An Optimal Method for Diffusion Parameters of Nonlinear Diffusion Problem of Drug Releasing in 2D-Disc Device by Separate Variable Method

Abstract

An optimization control model and the corresponding computational method drawing the diffusion parameters of the nonlinear problem for the drug releasing in the 2D-disc device were given in this paper. Firstly, based on the nonlinear diffusion equation of the drug releasing in the 2D-disc device, we used the linear diffusion problem to discrete the nonlinear diffusion problem with the discrete space and the discrete time. Then, by the separate variable method, the solution of the linear problem was given. Next, the least square method based on the separate variable idea (LSMSV) was used to estimate the nonlinear appropriate diffusion parameters. Finally, a numerical example was presented to show that the control model and the numerical method were valid for computing the diffusion coefficient of the nonlinear problem for the drug releasing in the 2D-disc device.

1. Introduction

In engineering fields, there exist many diffusion processes in many fields such as geomechanics engineering, biomedical science, civil engineering, water pollution, and soil engineering [1–5]. In order to simulate the diffusion processes to obtain their merits, it is important to draw the effective diffusiveness. There are many models for the simulation of the diffusion processes. Most of them are the nonlinear or linear models. For the linear models, most optimal control problems governed by the diffusion equations arose in many scientific and engineering applications such as the water pollution problems and the drug releasing fields [6–10]. There are many various techniques for the identification for the effective diffusiveness based on the linear models. These techniques are based on either empirical or semiempirical models from drug delivery mechanisms or on analytic solutions of the diffusion equation in 2D or in the special cases [6–8, 11, 12].

However, in the practice application, many diffusion processes are subjected to the nonlinear partial differential equations [6–12]. In order to illustrate the nonlinear diffusion processes in many fields, many nonlinear models are applied to estimate the properties. For many nonlinear diffusion fields, the diffusion coefficients are the functions of the diffusion concentration. Therefore, computing the diffusion parameters is mainly to determinate the parameters of the coefficient function called the diffusion parameters function. The diffusion parameter function of the concentration is considered as the main element to control the diffusion processes. Therefore, many researches were given to determine the parameter function to illustrate the diffusion processes. Many nonlinear optimal models drawing the diffusion parameters depended on both the lab technology and the shape of the container [9, 10, 12, 13]. In order to compute the diffusion parameters, many scientists and mathematicians provided some optimal methods to compute the diffusion parameters [9, 10, 12]. Most of them cost a lot of computing time and computing memory. Even some of them such as the different method or the finite element method cost more than one week [9, 14]. Therefore, in order to save the computing time and the computing memory, in our published papers, we had provided some numerical methods based on the separate variable method to compute the diffusion parameters of the linear process in the sphere device. For these reasons, in this paper, we will also propose a new numerical optimal method (the least square method based on the separate variable idea) to extract the diffusion parameters from the nonlinear diffusion problems.

The next two sections will give the nonlinear diffusion problem in the 2D-disc device based on the drug releasing property and the discrete method. Section 3 is devoted to providing the least square method based on the separate variable idea for the optimal control model of the nonlinear diffusion equation system governing the drug releasing process. In Section 4, the numerical example is presented to demonstrate the feasibility and the validity, the convergence of the model, and computing method. Finally, we will discuss the model and computing method and give the future work for the different control model.

2. The Nonlinear Diffusion Problem in the 2D-Disc Device and the Discrete Linear Problem

The nonlinear diffusion process in 2D-disc device is governed by the following partial differential equation: 𝜕𝐶𝑥,𝑦,𝑡(𝐷(𝐶))𝜕𝑡−∇⋅∇𝐶=0,𝑡>0,𝑥,𝑦∈Ω,𝜕𝐶𝑥,𝑦,𝑡𝐶,𝜕𝑛=0,𝑡>0,𝑥,𝑦∈𝜕Ω,𝑥,𝑦,0=𝐻𝑥,𝑦𝑥,𝑦∈Ω, (1) where the drug concentration is uniform in the device and zero in liquid for the initial condition at 𝑡=0; that is, 𝐻=𝑀𝑥,𝑦,0𝑉⁰_𝑑,𝑥,𝑦∈Ω₁,0,𝑥,𝑦∈Ω⧵Ω₁; (2)𝑀⁰ is the total mass of the diffusion material, 𝑉_𝑑 is the volume of Ω₁, the diffusion coefficient 𝐷(𝐶) is the function of the concentration, 𝐶(𝑥,𝑦,𝑡) is the drug releasing concentration, Ω₁ is the small disc containing the drug, and Ω is the big disc containing the liquid shown in Figure 1.

Figure 1 

Large disc Ω containing liquid and small disc Ω1 containing drug.

In order to solve the nonlinear diffusion process by the separate variable method, it is necessary to use some linear equations to replace the nonlinear equation (1). The discretion process is given as follows.

Time Discretion. Setting Δ𝑡_𝑖=𝑡_𝑖+1−𝑡_𝑖,𝑖=1,2,…,𝑁, the time [𝑇₀,𝑇₁] of the diffusion process is divided into many sections [𝑡_𝑖,𝑡_𝑖+1] and the concentration 𝐶_𝑖 is computed in [𝑡_𝑖,𝑡_𝑖+1].

Space Discretion. Setting Δ𝑟_𝑗=𝑟_𝑗+1−𝑟_𝑗, 𝑗=1,2,…,𝐾−1, 𝑅₁=𝑟₁<𝑟₂<⋯<𝑟_𝑘=𝑅₂, the radius [𝑅₁,𝑅₂] is divided into many intervals and the diffusion concentration in the section of [𝑟₁,𝑟₂],[𝑟₂,𝑟₃],…,[𝑟_𝑘−1,𝑟_𝑘] is obtained. So the nonlinear equation system (1) can be changed into (𝑘−1)∗𝑁 linear equations.

Using the linear diffusion processes to replace the nonlinear diffusion process in the time, it is easy to obtain the following linear equation: 𝜕𝐶_𝑖𝐶𝜕𝑡=𝐷_𝑖−1𝜕²𝐶_𝑖𝜕𝑥²+𝜕²𝐶_𝑖𝜕𝑦². (3)

Using the polar coordinate system to replace (3), the above equations can be changed as follows: 𝜕𝐶_𝑖𝐶𝜕𝑡=𝐷_𝑖−1𝜕²𝐶_𝑖𝜕𝑟²+1𝑟𝜕𝐶_𝑖.𝜕𝑟 (4)

From the above deduction, the nonlinear diffusion equation of the drug releasing in the section of [𝑡_𝑖,𝑡_𝑖+1] and [𝑟_𝑗,𝑟_𝑗+1] can be obtained as follows: 𝜕𝐶_𝑖𝑗𝐶𝜕𝑡=𝐷_{(𝑖−1)𝑗}𝜕²𝐶_𝑖𝑗𝜕𝑟²+1𝑟𝜕𝐶_𝑖𝑗,𝜕𝑟𝑖=1,2,…,𝑁,𝑗=1,2,…,𝐾−1. (5)

Because the length of the time and the range of space by the discretion are very small, 𝐷(𝐶_{𝑖−1,𝑗}) can be considered as a constant in [𝑡_𝑖,𝑡_𝑖+1]×[𝑟_𝑗,𝑟_𝑗+1].

In the section of [𝑡_𝑖,𝑡_𝑖+1], the linear equations can be represented as follows: 𝜕𝐶_𝑖𝑗𝐶𝜕𝑡=𝐷_{𝑖−1,𝑗}𝜕²𝐶_𝑖𝑗𝜕𝑟²+1𝑟𝜕𝐶_𝑖𝑗𝜕𝑟,𝑡_𝑖≤𝑡≤𝑡_𝑖+1,𝑖=1,…,𝑁,0<𝑟<𝑟_𝑗+1,𝑗=1,…,𝐾−1,𝜕𝐶_𝑖𝑗𝜕𝑟_{𝑟=𝑟𝑗+1}𝐶=0,_𝑖𝑗𝑟,𝑡_𝑖=𝑀_{𝑖−1,𝑗}𝑉_𝑑𝑗,0<𝑟<𝑟_𝑗,0,𝑟_𝑗<𝑟<𝑟_𝑗+1, (6) where 𝑉_𝑑𝑗=𝜋(𝑟²_𝑗+1−𝑟²_𝑗). Therefore, the concentrations 𝐶_𝑖𝑗 of this equation in [𝑡_𝑖,𝑡_𝑖+1]×[𝑟_𝑗,𝑟_𝑗+1] can be obtained and get 𝑀_𝑖𝑗=₀^{2𝜋𝑟𝑗+1}_𝑟𝑗𝐶_𝑖(𝑟,𝜃,𝑡)𝑑𝑟𝑑𝜃−₀^{2𝜋𝑟𝑗}_𝑟𝑗−1𝐶_𝑖−1(𝑟,𝜃,𝑡)𝑑𝑟𝑑𝜃,𝑖=1,2,…,𝑁,𝑗=1,2,…,𝐾. (7) If 𝑖=0 and 𝑗=1, 𝐶₀₁(𝑟,𝜃,0)=𝐻(𝑟,𝜃,0).

To solve (𝐾−1)×𝑁 linear equations, by the separation variable method, we can get the solutions of the nonlinear diffusion equation as follows: 𝐶_𝑖𝑗=𝑀_𝑖𝑗𝑉_𝑑𝑗+^∞_𝑛=12𝑀_𝑖𝑗𝑟_𝑗𝑉_𝑑𝑗𝑟_𝑗+1𝜇_𝑛𝐽₀²𝜇_𝑛𝐽₁𝑟_𝑗𝜇_𝑛𝑟_𝑗+1×exp−𝐷𝐶_{(𝑖−1)𝑗}𝜇_𝑛𝑟_𝑗+1²𝑡_𝑖𝐽₀𝜇_𝑛𝑟_𝑗+1𝑟,𝐶𝑖=1,2,…,𝑁,𝑗=1,2,…,𝐾,_𝑖=^𝑁𝑖_𝑗=1𝑙=1𝐶_𝑙𝑗(𝑟,𝜃,𝑡),𝑟∈𝑟_𝑗,𝑟_𝑗+1,𝑖=1,2,…,𝐾. (8)(9)

Equation (9) can be changed into the following formula: 𝐶_𝑖=^𝑁_𝑗=1^𝑖_𝑙=1𝑀_𝑙𝑗𝑉_𝑑+^∞_𝑛=12𝑀_𝑙𝑗𝑟_𝑗𝑉_𝑑𝑟_𝑗+1𝜇_𝑛⁽¹⁾𝐽²₀𝜇_𝑛⁽¹⁾𝐽₁𝑟_𝑗𝜇_𝑛𝑟_𝑗+1×exp−𝐷𝐶_{(𝑙−1)𝑗}𝜇_𝑛𝑟_𝑗+1²𝑡_𝑙×𝐽₀𝜇_𝑛𝑟_𝑗+1𝑟,𝑟∈𝑟₂,𝑟_𝐾,𝑖=1,2,…,𝐾−1, (10) where 𝜇_𝑛⁽¹⁾(𝑛=1,2,…) are the positive roots of 𝐽₁(𝑥) and Γ(𝑚+1)=𝑚!, 𝑚 is the positive integer and 𝐽₁𝜇_𝑛=^∞_𝑚=0(−1)^𝑚1Γ()Γ()𝜇𝑚+1𝑚+22^𝑛2𝑚+1,𝐽₀𝜇_𝑛𝑅₂𝑟=^∞_𝑚=0(−1)^𝑚1Γ()Γ()𝜇𝑚+1𝑚+1^𝑛𝑟2𝑅₂^2𝑚. (11)

3. Control Problem Drawing Nonlinear Diffusion Parameters

Problem 1. Search the coefficient functions 𝐷(𝑎₁,…,𝑎_𝐿,𝐶) to satisfy min_𝐷>0𝑀_𝑇1−𝑀⁰_𝑇1²+𝑀_𝑇2−𝑀⁰_𝑇2²+⋯+𝑀_𝑇𝑒−𝑀⁰_𝑇𝑒², (12) where 𝑀⁰_𝑇1,𝑀⁰_𝑇2,…,𝑀⁰_𝑇𝑒 are the given experimental data and 𝑀_𝑇1,𝑀_𝑇2,…,𝑀_𝑇𝑒 are the computed data by the following equations: 𝑀_𝑇𝑖=^𝐾_{𝑡𝑘≤𝑇𝑖,𝑗=2}𝑀_𝑘𝑗. (13)

4. Least Square Method Based on Separate Variable Method for Solving Optimal Control Problem

Let 𝐸(𝐷)=𝑀_𝑇1−𝑀⁰_𝑇1²+𝑀_𝑇2−𝑀⁰_𝑇2²+⋯+𝑀_𝑇𝑒−𝑀⁰_𝑇𝑒²=𝑀−𝑀^∗𝑇𝑀−𝑀^∗, (14) where 𝑀=(𝑀_𝑇1(𝑎₁,…,𝑎_𝐿), 𝑀_𝑇2(𝑎₁,…,𝑎_𝐿),…, 𝑀_𝑇𝑒(𝑎₁,…,𝑎_𝐿))^𝑇, and 𝑀^∗=(𝑀⁰_𝑇1,𝑀⁰_𝑇2,…,𝑀⁰_𝑇𝑒)^𝑇.

For an initial diffusion parameter point 𝐷₁(𝑎₁,…,𝑎_𝐿), Problem 1 can be solved iteratively by the following deduction. If 𝑎^𝑖₁,…,𝑎^𝑖_𝐿 are the 𝑖th approximation and 𝛿𝑎^𝑖₁,…,𝛿𝑎^𝑖_𝐿 are the 𝑖th increment of 𝑎₁,…,𝑎_𝐿, respectively, in each step, an increment 𝛿𝐷_𝑖(𝛿𝑎^𝑖₁,…,𝛿𝑎^𝑖_𝐿) will be computed as follows. To minimize 𝐸(𝐷_𝑖+𝛿𝐷_𝑖) with 𝛿𝐷_𝑖(𝛿𝑎^𝑖₁,…,𝛿𝑎^𝑖_𝐿), let 𝑋=(𝑥₁,…,𝑥_𝐿)=(𝑎₁,…,𝑎_𝐿), 𝑓_𝑇𝑖(𝑋)=𝑀_𝑇𝑖−𝑀⁰_𝑇𝑖, 𝐹(𝑋)=𝐸(𝐷), so 𝐹(𝑋)=𝐸(𝑋) and 𝛿𝐷(𝛿𝑋)=𝛿𝐷_𝑖(𝛿𝑎^𝑖₁,…,𝛿𝑎^𝑖_𝐿). If set 𝑓_𝑖(𝑋)=𝑀_𝑇𝑖−𝑀⁰_𝑇𝑖=₀^{2𝜋𝑟𝑁}_𝑟2𝐶_𝑇𝑖𝑟,𝜃,𝑇_𝑖𝑑𝑟𝑑𝜃−𝑀⁰_𝑖𝑖=1,2,…,𝑒, (15) where 𝐶_𝑇𝑖=^𝑁_𝑗=2^{max{𝑙≤𝑇𝑖}}_𝑙=1𝐶_𝑙𝑗(𝑟,𝜃,𝑡),𝑟∈𝑟_𝑗,𝑟_𝑗+1,𝑖=1,2,…,𝑒, (16) equation (15) can be written in the following form: 𝐹(𝑥)=^𝑒_𝑖=1𝑓²_𝑖(𝑋). (17)

To solve this problem, suppose 𝑋_(𝑘) is the 𝑘th approximation and let the function 𝑓_𝑖(𝑋) be the Taylor expansion function at 𝑋_(𝑘); the minimal point 𝑋_(𝑘+1) and the (𝑘+1)th approximation can be computed by the iterative method. The iterative formula is deduced in detail as follows.

Set 𝜙_𝑖(𝑋)=𝑓_𝑖𝑋_(𝑘)+∇𝑓_𝑖𝑋_(𝑘)^𝑇𝑋−𝑋_(𝑘)=∇𝑓_𝑖𝑋_(𝑘)^𝑇𝑋−∇𝑓_𝑖𝑋_(𝑘)^𝑇𝑋_(𝑘)−𝑓_𝑖𝑋_(𝑘),𝜙(𝑋)=𝑖=1,2,…,𝑒,_𝑒𝑖=1𝜙²_𝑖(𝑋). (18)(19)

We use 𝜙(𝑋) to replace 𝐹(𝑋) and compute the minimal point of 𝜙(𝑋) to estimate the function 𝐹(𝑋). The least square problem: min𝜙(𝑋) can be solved as follows. Set 𝐴=∇𝑓₁𝑋_(𝑘)^𝑇⋮∇𝑓_𝑒𝑋_(𝑘)^𝑇=𝜕𝑓₁𝑋_(𝑘)𝜕𝑥₁⋯𝜕𝑓₁𝑋_(𝑘)𝜕𝑥_𝐿⋮⋮𝜕𝑓_𝑒𝑋_(𝑘)𝜕𝑥₁⋯𝜕𝑓_𝑒𝑋_(𝑘)𝜕𝑥_𝐿,𝐵=∇𝑓₁𝑋_(𝑘)^𝑇𝑋_(𝑘)−𝑓₁𝑋_(𝑘)⋮∇𝑓_𝑒𝑋_(𝑘)^𝑇𝑋_(𝑘)−𝑓_𝑒𝑋_(𝑘)=𝐴𝑋_(𝑘)−𝑓_(𝑘),𝑓_(𝑘)=𝑓₁𝑋_(𝑘)𝑓₂𝑋_(𝑘)⋮𝑓_𝑒𝑋_(𝑘),𝜕𝑓_𝑖𝑋_(𝑘)𝜕𝑥_𝑙=^𝑟𝑁_𝑟20^𝑁2𝜋_𝑗=1^{𝑙≤𝑇𝑖}_𝑙=1𝑀_𝑙𝑗𝑉_𝑑+^∞_𝑛=12𝑀_𝑙𝑗𝑟_𝑗𝑉_𝑑𝑟_𝑗+1𝜇_𝑛⁽¹⁾𝐽²₀𝜇_𝑛⁽¹⁾×𝐽₁𝑟_𝑗𝜇_𝑛𝑟_𝑗+1×exp−𝐷𝑋_(𝑘),𝐶_{(𝑙−1),𝑗}×𝜇_𝑛𝑟_𝑗+1²𝑡_𝑙×𝐽₀𝜇_𝑛𝑟_𝑗+1𝑟𝜕𝐷𝑋_(𝑘),𝐶_{𝑙−1,𝑗}𝜕𝑥_𝑙×𝜇_𝑛𝑟_𝑗+1²𝑡_𝑙𝑑𝜃𝑑𝑟. (20)(21)

Equation (19) can be written as 𝜙(𝑋)=(𝐴𝑋−𝐵)^𝑇(𝐴𝑋−𝐵)=𝑋^𝑇𝐴^𝑇𝐴𝑋−2𝐵^𝑇𝐴𝑋+𝐵^𝑇𝐵. (22)

In order to search the stable point of 𝜙(𝑋), set ∇𝜙(𝑋)=2𝐴^𝑇𝐴𝑋−2𝐴^𝑇𝐵=0. (23)

Taking 𝐴 and 𝐵 into the above formula, there is 𝐴^𝑇𝐴𝑋=𝐴^𝑇𝐴𝑋_(𝑘)−𝑓_(𝑘). (24)

Moving the right 𝐴^Τ𝐴𝑋_(𝑘) to the left hand in the above equation, the following equation can be obtained: 𝐴^𝑇𝐴𝑋−𝑋_(𝑘)=−𝐴^𝑇𝑓_(𝑘). (25)

Obviously, this is a linear algebraic equation about the function value and the first order partial derivative 𝑓_𝑖(𝑋_(𝑘)) at the point 𝑋_(𝑘). If matrix 𝐴 is the full column rank, 𝐴^𝑇𝐴 is a symmetry positive matrix. Therefore, there exists (𝐴^𝑇𝐴)⁻¹. We can get the stable point of 𝜙(𝑋) by (19) 𝑋_(𝑘+1)=𝑋_(𝑘)−𝐴^𝑇𝐴⁻¹𝐴^𝑇𝑓_(𝑘). (26)

Set 𝑋_(𝑘+1) as the (𝑘+1)th approximation of stable point 𝐹(𝑋).

Algorithm 2 (diffusion parameter of nonlinear process). Step  1. Give the initial point 𝑋₍₁₎=(𝑎_1,(1),…,𝑎_𝑛,(1)) and set 𝑡₁=0, 𝑀₁=0,  𝑘=1, 𝑓₁(𝑋_(𝑘))=0 and the control error 𝜀>0, go to Step  2.
Step  2. Compute the drug concentration 𝑀_𝑖𝑗(𝑋_(𝑘)) and 𝐶_𝑖𝑗(𝑋_(𝑘)) at 𝑡_𝑖 in [𝑟_𝑗,𝑟_𝑗+1] by the formula (7) and (8), obtain the mass in all sections [𝑟_𝑗,𝑟_𝑗+1] at the time 𝑡_𝑖, and go to Step  3.
Step  3. Compute the mass 𝑀_𝑇𝑖(𝑋_(𝐾)), 𝑖=1,2,…,𝑒. at the outer container Ω⧵Ω₁ by the formula (13); go to Step  4.
Step  4. According to the formula (15), compute 𝑓_𝑖(𝑋_(𝑘))=𝑀_𝑇𝑖(𝑋_(𝑘))−𝑀⁰_𝑇𝑖; get the vector: 𝑓_(𝑘)=^{𝑓1(𝑋(𝑘))𝑓2(𝑋(𝑘))}_{⋮𝑓𝑒(𝑋(𝑘))}. Compute the first order partial derivative 𝑎_𝑖𝑗=𝜕𝑓_𝑖(𝑋_(𝑘))/𝜕𝑥_𝑗, 𝑖=1,2,…,𝑒, 𝑗=1,2,…,𝐿 by formula (21). Get the matrix 𝐴_𝑘=(𝑎_𝑖𝑗)_𝑒×𝐿 by the formula (20); go to Step  5.
Step  5. Compute 𝑋_(𝑘+1)=𝑋_(𝑘)−(𝐴^𝑇𝐴_𝑘)⁻¹𝐴^𝑇𝑓_(𝑘); go to Step  6.
Step  6. If ‖𝑋_(𝑘+1)−𝑋_(𝑘)‖_𝐿2≤𝜀, set 𝑋^∗=𝑋_(𝑘+1) and get the optimum coefficient (𝑎₁,…,𝑎_𝐿)=𝑋^∗, go to Step  7; otherwise, let 𝑘=𝑘+1, go to Step  2.
Step  7. Output the optimal diffusion (𝑎₁,…,𝑎_𝐿) and stop the algorithm.

5. The Numerical Example

To investigate the feasibility and the validity of the proposed scheme, a numerical example is given in this section. The cylinder device for the drug releasing is divided into the large and small cylinder structures where the small container includes the drug and the large container is filled with the liquid. The radius of small and large cylinder devices are 0.4800 dm and 2.8399 dm, respectively, and there is 100 g of drug in small disc vessel. When 𝑡=0, the inner concentration is 138.1553 g/dm² and the outer is 0 g/dm². After some diffusion process in a period of time, the inner and outer concentration will be equal and be 3.9468 g/dm²; it is 𝐶()=𝑀𝑟,0𝜋𝑅⁰²₁=138.1553,0<𝑟<𝑅₁,0,𝑅₁<𝑟<𝑅₂,𝐶𝑟,𝑇_𝑒=3.94680<𝑟<𝑅₂. (27)

In the numerical example, suppose 𝐷(𝐶)=𝑎𝐶+𝑏 where 𝑎 and 𝑏 are the unknown constants. We firstly suppose the given diffusion parameter (𝑎,𝑏) to be (0.0003,0.0003) then compute the diffusion process to obtain the computed data as the experiment data of the concentration shown in Table 1. Next, the above optimal model and the optimal method to estimate the diffusion parameter based on the optimal control model with the separation variable method of the drug releasing in the 2D-disc device are used to illustrate the feasibility and the validity of the LMSV for the nonlinear diffusion process. The following formula is given to compute the total error in order to estimate the convergence rate: totalerror(𝑘)=^𝑒_𝑗=1𝑀_𝑇𝑗−𝑀⁰_𝑇𝑗𝑘²,𝛿𝑎=𝑎_𝑘−𝑎_𝑘−1,𝛿𝑏=𝑏_𝑘−𝑏_𝑘−1. (28)

Table 1 

Diffusion qualities at the different times with diffusion parameters (0.0003 and 0.0003).

Supposing the space interval to be 0.012 dm and the time interval 20 seconds; getting 25 terms in the Bessel function and by the least square method algorithm, the optimal points and their error values in each optimal step are given in the Table 2. In order to illustrate the convergence of the least square algorithm, the optimal increment 𝛿𝑎,𝛿𝑏 and the optimal values (𝑎,𝑏) of the diffusion parameter are depicted in Figures 2, 3 and 4.

Table 2 

The different errors and optimization value based on initial value (0.0001 and 0.0001).

Figure 2 

The overall error with iterative times on the initial value for (0.0001 and 0.0001).

Figure 3 

The error value of 𝑎 on the initial value for (0.0001 and 0.0001).

Figure 4 

The error value of 𝑏 on the initial value for (0.0001 and 0.0001).

In order to illustrate the convergence of the algorithm for diffusion parameter of nonlinear process by the least square method by separate variables (LSMSV), we discuss the convergence data as follows: from the second column in Table 2 and Figure 3, the increments 𝛿𝑎 converges very fast because their values become from −3.0152 × 10⁻⁴ to 3.1408 × 10⁻⁶ and from Table 2 and Figure 4  𝛿𝑏 varies from 1.1501×10⁻³ to −6.1503 × 10⁻⁶. From the last column in Table 2, the optimized values 𝑎,𝑏 also become very fast from (−2.0152 × 10⁻⁴, 1.2502 × 10⁻³) to (2.9856 × 10⁻⁴, 3.0283×10⁻⁴) by only eight iterated steps. Therefore, from Table 2 and the error value in Figures 2, 3 and 4, the data show the error and the increment 𝛿𝑎,𝛿𝑏 convergent stately. It is easy to illustrate the convergence of the algorithm for diffusion parameter of nonlinear process by least square method by separate variables idea.

In order to test the convergent velocity, we obtain the computing time for the optimal parameters by the algorithm in the numerical examples. The computing time is 1 minute and 56 seconds by the algorithm for diffusion parameter of nonlinear process. In order to test the merits of the algorithm for diffusion parameter of nonlinear process by LSMSV, it is hard for us to use the algorithm in the paper [9] to compute the parameter values because the computed time is very long. In addition, from the computing processes, it is also easy to understand why the computing velocity becomes very high. Because the algorithm in this paper only computes some of the polynomial functions in each iteration to cost only less than 2 minutes, however, the algorithm in [9] needs to solve millions linear algebra equations in each iteration for the nonlinear diffusion process to cost more than one day. Therefore, from the computing convergent time and the computational theory, we can obtain the conclusion that the convergent velocity of algorithm for diffusion parameter of nonlinear process by LSMSV is very fast.

In order to estimate the validity of the algorithm for diffusion parameter of nonlinear process by LSMSV, the total error values between the experiment values and the computed values in this paper are shown in Figure 5 and the first column in Table 2. The optimal computed drug mass depending on the different optimal diffusion parameters and the experiment mass are depicted in Figure 6. From the first column in Table 2, the errors of the optimized values 𝑎,𝑏 and the parameters are only 0.0156 × 10⁻⁴ and 0.213 × 10⁻⁴, the error between the optimal computed drug mass 100 g and the experiment mass data 97.1432 g is only 2.8568 g, and the relative error of the mass is only 2.9%. The error result shows the algorithm for diffusion parameter of nonlinear process by LSMSV being valid to extract the diffusion parameter of the drug releasing in the disc devices for the nonlinear diffusion process. From Figures 5 and 6 and the last optimal diffusion parameter value (2.9856×10⁻⁴, 3.0283×10⁻⁴), the algorithm is valid to extract the diffusion parameter of the drug releasing for the nonlinear diffusion process in the 2D-disc devices. From the numerical example, comparing the data of all iterative steps of the optimization values and the experimental data, it is easy to get the conclusion that the algorithm is the convergence and effectiveness to extract the diffusion parameters of the nonlinear drug releasing in the 2D-disc devices.

Figure 5 

The optimization increments of (𝑎,𝑏).

Figure 6 

Computed data in each iteration and experimental data.

6. Conclusion

In this paper, we propose an optimal method to extract the diffusion parameters of the nonlinear diffusion process of the drug releasing in the 2D-disc device based on the separation variable method with discrete time and discrete space. The numerical result given in the previous section demonstrates the feasibility and validity of this algorithm for diffusion parameter of nonlinear process by LSMSV. The effectiveness of this optimal control model to estimate the diffusion parameters for the nonlinear drug releasing in the 2D-disc device is also discussed. How to establish the theorem of the algorithm is our future work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (NSFC) Grants (no. 11072041 and 61202496), by State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin (IWHRSKL201205), by State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology (GZ1005), by China Postdoctoral Science Foundation (20100480944 and 2012T50692), and by Hunan Provincial Natural Science Foundation of China (13JJ3070, 13JJ2031).