- Research
- Open Access
A Ritz-Galerkin approximation to the solution of parabolic equation with moving boundaries
- Jianrong Zhou^{1} and
- Heng Li^{2}Email author
- Received: 7 August 2015
- Accepted: 3 December 2015
- Published: 15 December 2015
Abstract
The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.
Keywords
- Ritz-Galerkin method
- Bernstein polynomial basis
- parabolic equation
- moving boundaries
- initial boundary value problem
- approximation solution
- ductal carcinoma in situ (DCIS) model
MSC
- 35K05
- 35K15
- 35K20
- 35Q68
1 Introduction
So far, there are many publications about parabolic equations with fixed value boundary condition [15, 16], but to the best knowledge of the authors, this is the first time that Ritz-Galerkin method is used for moving boundary value problem presented here. Therefore, it is also significant mathematically.
The Ritz-Galerkin method in Bernstein polynomials basis is the method to convert a continuous operator problem to a discrete problem, which essentially converts the equation to a weak formulation, and then apply some constraints on the function space to characterize the space with a finite set of basis functions. It has been widely used in many areas of mathematics, especially in the field of numerical analysis [17–21].
In this paper, we obtain the existence and uniqueness of the one-dimensional nutrient concentration DCIS model (1.1)-(1.6). Furthermore, this is the first time the Ritz-Galerkin method in Bernstein polynomials basis is employed to give an approximate solution of the parabolic equation with moving boundaries. Illustrative examples are included to demonstrate the validity and applicability of our technique.
The paper is divided as follows. In Section 2, we present an equivalent form of original problem. Section 3 is devoted to the existence and uniqueness of a solution. The properties of Bernstein polynomials are presented in Section 4. The numerical schemes for the solution of equations (5.1)-(5.6) are described in Section 5. Section 6 presents two test examples to support the new method. Finally, conclusions are made in Section 7.
2 Equivalent problems
In this section, we introduce two transformations to convert our problem (1.1)-(1.6) to two equivalent forms. Therefore, we may apply the Ritz-Gelerkin method to the second equivalent form to get approximation solution of problem.
Then the variable \(x\in[\varphi_{1}(t),\varphi_{2}(t)]\) makes \(\xi\in[0,1]\).
3 Existence and uniqueness
In this section, the existence and uniqueness of a solution of problem (1.1)-(1.6) are discussed.
Assumption
- (a)The function \(F(x,\eta,w,p)\) is defined and continuous on the set$$\Omega=\bigl\{ (x,\eta,w,p)| (x,\eta)\in[0,1]\times[0,1],-\infty< w< \infty, - \infty< p< \infty\bigr\} ; $$
- (b)
For each \(C>0\) and for \(|w|,|p|< C\), the function \(F(x,\eta,w,p)\) is uniformly Hölder continuous in x and η for each compact subset of \(D_{T}=\{(x,\eta)| (x,\eta)\in(0,1)\times(0,1]\}\);
- (c)There exists a constant \(C_{F}\) such thatfor all \((w_{i},p_{i})\), \(i=1,2\).$$\bigl|F(x,\eta,w_{1},p_{1})\bigr|-\bigl|F(x,\eta,w_{2},p_{2})\bigr| \leq C_{F}\bigl[|w_{1}-w_{2}|+|p_{1}-p_{2}|\bigr] $$
Applying the results of Cannon [22], p.351, Thm. 20.3.3, to the initial boundary value problem given by equations (3.7)-(3.12), we have the following theorem, which gives the existence and uniqueness of its solution.
Theorem 1
According to the relationship of functions \(u(x,t)\), \(v(x,t)\), and \(w(x,\eta)\), we can easily get the following theorem.
Theorem 2
4 Bernstein polynomials and their properties
Remarks
- (1)
Space \(\operatorname{Span}\{L_{0}(x),L_{1}(x),\ldots,L_{m}(x)\} = \operatorname{Span}\{B_{0,m}(x),B_{1,m}(x),\ldots,B_{m,m}(x)\}:=Y \subset V \) and \(B_{1,m}(x),B_{2,m}(x),\ldots,B_{m,m}(x)\) are bases of the subspace Y of V.
- (2)Let \(f(x)\in V=L^{2}[0,1]\). Then there exists a unique best approximation to \(f(x)\) out of Y such that \(y_{0}(x)\in Y\); that is, if \(y(x)\in Y\), thenmoreover,$$ \bigl\| y_{0}(x)-f(x)\bigr\| \leq \bigl\| y(x)-f(x)\bigr\| ; $$(4.13)where the coefficient matrix \(C^{T}\) can be obtained by$$ y_{0}(x)=\sum_{k=0}^{m}c_{k}B_{k,m}=(c_{0},c_{1}, \ldots ,c_{m}) \bigl(B_{0,m}(x),B_{1,m}(x), \ldots,B_{m,m}(x)\bigr)^{T}:=C^{T}\phi, $$(4.14)$$ C^{T}=\bigl\langle f,\phi^{T}\bigr\rangle \bigl\langle \phi,\phi^{T}\bigr\rangle ^{-1}. $$(4.15)
5 Bernstein Ritz-Galerkin method
In this section, we apply the Ritz-Galerkin method to the second equivalent problem (2.21)-(2.26) in Section 2. Then an approximate solution of the original problem can be easily obtained by (2.29).
6 Numerical application
In this section, we perform two numerical examples with the Ritz-Galerkin methods described in previous sections. The validity and efficiency of our numerical scheme are demonstrated by comparing the approximate result with the exact solution.
Example 1
We applied the method presented in this paper with \(N=2\), \(M=4\) and solved equation (6.8).
Similarly, we can get approximate solutions of problems (6.8)-(6.11) and (1.1)-(1.6) for different values of N and M.
The absolute error for \(\pmb{H(x,t)}\) in Example 1
( x , t ) | N = 2, M = 4 | N = 2, M = 6 | N = 2, M = 8 |
---|---|---|---|
(0,0) | 0 | 0 | 0 |
(0.1,0.1) | 4.51 × 10^{−6} | −9.23 × 10^{−7} | 9.09 × 10^{−9} |
(0.2,0.2) | 8.98 × 10^{−5} | 1.00 × 10^{−6} | −3.19 × 10^{−8} |
(0.3,0.3) | −7.04 × 10^{−6} | 2.68 × 10^{−6} | 6.72 × 10^{−8} |
(0.4,0.4) | −1.97 × 10^{−4} | −3.49 × 10^{−6} | −3.04 × 10^{−8} |
(0.5,0.5) | −1.60 × 10^{−4} | −4.19 × 10^{−6} | −8.21 × 10^{−8} |
(0.6,0.6) | 1.23 × 10^{−4} | 4.37 × 10^{−6} | 1.02 × 10^{−7} |
(0.7,0.7) | 2.80 × 10^{−4} | 4.62 × 10^{−6} | 9.62 × 10^{−9} |
(0.8,0.8) | 5.62 × 10^{−5} | −4.14 × 10^{−6} | −7.34 × 10^{−8} |
(0.9,0.9) | −1.46 × 10^{−4} | −4.27 × 10^{−7} | 5.06 × 10^{−8} |
(1,1) | 0 | 0 | 0 |
The absolute error for \(\pmb{u (\frac{x}{2-t},t )}\) in Example 1
( x , t ) | N = 2, M = 4 | N = 2, M = 6 | N = 2, M = 8 |
---|---|---|---|
(0,0) | 0 | 0 | 0 |
(0.1,0.1) | 7.80 × 10^{−6} | −1.64 × 10^{−7} | 1.62 × 10^{−8} |
(0.2,0.2) | 1.37 × 10^{−4} | 1.52 × 10^{−6} | −4.89 × 10^{−8} |
(0.3,0.3) | −9.19 × 10^{−6} | 3.39 × 10^{−6} | 8.52 × 10^{−8} |
(0.4,0.4) | −2.02 × 10^{−4} | −3.61 × 10^{−6} | −3.25 × 10^{−8} |
(0.5,0.5) | −1.23 × 10^{−4} | −3.19 × 10^{−6} | −6.21 × 10^{−8} |
(0.6,0.6) | 7.91 × 10^{−5} | 2.70 × 10^{−6} | 6.13 × 10^{−8} |
(0.7,0.7) | 1.12 × 10^{−4} | 1.69 × 10^{−6} | −8.15 × 10^{−10} |
(0.8,0.8) | 8.81 × 10^{−6} | −1.10 × 10^{−6} | −1.74 × 10^{−8} |
(0.9,0.9) | −1.68 × 10^{−5} | −1.12 × 10^{−8} | 5.72 × 10^{−9} |
(1,1) | 0 | 0 | 0 |
The \(\pmb{L^{2}}\) norm errors for functions \(\pmb{H(x,t)-\widetilde{H}(x,t)}\) and \(\pmb{u(x,t)-\widetilde{u}(x,t)}\) in Example 1
( N , M ) | \(\boldsymbol {\|H(x,t)-\widetilde{H}(x,t)\|_{L^{2}([0,1]\times[0,1])}}\) | \(\boldsymbol {\|u(x,t)-\widetilde{u}(x,t)\|_{L^{2}([\varphi_{1}(t),\varphi _{2}(t)]\times[0,1])}}\) |
---|---|---|
(2,3) | 3.34 × 10^{−7} | 2.74 × 10^{−7} |
(2,4) | 1.92 × 10^{−8} | 1.51 × 10^{−8} |
(2,5) | 2.00 × 10^{−10} | 1.61 × 10^{−10} |
(2,6) | 9.25 × 10^{−12} | 7.07 × 10^{−12} |
(2,7) | 1.06 × 10^{−13} | 8.30 × 10^{−14} |
(2,8) | 2.81 × 10^{−15} | 2.09 × 10^{−15} |
Example 2
We applied the method presented in this paper with \(N=2\), \(M=4\) and solved equation (6.24).
Similarly, we get approximate solutions of problems (6.8)-(6.11) and (1.1)-(1.6) for different values of N and M.
The absolute error for \(\pmb{H(x,t)}\) in Example 2
( x , t ) | N = 2, M = 2 | N = 2, M = 4 | N = 2, M = 6 |
---|---|---|---|
(0,0) | 0 | 0 | 0 |
(0.1,0.1) | 1.69 × 10^{−3} | −1.36 × 10^{−5} | −3.93 × 10^{−7} |
(0.2,0.2) | 1.40 × 10^{−3} | 1.14 × 10^{−4} | −5.00 × 10^{−7} |
(0.3,0.3) | −2.74 × 10^{−3} | 9.58 × 10^{−5} | 2.47 × 10^{−6} |
(0.4,0.4) | −7.35 × 10^{−3} | −1.48 × 10^{−4} | 4.35 × 10^{−7} |
(0.5,0.5) | −7.96 × 10^{−3} | −2.81 × 10^{−4} | −4.25 × 10^{−6} |
(0.6,0.6) | −2.92 × 10^{−3} | −5.10 × 10^{−5} | −5.68 × 10^{−7} |
(0.7,0.7) | 4.58 × 10^{−3} | 2.65 × 10^{−4} | 4.70 × 10^{−6} |
(0.8,0.8) | 7.91 × 10^{−3} | 1.70 × 10^{−4} | −9.47 × 10^{−7} |
(0.9,0.9) | 3.18 × 10^{−3} | −1.25 × 10^{−4} | −1.45 × 10^{−6} |
(1,1) | 0 | 0 | 0 |
The absolute error for \(\pmb{u(x+\sin(\frac{\pi}{2} t),t)}\) in Example 2
( x , t ) | N = 2, M = 2 | N = 2, M = 4 | N = 2, M = 6 |
---|---|---|---|
(0,0) | 0 | 0 | 0 |
(0.1,0.1) | −1.10 × 10^{−3} | 8.91 × 10^{−6} | 2.48 × 10^{−7} |
(0.2,0.2) | −1.05 × 10^{−3} | −7.88 × 10^{−5} | 3.76 × 10^{−7} |
(0.3,0.3) | 1.87 × 10^{−3} | −7.79 × 10^{−5} | −1.83 × 10^{−6} |
(0.4,0.4) | 5.66 × 10^{−3} | 1.04 × 10^{−4} | −5.49 × 10^{−7} |
(0.5,0.5) | 6.76 × 10^{−3} | 2.31 × 10^{−4} | 3.37 × 10^{−6} |
(0.6,0.6) | 3.35 × 10^{−3} | 7.52 × 10^{−5} | 9.43 × 10^{−7} |
(0.7,0.7) | −2.45 × 10^{−3} | −1.85 × 10^{−4} | −3.68 × 10^{−6} |
(0.8,0.8) | −5.56 × 10^{−3} | −1.56 × 10^{−4} | 1.55 × 10^{−7} |
(0.9,0.9) | −2.82 × 10^{−3} | 6.30 × 10^{−5} | 1.34 × 10^{−6} |
(1,1) | 0 | 0 | 0 |
The \(\pmb{L^{2}}\) norm error for the functions \(\pmb{H(x,t)-\widetilde{H}(x,t)}\) and \(\pmb{u(x,t)-\widetilde{u}(x,t)}\) in Example 2
( N , M ) | \(\boldsymbol {\|H(x,t)-\widetilde{H}(x,t)\|_{L^{2}([0,1]\times[0,1])}}\) | \(\boldsymbol {\|u(x,t)-\widetilde{u}(x,t)\|_{L^{2}([\varphi_{1}(t),\varphi_{2}(t)]\times[0,1])}}\) |
---|---|---|
(N = 2, M = 1) | 1.12 × 10^{−4} | 1.12 × 10^{−4} |
(N = 2, M = 2) | 2.18 × 10^{−5} | 2.18 × 10^{−5} |
(N = 2, M = 3) | 7.24 × 10^{−8} | 7.24 × 10^{−8} |
(N = 2, M = 4) | 2.24 × 10^{−8} | 2.24 × 10^{−8} |
(N = 2, M = 5) | 5.96 × 10^{−10} | 5.96 × 10^{−10} |
(N = 2, M = 6) | 4.95 × 10^{−12} | 4.95 × 10^{−12} |
Remark
7 Conclusion
In this paper, the Ritz-Galerkin method in Bernstein polynomial basis is implemented to obtain an approximate solution of a nonclassical parabolic equation subject to given initial and moving boundary conditions. Also, the existence and uniqueness of a solution are discussed. The properties of Bernstein polynomials and the Ritz-Galerkin method are first presented’ then the Ritz-Galerkin method is used to reduce the parabolic equation with moving boundaries to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of new numerical technique developed.
Declarations
Acknowledgements
The present investigation was supported in part by the National Natural Science Foundation of the People’s Republic of China under grant numbers 11201070 and the Science Research Fund of Department of Guangdong Province of the People’s Republic of China under grant numbers Yq2013161.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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