Abstract
In this article, we develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.
1 Introduction
Fourth-order problems have been widely used in many science and engineering [1,2], and the numerical solutions of many complicated and nonlinear equations are ultimately attributed to solving a fourth-order problem repeatedly, such as transmission eigenvalue problem and Cahn-Hilliard equation [3–7]. There have been many results of theoretical analysis and numerical computing on the fourth-order problems, mainly including the finite element methods [8–10], spectral Galerkin methods [11–17], and some finite-difference methods [18].
In recent years, the second problems with singular potential have attracted more and more scholars’ attention because of its challenging theoretical analysis and numerical calculation [19–23]. However, they usually solve these problems using the numerical methods with a direct dividing of the computational domain, which will generate a large number of degrees of freedom and spend a lot of calculation time and memory capacity to obtain high-accuracy numerical solutions [18,24,25]. Especially for the fourth-order problems with singular potential and simply supported plate (SSP) boundary condition in some special regions, both theoretical analysis and numerical calculation are challenging. The principal reason is that the singularity and complex boundary conditions are introduced by the polar transformation [11,26–28], which brings some difficulties in algorithm implementation and error estimation. As we know, there are few studies on the spectral approximation method based on a reduced-dimension scheme for the fourth-order equations with inverse square singular potential and SSP boundary condition in a circular domain.
Thus, the aim of this study is to develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for fourth-order equation with singular potential and SSP boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.
The rest of this article is arranged as follows. We deduce in Section 2 the equivalent reduced-dimension scheme. We prove in Section 3 the existence and uniqueness for the weak solution and its approximate solutions. We give in Section 4 the error estimation of the approximate solutions. We describe in Section 5 the effective implementation process of the algorithm in detail. We carry on in Section 6 some numerical tests. Finally, we make in Section 7 some concluding remarks.
2 Equivalent dimension reduction scheme
We consider in this study the following fourth-order equation:
where
Let
where
Plugging (2.6) into (2.3), we obtain that
Thus, for the well posedness of equation (2.4), the following essential polar conditions need to be imposed, i.e.,
By discussing
Let
3 Existence and uniqueness of the solution
Start with deriving the weak formulation and its discrete scheme for problems (2.10)–(2.12). Then, we further give the proof for the existence and uniqueness of the solution.
3.1 Weak formulation and discrete scheme
Let
with the inner product and norm as follows:
where
equipped with the following inner product and norm:
Thus, a weak formulation of equations (2.10)–(2.12) is: find
where
Denote by
3.2 Existence and uniqueness of the solution
We first introduce a symbol that will be used frequently later. Namely,
Lemma 1
For any
with
with
Proof
When
We derive from (5.8)–(3.7) that
Then, (3.3) follows. When
Then, (3.4) follows.□
Lemma 2
For any
Proof
By the boundary condition
Then, (3.8) follows. Next, we prove Inequality (3.9). When
When
Then, we derive from (3.3) and (3.10) that
Then, (3.9) follows.□
Lemma 3
Proof
We derive from Lemmas 1–2 and Cauchy-Schwarz inequality that
On the other hand, we can derive that
Theorem 1
If
Proof
Note that if
In fact, from the Cauchy-Schwarz inequality and Lemma 2, we derive that
Thus, it can be known from the Lax-Milgram theorem and Lemma 3 that the desired result follows.□
4 Error estimation
We shall give in this section the error estimates of the approximate solution.
Lemma 4
Let
Proof
It follows from equations (3.1) and (3.2) that
Thus, we have
By Lemma 3 and (4.1), we derive that
Namely,
Note that
In fact, using Hardy inequality (cf. B8.6 in [29]), we derive that
with
When
On the other hand, we derive that
Then, from Cauchy-Schwarz inequality and (4.5), we derive that
Then, (4.2) follows. This finishes our proof.□
Let
and the corresponding inner product and norm are as follows:
Define
with the following inner product and norm:
Let
Define an orthogonal projection operator:
According to Theorem 1.8.2 in [30], we have the following lemma:
Lemma 5
For any
Theorem 2
There exists an operator
where
Proof
Let
Then, it follows that
Note that
Namely,
Similarly, we can derive that
For
which means
By Lemma 5, we have
Combined with equations (4.6), (4.7), and (4.8), the desired result follows.□
Following the arguments of the proof in Theorem 2, we have the following theorem:
Theorem 3
There exists an operator
where
We now give the main theorem as follows:
Theorem 4
Let
Proof
From Lemma 4 and Theorems 2–3, we can obtain the desired result.□
5 Efficient implementation of the algorithm
We will describe in this section the implementation process of the algorithm in detail. Let us begin with constructing a set of basis functions in the approximation space. Denote by
We can check that
where
Setting
We shall look for
Plugging equations (5.6) and (5.7) into equation (3.2) and taking
where
Solving equations (5.8) and (5.9), one can obtain the expansion coefficients
Remark 2
We mainly use Gaussian numerical integration to calculate the elements of the discretized matrix. In addition, we know from the orthogonality of Legendre polynomial that the matrices A, B, C, D, E, G, H, K, P, and W are symmetric ribbon matrix for the basis functions
6 Numerical experiments
We shall carry out some numerical experiments to illustrate the validity and high accuracy of our algorithm. We operate our programs in MATLAB 2019b.
Let
we have
Define an error between
Example 1. We take
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20 | 0.0059 |
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0.0015 |
25 | 0.0060 |
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30 | 0.0060 |
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35 | 0.0060 |
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40 | 0.0060 |
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It can be seen from Table 1 that the
Example 2. We take
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20 | 0.0078 |
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0.0019 |
25 | 0.0078 |
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30 | 0.0078 |
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35 | 0.0078 |
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40 | 0.0078 |
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It can be seen from Table 2 that the
Example 3. We take
Similarly, the error
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20 |
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25 |
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30 |
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35 |
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40 |
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It can be seen from Table 3 that the
7 Conclusions
An efficient Legendre-Galerkin spectral approximation method is developed to solve the fourth-order equation with inverse square singular potential and SSP boundary conditions in a circular domain. First, we establish a weak formulation and its discrete scheme based on a reduced-dimension scheme; furthermore, we prove the existence and uniqueness of the weak solution and approximation solutions and the error estimation between them. In particular, the essential pole conditions we introduce overcome the difficulty of pole singularity. In addition, our approximation scheme is a high-order numerical method with spectral accuracy, which only need a few degrees of freedom to achieve high-precision numerical solutions. Finally, the method developed in this study can be extended to some complicated fourth-order singular nonlinear Schrödinger equations in a unbounded domain, which will be our goal in the future.
Acknowledgement
We are very grateful to Xiaohu Yang and Hui Zhang for their contributions to the revision of this manuscript.
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Funding information: The work of Shuimu Zou was supported by the Research Foundation for Scientific Scholars of Moutai Institute (No. [2022]140). The work of Jun Zhang is supported by the National Natural Science Foundation of China (Nos. 12261017, 12261018), the Academic Project of Guizhou University of Finance and Economics (No. 2022ZCZX077) and the Universities Key Laboratory of Mathematical Modeling and Data Mining in Guizhou Province (No. 2023013).
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Author contributions: Shuimu Zou did the analyses and wrote this article. Jun Zhang developed the idea for the study. The authors approved the final manuscript.
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Conflict of interest: The authors state no conflicts of interest.
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Data availability statement: The data are available from the corresponding author on reasonable request.
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