Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method

Abstract

In this paper, we examine a system of two coupled nonlinear differential equations that relates the concentrations of carbon dioxide CO$_2$ and phenyl glycidyl ether in solution. This system is subject to a set of Dirichlet boundary conditions and a mixed set of Neumann and Dirichlet boundary conditions. We apply the Adomian decomposition method combined with the Duan–Rach modified recursion scheme to analytically treat this system of coupled nonlinear boundary value problems. The rapid convergence of our analytic approximate solutions is demonstrated by graphs of the objective error analysis instead of comparison to an alternate solution technique alone. The Adomian decomposition method yields a rapidly convergent, easily computable, and readily verifiable sequence of analytic approximate solutions that is suitable for numerical parametric simulations. Thus our sequence of approximate solutions are shown to identically satisfy the original set of model equations as closely as we please.

Appendices

Appendix 1: MATHEMATICA code for the two-variable Adomian polynomials based on Theorem 1 [14]

Appendix 2: The technique used in [17]

Consider the nonlinear differential equations in Eqs. (5) and (6),

$$\begin{aligned} Lu&= \alpha _1 f\left( {u\left( x \right) , v\left( x \right) } \right) ,\end{aligned}$$

(31)

$$\begin{aligned} Lv&= \alpha _2 f\left( {u\left( x \right) , v\left( x \right) } \right) , \end{aligned}$$

(32)

where the linear differential operator $L$ and the composite nonlinearity are

$$\begin{aligned} L\left( \cdot \right) = \frac{d^2}{dx^2}\left( \cdot \right) ,\ \ f\left( {u\left( x \right) ,v\left( x \right) } \right) = \frac{u\left( x \right) v\left( x \right) }{1 + \beta _1 u\left( x \right) + \beta _2 v\left( x \right) }. \end{aligned}$$

(33)

In the double decomposition method, the inverse linear operator $L^{-1}$ is taken as a two-fold indefinite integration for second-order differential equations [7, 8], i.e.

$$\begin{aligned} L^{-1}(\cdot ) =C_0+C_1 x+I_x^2(\cdot )= C_0+C_1 x+ \int \int (\cdot )\,dxdx, \end{aligned}$$

(34)

where $C_0$ and $C_1$ are the constants of integration, which are called the matching coefficients, and where $I_x^2(\cdot )=\int \int (\cdot )\,dxdx$ denotes pure integrations. Applying the operator $L^{-1}$ to both sides of Eqs. (31) and (32) yields the system of coupled nonlinear integral equations

$$\begin{aligned} u(x)=C_0+C_1 x+ \alpha _1 I_x^{2} f\left( {u\left( x \right) , v\left( x \right) }\right) , \end{aligned}$$

(35)

$$\begin{aligned} v(x)=D_0+D_1 x+ \alpha _2 I_x^{2}f\left( {u\left( x \right) , v\left( x \right) }\right) , \end{aligned}$$

(36)

where $C_0$, $C_1$, $D_0$, $D_1$ are arbitrary constants of integration to be determined by decomposition and matching at the boundaries for each stage of approximation.

The double decomposition method decomposes the solution $u(x), v(x)$, the nonlinearity $f(u,v)$, and the matching coefficients $C_i$ and $D_i$ as

$$\begin{aligned}&u(x)=\sum _{n=0}^\infty u_n(x),\ v(x)=\sum _{n=0}^\infty v_n(x),\ f(u,v)=\sum _{n=0}^\infty A_n,\end{aligned}$$

(37)

$$\begin{aligned}&C_i=\sum _{n=0}^\infty C_{i,n},\ D_i=\sum _{n=0}^\infty D_{i,n}, \ \ i=0,1. \end{aligned}$$

(38)

Upon substitution of these series into Eqs. (35) and (36), we can design the recursion scheme as

$$\begin{aligned} u_0&= C_{0,0}+C_{1,0}x,\ v_0=D_{0,0}+D_{1,0}x, \nonumber \\ u_{n+1}&= C_{0,n+1}+C_{1,n+1}x +\alpha _1 I_x^{2}A_{n},\ n\ge 0,\nonumber \\ v_{n+1}&= D_{0,n+1}+D_{1,n+1}x +\alpha _2 I_x^{2}A_{n},\ n\ge 0. \end{aligned}$$

(39)

Note that these necessary matching coefficients for $u_{n+1}$ and $v_{n+1}$ were omitted from recurrence relations in [17].

The constants $C_{i,n}$ and $D_{i,n}$, $i=0,1$, are determined by matching each of the partial sums $\phi _n(x)$ and $\psi _n(x)$, $n=1,2,\dots$, to their respective boundary values [5, 7, 8] in Eqs. (3) and (4). This procedure can be carried out by matching $u_0,v_0$ to the given boundary values in Eqs. (3) and (4) to determine the values of $C_{i,0}$ and $D_{i,0}$, $i=0,1$, matching $u_1, v_1$ to the corresponding homogeneous boundary values $u_1(0)=u_1(1)=v_1'(0)=v_1(1)=0$ to determine the values of $C_{i,1}$ and $D_{i,1}$, $i=0,1$, $\dots$, and matching $u_n, v_n$ to the corresponding homogeneous boundary values $u_n(0)=u_n(1)=v_n'(0)=v_n(1)=0$ to determine the values of $C_{i,n}$ and $D_{i,n}$, $i=0,1$.

Matching $u_0$ and $v_0$ to the boundary values in Eqs. (3) and (4) determines

$$\begin{aligned} u_0=1+(k-1)x,\ v_0=1. \end{aligned}$$

Calculating $u_1$ and $v_1$ using (39) and matching $u_1$ and $v_1$ to the corresponding homogeneous boundary values $u_1(0)=u_1(1)=v_1'(0)=v_1(1)=0$ determine that

$$\begin{aligned} u_1&= \frac{\alpha _1 (x-1) x}{2 \beta _1}+\frac{\alpha _1 \left( \beta _2+1\right) x}{\beta _1^3 (k-1)^2} \left( \beta _2+\beta _1 k+1\right) \log \left( \beta _2+\beta _1 k+1\right) \\&\quad -\,\frac{\alpha _1 \left( \beta _2+1\right) }{\beta _1^3 (k-1)^2} \left( \beta _1+\beta _2+\beta _1 (k-1) x+1\right) \log \left( \beta _1+\beta _2+\beta _1 (k-1) x+1\right) \\&\quad -\,\frac{\alpha _1 \left( \beta _2+1\right) }{\beta _1^3 (k-1)^2} (x-1) \left( \beta _1+\beta _2+1\right) \log \left( \beta _1+\beta _2+1\right) , \\ v_1&= \frac{(x-1) \alpha _2 }{2 (k-1) \beta _1^2}\left( (k-1) (1+x) \beta _1+2 \left( 1+\beta _2\right) \right) +\frac{(x-1) \alpha _2 \left( 1+\beta _2\right) }{(k-1) \beta _1^2}\\&\quad \times \,\log \left( \beta _1+\beta _2+1\right) +\frac{\alpha _2 \left( 1+\beta _2\right) }{(k-1)^2 \beta _1^3} \left( \beta _2+\beta _1 k+1 \right) \log \left( \beta _2+\beta _1 k+1\right) \\&\quad -\,\frac{\alpha _2 \left( 1+\beta _2\right) }{(k-1)^2 \beta _1^3} \left( \beta _1+\beta _2+\beta _1 (k-1) x+1\right) \log \left( \beta _1+\beta _2+\beta _1 (k-1) x\!+\!1 \right) . \end{aligned}$$

Note that there are several errors in the expressions (B.13) and (B.15) as published in [17]. We have checked by using MATHEMATICA that it is quite time-consuming to calculate $u_2$ and $v_2$ and it is not at all feasible to calculate $u_3$ and $v_3$ if the results are parametrized by $\beta _1$ and $\beta _2$.