An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

Ji-Huan He; Shuai-Jia Kou; Hamid M. Sedighi; Ji-Huan He; Shuai-Jia Kou; Hamid M. Sedighi

doi:10.3934/mmc.2021016

Mathematical Modelling and Control

2021, Volume 1, Issue 4: 172-176. doi: 10.3934/mmc.2021016

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Research article

An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

1.
Xi'an University of Architecture and Technology, Xi'an, China
2.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China
3.
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou, China
4.
Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Received: 26 October 2021 Accepted: 09 December 2021 Published: 28 December 2021

Taylor series method is simple, and an infinite series converges to the exact solution for initial condition problems. For the two-point boundary problems, the infinite series has to be truncated to incorporate the boundary conditions, making it restrictively applicable. Here is recommended an ancient Chinese algorithm called as Ying Buzu Shu, and a nonlinear reaction diffusion equation with a Michaelis-Menten potential is used as an example to show the solution process.

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Citation: Ji-Huan He, Shuai-Jia Kou, Hamid M. Sedighi. An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics[J]. Mathematical Modelling and Control, 2021, 1(4): 172-176. doi: 10.3934/mmc.2021016

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Abstract

1. Introduction

Nonlinear equations arise always in electroanalytical chemistry with complex and esoteric nonlinear terms^[1,2], though there are some advanced analytical methods to deal with nonlinear problems, for examples, the Gamma function method^[3], Fourier spectral method^[4], the reproducing kernel method^[5], the perturbation method^[6], the homotopy perturbation method^[7,8], He's frequency formulation^[9,10,11] and the dimensional method^[12], chemists are always eager to have a simple one step method for nonlinear equations. This paper introduces an ancient Chinese algorithm called as the Ying Buzu algorithm^[13] to solve nonlinear differential equations.

2. Taylor series solution

We first introduce the Taylor series method^[14]. Considering the nonlinear differential equation:

$\begin{equation} \frac{d^{2}u}{dx^{2}}+F(u) = 0. \end{equation}$

(0.1)

The boundary conditions are

$\begin{equation} \frac{du}{dx}(a) = \alpha, \end{equation}$

(0.2)

$\begin{equation} u(b) = \beta. \end{equation}$

(0.3)

If $u(a)$ is known, we can use an infinite Taylor series to express the exact solution^[14]. We assume that

$\begin{equation} u(a) = c. \end{equation}$

(0.4)

From (0.1), we have

$\begin{equation*} u^{''}(a) = -F(u(a)) = -F(c), \end{equation*}$

$\begin{equation*} u^{'''}(a) = -\frac{\partial F(c)}{\partial u}u^{'}(a) = -\alpha\frac{\partial F(c)}{\partial u}. \end{equation*}$

Other higher order derivatives can be obtained with ease, and its Taylor series solution is

$\begin{align*} u(x) = &u(a)+(x-a)u^{'}(a)+\frac{1}{2!}(x-a)^{2}u^{''}(a)\\ &+\frac{1}{3!}(x-a)^{3}u^{'''}(a)+...+\frac{1}{N!}(x-a)^{N}u^{(N)}(a), \end{align*}$

the constant $c$ can be determined by the boundary condition of (0.3).

3. **The Ying Buzu algorithm**

The Ying Buzu algorithm^[15,16] was used to solve differential equations in 2006^[13], it was further developed to He's frequency formulation for nonlinear oscillators^{[13,17,18,19,20,21,22,23]} and Chun-Hui He's algorithm for numerical simulation^[24].

As $c$ in (0.4) is unknown, according to the Ying Buzu algorithm^[13,15,16], we can assume two initial guesses:

$\begin{equation} u_{1}(a) = c_{1}, \; u_{2}(a) = c_{2}. \end{equation}$

(0.5)

where $c_{1}$ and $c_{2}$ are given approximate values.

Using the initial conditions given in (0.2) and (0.5), we can obtain the terminal values:

$\begin{equation*} u(b, c_{1}) = \beta_{1}, \; u(b, c_{2}) = \beta_{2}. \end{equation*}$

According to the Ying Buzu algorithm^{[6,7,8,9,10,11,12]}, the initial guess can be updated as

$\begin{equation*} u(a)_{est} = c_{3} = \frac{c_{1}(\beta-\beta_{2})-c_{2}(\beta-\beta_{1})}{(\beta-\beta_{2})-(\beta-\beta_{1})}, \end{equation*}$

and its terminal value can be calculated as

$\begin{equation*} u(b, c_{3}) = \beta_{3}. \end{equation*}$

For a given small threshold, $\varepsilon$ , $|\beta-\beta_{3}|\le\varepsilon$ , we obtain $u(a) = c_{3}$ as an approximate solution.

4. Application

Here, we take Michaelis Menten dynamics as an example to solve the equation. Michaelis Menten reaction diffusion equation is considered as follows^[25,26]:

$\begin{equation} \frac{d^{2}u}{dx^{2}}-\frac{u}{1+u} = 0. \end{equation}$

(0.6)

The boundary conditions of it are as follows:

$\begin{equation} \frac{du}{dx}(0) = 0, u(1) = 1. \end{equation}$

(0.7)

We assume

$\begin{equation*} u(0) = c. \end{equation*}$

From (0.6), we have

$\begin{equation*} u^{''}(0) = \frac{c}{1+c}, \end{equation*}$

$\begin{equation} u^{'''}(0) = 0, \end{equation}$

(0.8)

$\begin{equation*} u^{(4)} = \frac{c}{(1+c)^{3}}. \end{equation*}$

The $2^{nd}$ order Taylor series solution is

$\begin{equation*} u(x) = u(0)+\frac{u^{'}(0)}{1!}x+\frac{u^{''}(0)}{2!}x^{2} = c+\frac{c}{2(1+c)}x^{2}. \end{equation*}$

In view of the boundary condition of (0.7), we have

$\begin{equation} u(1) = c+\frac{c}{2(1+c)} = 1, \end{equation}$

(0.9)

solving $c$ from (0.9) results in

$\begin{equation*} c = 0.7808. \end{equation*}$

So we obtain the following approximate solution

$\begin{equation*} u(x) = 0.7808+0.2192x^{2}. \end{equation*}$

Similarly the fourth order Taylor series solution is

$\begin{equation*} u(x) = c+\frac{c}{2!(1+c)}x^{2}+\frac{c}{4!(1+c)^{3}}x^{4}. \end{equation*}$

Incorporating the boundary condition, $u(1) = 1$ , we have

$\begin{equation} c+\frac{c}{2!(1+c)}+\frac{c}{4!(1+c)^{3}} = 1. \end{equation}$

(0.10)

We use the Ying Buzu algorithm to solve $c$ , and write (0.10) in the form

$\begin{equation*} R(c) = c+\frac{c}{2(1+c)}+\frac{c}{24(1+c)^{3}}-1. \end{equation*}$

Assume the two initial solutions are

$\begin{equation*} c_{1} = 0.8, \; c_{2} = 0.5. \end{equation*}$

We obtain the following residuals

$\begin{equation*} R_{1}(0.8) = 0.0279, \; R_{2}(0.5) = -0.3271. \end{equation*}$

By the Ying Buzu algorithm, $c$ can be calculated as

$\begin{equation*} c = \frac{R_{2}c_{1}-R_{1}c_{2}}{R_{2}-R_{1}} = \frac{0.0279\times 0.5+0.3271\times 0.8}{0.0279+0.3271} = 0.7764. \end{equation*}$

The exact solution of (0.10) is

$\begin{equation*} c = 0.7758. \end{equation*}$

The $4^{th}$ order Taylor series solution is

$\begin{equation*} u(x) = 0.7758+0.2192x^{2}+0.0057x^{4}. \end{equation*}$

shows the Taylor series solutions, which approximately meet the requirement of the boundary condition at $x = 1$ .

Figure 1. Taylor series solution.

DownLoad: Full-Size Img PowerPoint

Now we use the Ying Buzu algorithm by choosing two initial guesses

$\begin{equation*} u_{1}(0) = 0.5, \; u_{2}(0) = 1, \end{equation*}$

which lead to $u_{1} = 0.6726$ and $u_{2} = 1.2550$ , respectively, see Figure 2 (a) and (b).

Figure 2. The shooting processes with different initial guesses.

DownLoad: Full-Size Img PowerPoint

It is obvious that the terminal value at $x = 1$ deviates from $u(1) = 1$ for each guess, according to the Ying Buzu algorithm, the initial guess can be updated as

$\begin{equation} u_{3}(0) = \frac{0.5\times(1-1.2550)-1\times(1-0.6726)}{(1-1.2550)-(1-0.6726)} = 0.7810. \end{equation}$

(0.11)

The shooting process using (0.11) results in

$\begin{equation*} u_{3}(1) = 1.0058, \end{equation*}$

which deviates the exact value of $u(1) = 1$ with a relative error of $0.5\%$ , see Figure 3.

Figure 3. The shooting processes with an updated initial guess of

$u(0) = 0.7810.$ .

DownLoad: Full-Size Img PowerPoint

We can continue the iteration process to obtain a higher accuracy by using two following two guesses $u_{1}(0) = 0.5$ , $u_{3}(0) = 0.7810$ :

$\begin{align} u_{4}(0) & = \frac{0.5\times(1-1.0058)-0.7810\times(1-0.6726)}{(1-1.0058)-(1-0.6726)} \\ & = 0.7761. \end{align}$

Using this updated initial value, the shooting process leads to the result

$\begin{equation*} u(1) = 1.0001, \end{equation*}$

so the approximate $u(0) = 0.7761$ has only a relative error of $0.01\%$ .

The above solution process couples the numerical method, and the ancient method can also be solved independently.

We assume that solution is

$\begin{equation} u(x) = c+(1-c)x^{2}. \end{equation}$

(0.12)

Equation (0.12) meets all boundary conditions.

The residual equation is

$\begin{equation*} R(x) = \frac{d^{2}u}{dx^{2}}-\frac{u}{1+u}. \end{equation*}$

It is easy to find that

$\begin{equation*} R(0) = 2(1-c)-\frac{c}{1+c}. \end{equation*}$

We choose two guesses:

$\begin{equation*} c_{1} = 0.5, c_{2} = 1. \end{equation*}$

We obtain the following residuals

$\begin{equation*} R_{1}(0) = 2(1-0.5)-\frac{0.5}{1+0.5} = \frac{2}{3}, \end{equation*}$

$\begin{equation*} R_{2}(0) = 2(1-1)-\frac{1}{1+1} = -\frac{1}{2}. \end{equation*}$

The Ying Buzu algorithm leads to the updated result:

$\begin{equation*} c = \frac{c_{2}R_{1}(0)-c_{1}R_{2}(0)}{R_{1}(0)-R_{2}(0)} = \frac{\frac{2}{3}\times 1+\frac{1}{2}\times 0.5}{\frac{2}{3}+\frac{1}{2}} = 0.7857. \end{equation*}$

The relative error is $1.2\%$ , and the process can continue if a higher accuracy is still needed.

5. Discussion and Conclusion

The ancient Chinese algorithm provides a simple and straightforward tool to two-point boundary value problems arising in chemistry, and it can be used for fast insight into the solution property of a complex problem.

6. Conflict of interest

The authors declare that they have no conflicts of interest to this work.

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Mathematical Modelling and Control

An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

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Abstract

1. Introduction

2. Taylor series solution

3. **The Ying Buzu algorithm**

4. Application

5. Discussion and Conclusion

6. Conflict of interest

References

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Mathematical Modelling and Control

An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

Related Papers:

Abstract

1. Introduction

2. Taylor series solution

3. The Ying Buzu algorithm

4. Application

5. Discussion and Conclusion

6. Conflict of interest

References

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3. **The Ying Buzu algorithm**