (U (x_{j}) - \hat{U} (x_{j})) | + | g (x_{j}, U (x_{j})) - \hat{g} (x_{j}, \hat{U} (x_{j}))| .

From the Eq. (31) and using Theorem 3.1, we can obtain,

(32)

‖ ϕ - \hat{ϕ} ‖ ⩽ b φ_{2} h^{2} + a φ_{1} h^{3} + ‖ \frac{1}{γ k} ‖ φ_{0} h^{4} + R | U (x_{j}) - \hat{U} (x_{j}) | ⩽ b φ_{2} h^{2} + a φ_{1} h^{3} + ‖ \frac{1}{γ k} ‖ φ_{0} h^{4} + R φ_{0} h^{4},

where

‖ g^{'} (z) ‖ ⩽ R, z \in R^{3}

. By using the Eq. (32) we can obtain:

(33)

‖ ϕ - \hat{ϕ} ‖ ⩽ R_{1} h^{2}, R_{1} = b φ_{2} + a φ_{1} h + φ_{0} h^{2} (‖ \frac{1}{γ k} ‖ + R) .

Now the bound on $h^{2} (G - \hat{G})$ is, $h^{2} (G - \hat{G}) = h^{2} ((\begin{matrix} g (x_{0}, Q_{0} (t)) \\ ⋮ \\ g (x_{i}, \frac{4 - λ}{24} c_{i - 1} + \frac{8 + λ}{12} c_{i} + \frac{4 - λ}{24} c_{i + 1}) \\ ⋮ \\ g (x_{N}, Q_{1} (t)) \end{matrix}) - (\begin{matrix} \hat{g} (x_{0}, Q_{0} (t)) \\ ⋮ \\ \hat{g} (x_{i}, \frac{4 - λ}{24} {\hat{c}}_{i - 1} + \frac{8 + λ}{12} {\hat{c}}_{i} + \frac{4 - λ}{24} {\hat{c}}_{i + 1}) \\ ⋮ \\ \hat{g} (x_{N}, Q_{1} (t)) \end{matrix})),$ by using the mean value theorem we have the following relation,

(34)

h^{2} (G - \hat{G}) = h^{2} (\frac{\partial g}{\partial u} (ζ_{1})) L (C - \hat{C}),

where

ζ_{1} \in (0, 1)

and L is defined as:

L = (\begin{matrix} 0 & 0 \\ \frac{4 - λ}{24} & \frac{8 + λ}{12} & \frac{4 - λ}{24} \\ ⋱ & ⋱ & ⋱ \\ \frac{4 - λ}{24} & \frac{8 + λ}{12} & \frac{4 - λ}{24} \\ 0 & 0 \end{matrix}),

substituting (30) into (34) we get

(35)

F (C - \hat{C}) = h^{2} (ϕ - \hat{ϕ}),

where

F = A + h^{2} (\frac{\partial g}{\partial u} (ζ_{1})) L

. The matrix F, for

λ ⩾ - 2

is a strictly diagonally dominant matrix so it is non-singular, thus we can rewrite (35) in the following form:

(36)

(C - \hat{C}) = h^{2} F^{- 1} (ϕ - \hat{ϕ}),

by taking norm from (36) we get

(37)

‖ C - \hat{C} ‖ ⩽ h^{2} ‖ F^{- 1} ‖ ‖ ϕ - \hat{ϕ} ‖ ⩽ h^{2} ‖ F^{- 1} ‖ h^{2} R_{1} = h^{4} R_{1} ‖ F^{- 1} ‖ .

Now we calculate the sum of each row of matrix F, let that $χ_{j}, 0 ⩽ j ⩽ N$ is the summation of the jth row of the matrix $F = {[f_{j, i}]}_{(N + 1) \times (N + 1)}$ , then we get the following relations:

(38)

χ_{0} = \sum_{i = 0}^{N} f_{0, i} = \frac{- 12 b}{h^{2}} (\frac{2 + λ}{4 - λ}) - \frac{a}{h} (\frac{8 + λ}{4 - λ}), j = 0,

(39)

χ_{j} = \sum_{i = 0}^{N} f_{j, i} = \frac{1}{γ k} + h^{2} \frac{\partial g}{\partial u}, 1 ⩽ j ⩽ N - 1,

(40)

χ_{N} = \sum_{i = 0}^{N} f_{N, i} = \frac{- 12 b}{h^{2}} (\frac{2 + λ}{4 - λ}) + \frac{a}{h} (\frac{8 + λ}{4 - λ}), j = N .

Noticing to the theory of matrices Varga (1962), we have:

(41)

\sum_{j = 0}^{N} f_{l, j}^{- 1} χ_{j} = 1, l = 0, \dots, N,

where

f_{l, j}^{- 1}

are the elements of

F^{- 1}

, we have

(42)

‖ F^{- 1} ‖ = \sum_{j = 0}^{N} | f_{l, j}^{- 1} | ⩽ \frac{1}{\min χ_{j}} = \frac{1}{h^{2} σ_{ℓ}} ⩽ \frac{1}{h^{2} | σ_{ℓ} |}, 0 ⩽ j ⩽ N .

By substituting (42) into (37) we get,

(43)

‖ C - \hat{C} ‖ ⩽ \frac{R_{1} h^{4}}{h^{2} | σ_{ℓ} |} ⩽ R_{2} h^{2}, R_{2} = \frac{R_{1}}{| σ_{ℓ} |} .

Now we have:

(44)

U (x) - \hat{U} (x) = \sum_{j = - 1}^{N + 1} (c_{j} - {\hat{c}}_{j}) B_{j} (x, λ),

by taking norm from (44) and using (43) and (27), we obtain

(45)

‖ U (x) - \hat{U} (x) ‖ = ‖ \sum_{j = - 1}^{N + 1} (c_{j} - {\hat{c}}_{j}) B_{j} (x, λ) ‖ ⩽ | \sum_{j = - 1}^{N + 1} B_{j} (x, λ) | ‖ c_{j} - {\hat{c}}_{j} ‖ ⩽ \frac{7}{4} R_{2} h^{2} .

By using Theorem 3.1 we have:

(46)

‖ \hat{u} (x) - U (x) ‖ ⩽ φ_{0} h^{4},

by using Eqs. (45) and (46) we get

‖ \hat{u} (x) - \hat{U} (x) ‖ ⩽ ‖ \hat{u} (x) - U (x) ‖ + ‖ U (x) - \hat{U} (x) ‖ ⩽ φ_{0} h^{4} + \frac{7}{4} R_{2} h^{2} = β h^{2},

where

β = φ_{0} h^{2} + \frac{7}{4} R_{2}

. □

If we assume $u (x, t)$ be the exact solution of Eq. (1) and $U (x, t)$ be the numerical approximation, then we have:

$‖ u (x, t) - U (x, t) ‖ ⩽ ρ (k^{2} + h^{2})$ , $ρ = Max {φ_{0} h^{2} + \frac{7}{4} R_{2}, ϑ}$ , where $ρ$ is a constant.

4 Numerical illustrations

In this section, we consider five examples of linear and non-linear Convection-Reaction-Diffusion equation. Our numerical results are compared with results in Biazar and Mehrlatifan (2018), Dehghan (2004), Ismail et al. (2004), Macias Diaz and Puri (2012), Mittal and Jain (2012), Rashidinia and Shekarabi (2016). The absolute errors in the solutions are tabulated in Tables 1–6 . Our results demonstrate the accurate nature of the proposed method.

Example 1

Consider the following linear Convection-Reaction-Diffusion equation (Ismail et al., 2004): $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = b \frac{\partial^{2} u}{\partial x^{2}}, x \in [0, 1], t ⩾ 0,$

with initial and boundary conditions

u (x, 0) = \exp (γ x), u (0, t) = \exp (δ t), u (1, t) = \exp (γ + δ t) .

The exact solution is given by

u (x, t) = \exp (γ x + δ t)

Table 1 Maximum absolute error for example 1.

x	Method in Ismail et al. (2004)
	$t = 1$	$t = 2$	$t = 5$
$0.10$	$2.56 (- 10)$	$2.38 (- 10)$	$5.65 (- 10)$
$0.50$	$8.93 (- 10)$	$1.38 (- 9)$	$1.91 (- 9)$
$0.90$	$1.33 (- 9)$	$2.83 (- 9)$	$3.97 (- 9)$

x	Douglas method Rashidinia and Shekarabi (2016)
	$t = 1$	$t = 2$	$t = 5$
$0.10$	$2.56 (- 7)$	$2.37 (- 7)$	$5.63 (- 7)$
$0.50$	$8.37 (- 7)$	$1.38 (- 6)$	$1.90 (- 6)$
$0.90$	$1.33 (- 6)$	$2.82 (- 6)$	$3.95 (- 6)$

x	Present method $(λ = - 2)$
	$t = 1$	$t = 2$	$t = 5$
$0.10$	$1.99181 (- 11)$	$1.86257 (- 11)$	$6.75869 (- 11)$
$0.50$	$6.85145 (- 11)$	$6.69412 (- 11)$	$4.78064 (- 11)$
$0.90$	$4.00414 (- 10)$	$4.96767 (- 9)$	$1.33356 (- 9)$

Table 2 Maximum absolute error for different values of

λ

for example 1.

x	$λ = 0$	$λ = - 1.5$	$λ = - 2$
$0.1$	$6.95357 (- 7)$	$8.91931 (- 6)$	$6.75898 (- 9)$
$0.2$	$1.86257 (- 7)$	$4.83274 (- 6)$	$6.73565 (- 9)$
$0.3$	$4.96767 (- 8)$	$2.50392 (- 6)$	$5.75884 (- 9)$
$0.4$	$1.24526 (- 8)$	$1.08594 (- 6)$	$5.65548 (- 9)$
$0.5$	$1.33356 (- 8)$	$6.29531 (- 8)$	$4.75869 (- 9)$
$0.6$	$1.19181 (- 8)$	$9.36954 (- 7)$	$6.41047 (- 9)$
$0.7$	$4.78064 (- 8)$	$2.27749 (- 6)$	$8.60556 (- 9)$
$0.8$	$1.79305 (- 7)$	$4.44594 (- 6)$	$5.84587 (- 9)$
$0.9$	$6.69412 (- 7)$	$8.23082 (- 6)$	$3.75839 (- 9)$

Table 3 Maximum absolute error for example 2.

x	Method in Dehghan (2004)	Method in Mittal and Jain (2012)	Present method $(λ = - 2)$
	$t = 1$	$t = 1$	$t = 1$
$0.10$	$9.06 (- 10)$	$9.96 (- 9)$	$9.79263 (- 10)$
$0.20$	$1.54 (- 9)$	$1.91 (- 8)$	$9.03905 (- 9)$
$0.30$	$1.84 (- 9)$	$2.70 (- 8)$	$8.62793 (- 9)$
$0.40$	$1.77 (- 9)$	$3.33 (- 8)$	$9.20736 (- 9)$
$0.50$	$1.33 (- 9)$	$3.78 (- 8)$	$8.45641 (- 9)$
$0.60$	$5.59 (- 10)$	$4.02 (- 8)$	$9.37931 (- 10)$
$0.70$	$3.83 (- 10)$	$3.99 (- 8)$	$8.28461 (- 10)$
$0.80$	$1.18 (- 9)$	$3.60 (- 8)$	$9.54806 (- 9)$
$0.90$	$1.31 (- 9)$	$2.55 (- 8)$	$8.11957 (- 10)$

Table 4 Maximum absolute error for example 3.

x	Method in Rashidinia and Shekarabi (2016)		Present method
	$δ = 1, a = 100$	$δ = 100, a = 5$	$δ = 1, a = 100$	$δ = 100, a = 5$
$0.10$	$5.41124 (- 8)$	$1.38435 (- 3)$	$7.21942 (- 8)$	$2.91738 (- 6)$
$0.20$	$3.53667 (- 7)$	$2.85513 (- 4)$	$5.11234 (- 7)$	$1.84315 (- 6)$
$0.30$	$7.22626 (- 7)$	$5.86763 (- 5)$	$8.33893 (- 7)$	$3.25879 (- 6)$
$0.40$	$1.20347 (- 6)$	$1.02727 (- 5)$	$2.25439 (- 7)$	$2.46877 (- 6)$
$0.50$	$1.71858 (- 6)$	$2.48386 (- 6)$	$3.65431 (- 7)$	$3.71574 (- 6)$

Table 5 Maximum absolute error for example 4.

x	Method in Mittal and Jain (2012)		Present method
	$t = 1$	$t = 2$	$t = 1$	$t = 2$
$0.10$	$1.06 (- 6)$	$1.01 (- 13)$	$6.83553 (- 7)$	$7.56433 (- 13)$
$0.20$	$5.02 (- 6)$	$3.65 (- 12)$	$6.52334 (- 7)$	$7.01161 (- 13)$
$0.30$	$1.82 (- 5)$	$7.56 (- 11)$	$6.65116 (- 7)$	$6.91534 (- 12)$
$0.40$	$1.08 (- 5)$	$1.06 (- 9)$	$6.98728 (- 7)$	$6.463282 (- 11)$
$0.50$	$4.63 (- 5)$	$1.04 (- 8)$	$7.22451 (- 7)$	$1.38677 (- 9)$
$0.60$	$4.17 (- 6)$	$7.27 (- 8)$	$7.41511 (- 7)$	$1.12578 (- 9)$
$0.70$	$3.78 (- 5)$	$3.53 (- 7)$	$7.39468 (- 7)$	$1.01969 (- 9)$
$0.80$	$7.10 (- 6)$	$1.14 (- 6)$	$6.93217 (- 7)$	$2.00434 (- 8)$
$0.90$	$8.98 (- 6)$	$2.12 (- 6)$	$6.10194 (- 7)$	$1.49526 (- 8)$

Table 6 Maximum absolute error for example 5.

	Method in Biazar and Mehrlatifan (2018)	Method in Macias Diaz and Puri (2012)	Present method
$h = 0.0125, k = 0.001$	$1.09 (- 4)$	$3.1835 (- 2)$	$1.25216 (- 6)$
$h = 0.025, k = 0.0001$	$2.8 (- 3)$	$3.1835 (- 2)$	$1.9957 (- 6)$

The absolute errors in the solutions for different time levels with $a = 3.5, b = 0.022, γ = 0.02854797991928, δ = - 0.0999, h = 0.01$ and $k = 0.001$ are tabulated in Table 1, and compared with the results (Ismail et al., 2004), these results verify the accurate value of our scheme. The maximum absolute errors in the solution for different $λ$ , with $t = 2, h = 0.1$ and $k = 0.01$ , are tabulated in Table 2. We observe that, the different values of $λ$ , satisfied in the condition $λ ⩾ - 2$ .

Example 2

Consider the following linear Convection-Reaction-Diffusion equation (Dehghan, 2004): $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = b \frac{\partial^{2} u}{\partial x^{2}}, x \in [0, 1], t ⩾ 0,$ with initial and boundary conditions $u (x, 0) = \exp (- \frac{{(x - 2)}^{2}}{8}), u (0, t) = \sqrt{\frac{20}{20 + t}} \exp (\frac{{(2 + 0.8 t)}^{2}}{0.4 t + 8}), u (1, t) = \sqrt{\frac{20}{20 + t}} \exp (\frac{{(1 + 0.8 t)}^{2}}{0.4 t + 8})$ .

The exact solution is given by $u (x, t) = \sqrt{\frac{20}{20 + t}} \exp (- \frac{{(x - 2 - 0.8 t)}^{2}}{0.4 t + 8})$ .

The absolute errors in the solutions with $a = 0.8, b = 0.1, h = 0.01, k = 0.001$ and in time $t = 1$ are tabulated in Table 3, and compared with the results in Dehghan (2004), Mittal and Jain (2012), these results verify the accurate nature of our scheme.

Example 3

Consider the nonlinear Convection-Reaction-Diffusion equation (Hundsdorfer, 2000): $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = b \frac{\partial^{2} u}{\partial x^{2}} + δ u^{2} (1 - u), x \in [0, 1], t ⩾ 0,$ with initial and boundary conditions $u (x, 0) = {(1 + \exp (α x + γ))}^{- 1}, u (0, t) = {(1 + \exp (α (- β t) + γ))}^{- 1}, u (1, t) = {(1 + \exp (α (1 - β t) + γ))}^{- 1} .$

The exact solution is given by $u (x, t) = {(1 + \exp (α (x - β t) + γ))}^{- 1}$ , where $α = \sqrt{\frac{δ}{4 b}}, β = 2 a + \sqrt{δ b}, γ = α (β - 1)$ . The absolute errors in the solutions for different $δ$ and appropriate parameter $λ = - 1.99$ with $b = 0.1, h = k = 0.01$ , and $t = 0.9$ are tabulated in Table 4, and compared with the results in Rashidinia and Shekarabi (2016), these results show that our presented method is more accurate.

Example 4

Consider the linear Convection-Reaction-Diffusion equation $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = b \frac{\partial^{2} u}{\partial x^{2}}, x \in [0, 1], t ⩾ 0,$ with the following initial condition $u (x, 0) = \exp (- \frac{{(x + 0.5)}^{2}}{0.00125})$ . The exact solution is given by $u (x, t) = \frac{0.025}{\sqrt{0.000625 + 0.02 t}} \exp (- \frac{{(x + 0.5 - t)}^{2}}{0.00125 + 0.04 t})$ . The boundary conditions can be derived from the exact solution.

For this example, we put $λ = - 1.55, a = 1$ and $b = 0.01$ . We take $h = 0.001$ and $k = 0.0001$ and the maximum absolute error in the solution for $t = 1$ and $t = 2$ are computed. The results are reported in Table 5 and compared with (Mittal and Jain, 2012), our numerical results these results show that our presented method is more accurate.

Example 5

Consider the linear Convection-Reaction-Diffusion equation $\frac{\partial u}{\partial t} = b \frac{\partial^{2} u}{\partial x^{2}} - cu, x \in [0, 1], t ⩾ 0,$ with initial and boundary conditions $u (x, 0) = Sin (n π x), u (0, t) = u (1, t) = 0$ . The exact solution is given by $u (x, t) = \exp ((c - b π^{2} n^{2}) t) Sin (n π x)$ .

For this example, we put $λ = - 2, b = 0.1, c = - 0.5$ and $n = 1$ . We take $h = 0.0125, h = 0.025$ and $k = 0.001, k = 0.0001$ and the maximum absolute error in the solution for $t = 0.8$ are computed. The results are reported in Table 6 and compared with Biazar and Mehrlatifan (2018), Macias Diaz and Puri (2012), our numerical results these results show that our presented method is more accurate.

5 Conclusions

In this work, a new numerical method has been developed to the approximate solution of Convection-Reaction-Diffusion equation. Our method is based on the collocation method with extended cubic B-spline basis functions. The stability and convergence analysis of the method is described, we established unconditionally two level explicit finite difference scheme that it can be applied as a suitable scheme for modelling the behaviour of the similar problems. The applicability of our method has been considered by various examples. The numerical results show that our method provides more accurately than the recent existing methods. The value of an arbitrary $λ$ is varied systemically, our scheme is unconditionally stable for $λ ⩾ - 2$ , the values of $λ$ in our examples are different but they satisfied in this condition.

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Introduction

Description of the method

Temporal discretization
Redefined extended cubic B-spline

Stability and convergence analysis

Stability analysis
Convergence analysis

Numerical illustrations

Conclusions

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[1] Jafar Biazar, Mehrlatifan Bagher, . A compact finite difference scheme for reaction-convection-diffusion equation. Chiang Mai J. Sci.. 2018;45:1559-1568.
[Google Scholar]

[2] Cengel Y.A., Cimbala J.M., . Fluid Mechanics: Fundamental and Application. McGraw-Hill Higher Education; 2006.

[3] Cui M., . Compact exponential scheme for the time fractional convection-diffusion-reaction equation with variable coefficients. J. Comput. Phys.. 2015;280:143-163.
[Google Scholar]

[4] Cunningham A.B., Charaklis W.G., Abedeen F., Crawford D., . Influence of biofilm accumulation on porous media hydrodynamics. Environ. Sci. Tech.. 1991;25:1305-1310.
[Google Scholar]

[5] Dehghan M., . Weighted finite difference techniques for the one-dimensional advection diffusion equation. Appl. Math. Comput.. 2004;147:307-319.
[Google Scholar]

[6] Duan H., Hsieh P., Tan R., Yang S., . Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations with a large reaction coefficient. Comput. Meth. Appl. Mech. Eng 2012:15-36.
[Google Scholar]

[7] Hall C.A., . On error bounds for spline interpolation. J. Approximation Theory. 1968;1:209-218.
[Google Scholar]

[8] Henrici P., . Discrete Variable Methods in Ordinary Differential Equation. New york: Wiley; 1962.

[9] Hsieh P., Yang S., . A new stabilized linear finite element method for solving reaction-convection-diffusion equations. Comput. Meth. Appl. Mech. Eng.. 2016;307:362-382.
[Google Scholar]

[10] Hundsdorfer W., . Numerical Solution of Advection-Diffusion-Reaction Equations. Lecture Notes. Thomas Stieltjes Institute; 2000.

[11] Isenberg J., Gutfinge C., . Heat transfer to a draining film. Int. J. Heat Transfer. 1972;16:505-512.
[Google Scholar]

[12] Ismail H.N.A., Elbarbary E.M., Salem G.S.E., . Restrictive Taylor approximation for solving convection-diffusion equation. Appl. Math. Comput.. 2004;147:355-363.
[Google Scholar]

[13] Kaya A., . Finite difference approximations of multidimensional unsteady convection-diffusion-reaction equations. J. Comput. Phys.. 2015;285:331-349.
[Google Scholar]

[14] Macias Diaz J.E., Puri A., . An explicit positivity-preserving finite-difference scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation Appl. Math. Comput.. 2012;218:5829-5837.
[Google Scholar]

[15] Makungu M., Haario H., Mahera W.C., . A generalised 1-dimensional particle transport method for convection diffusion reaction. Afr Math Un.. 2012;23:21-39.
[Google Scholar]

[16] McRea G.J., Goodin W.R., Seinfeld J.H., . Numerical solution of atmospheric diffusion for chemically reacting flows. J. Comput. Phys.. 1982;77:1-42.
[Google Scholar]

[17] Mittal R.C., Jain R.K., . Redefined cubic B-splines collocation method for solving convection diffusion equations. Appl. Math. Modell.. 2012;36:5555-5573.
[Google Scholar]

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Collocation method for Convection-Reaction-Diffusion equation

Abstract

Keywords

65M06

65M12

65M15

65M70

Convection-Reaction-Diffusion equation

Redefined extended cubic B-spline

Stability and convergence analysis

1 Introduction

2 Description of the method

2.1 Temporal discretization

2.2 Redefined extended cubic B-spline

3 Stability and convergence analysis

3.1 Stability analysis

3.2 Convergence analysis

4 Numerical illustrations

5 Conclusions

References

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