An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition

Shuimu Zou; Jun Zhang

doi:10.1515/math-2023-0128

Article Open Access

An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition

Shuimu Zou and Jun Zhang

Published/Copyright: December 22, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

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.

Keywords: fourth-order equation; singular potential; SSP boundary condition; Legendre-Galerkin approximation; error estimation

MSC 2010: 15A18; 42C10; 65G50

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:

(2.1) Δ 2 σ − α Δ σ + β ∣ x ∣ 2 σ = g , in Ω ,

(2.2) σ = Δ σ = 0 , on ∂ Ω ,

where Δ and Δ 2 denote the Laplacian operator and biharmonic operator, respectively, σ is the solution satisfying equations (2.1)–(2.2), g is the source term, both α and β are non-negative constants, and Ω = { x ∈ R 2 : ∣ x ∣ < R } with ∣ x ∣ = x 1 2 + x 2 2 . Using the polar coordinate transformation x 1 = r cos θ and x 2 = r sin θ , the Laplace operators in polar coordinates can be derived:

(2.3) ℒ σ = 1 r ∂ ∂ r r ∂ σ ∂ r + 1 r 2 ∂ 2 σ ∂ θ 2 .

Let ϱ ( r , θ ) = σ ( r cos θ , r sin θ ) and f ( r , θ ) = g ( r cos θ , r sin θ ) . Then, equations (2.1) and (2.2) can be rewritten as:

(2.4) ℒ 2 ϱ − α ℒ ϱ + β r 2 ϱ = f , ( r , θ ) ∈ D = ( 0 , R ) × [ 0 , 2 π ) ,

(2.5) ϱ ( R , θ ) = ℒ ϱ ( R , θ ) = 0 , θ ∈ [ 0 , 2 π ) ,

where ϱ is periodic in θ direction. Note that when β = 0 , equation (2.1) degenerates into a general fourth-order problem. Therefore, we might as well assume that β > 0 . According to the periodicity of ϱ in θ direction, we have the following Fourier basis function expansion:

(2.6) ϱ ( r , θ ) = ∑ ∣ m ∣ = 0 ∞ ϱ m ( r ) e i m θ , f ( r , θ ) = ∑ ∣ m ∣ = 0 ∞ f ˆ m ( r ) e i m θ .

Plugging (2.6) into (2.3), we obtain that

ℒ ϱ = ∑ ∣ m ∣ = 0 ∞ 1 r d d r r d ϱ m d r − m 2 r ϱ m e i m θ .

Thus, for the well posedness of equation (2.4), the following essential polar conditions need to be imposed, i.e.,

(2.7) ϱ m ( 0 ) = 0 , d d r r d ϱ m d r − m 2 r ϱ m ∣ r = 0 = 0 .

By discussing m , (2.7) can be further simplified as follows:

(2.8) ( 1 ) ϱ m ( 0 ) = 0 , ( ∣ m ∣ = 1 ) ;

(2.9) ( 2 ) ϱ m ( 0 ) = 0 , ϱ m ′ ( 0 ) = 0 , ( ∣ m ∣ ≠ 1 ) .

Let r = ζ + 1 2 R , σ m ( ζ ) = ϱ m ( ζ + 1 2 R ) , f m ( ζ ) = f ˆ m ( ζ + 1 2 R ) , and ℒ m σ m = 1 ζ + 1 ∂ ζ [ ( ζ + 1 ) ∂ ζ σ m ] − m 2 ( ζ + 1 ) 2 σ m . Substituting (2.6) into (2.4)–(2.5), using the orthogonality of Fourier basis functions and (2.8)–(2.9), we derive that

(2.10) ℒ m 2 σ m − α R 2 4 ℒ m σ m + β R 2 4 σ m ( ζ + 1 ) 2 = R 4 16 f m , ζ ∈ ( − 1 , 1 ) ,

(2.11) ( 1 ) σ m ( ± 1 ) = 0 , ℒ m σ m ( 1 ) = 0 , ( ∣ m ∣ = 1 ) ;

(2.12) ( 2 ) σ m ( ± 1 ) = 0 , σ m ′ ( − 1 ) = 0 , ℒ m σ m ( 1 ) = 0 , ( ∣ m ∣ ≠ 1 ) .

Remark 1

Note that for different Fourier modules m , systems (2.10)–(2.12) are decoupled and can be solved in parallel. In addition, by making R → ∞ , we can similarly deduce the polar conditions and dimensionality reduction scheme for the fourth-order equations in a unbounded domain.

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 L ω 2 ( I ) be a usual weighted Sobolev space, i.e.,

L ω 2 ( I ) ≔ σ m : ∫ I ω σ m 2 d ζ < ∞ ,

with the inner product and norm as follows:

( σ m , ς m ) ω = ∫ I ω σ m ς m d ζ and ‖ σ m ‖ w = ∫ I ω σ m 2 d ζ 1 2 ,

where ω = 1 + ζ and I = ( − 1 , 1 ) . Define a class of non-uniformly weighted Sobolev spaces H m 2 ( I ) :

H m 2 ( I ) ≔ { σ m : ℒ m σ m ∈ L ω 2 ( I ) , σ m ( ± 1 ) = ( 1 − m 2 ) σ m ′ ( − 1 ) = 0 } ,

equipped with the following inner product and norm:

( σ m , ς m ) 2 , ω , m = ( ℒ m σ m , ℒ m ς m ) ω and ‖ σ m ‖ 2 , ω , m = [ ( σ m , σ m ) 2 , ω , m ] 1 2 .

Thus, a weak formulation of equations (2.10)–(2.12) is: find σ m ∈ H m 2 ( I ) , such that

(3.1) A m ( σ m , ς m ) = ℱ m ( ς m ) , ∀ ς m ∈ H m 2 ( I ) ,

where

A m ( σ m , ς m ) = ∫ I ( ζ + 1 ) ℒ m σ m ℒ m ς m d ζ + α R 2 4 ∫ I ( ζ + 1 ) σ m ′ ς m ′ + m 2 1 1 + ζ σ m ς m d ζ + β R 2 4 ∫ I 1 1 + ζ σ m ς m d ζ , ℱ m ( ς m ) = R 4 16 ∫ I ( ζ + 1 ) f m ς m d ζ .

Denote by P N an N -degree polynomial space. Let X N ( m ) = P N ∩ H m 2 ( I ) , then X N ( m ) is an approximation space of H m 2 ( I ) . Thus, a discrete scheme associated with (3.1) is: find σ m N ∈ X N ( m ) , such that

(3.2) A m ( σ m N , ς m N ) = ℱ m ( ς m N ) , ∀ ς m N ∈ X N ( m ) .

3.2 Existence and uniqueness of the solution

We first introduce a symbol that will be used frequently later. Namely, a 1 ≲ a 2 means that a 1 ≤ C a 2 with C being a positive constant.

Lemma 1

For any σ m , ς m ∈ H m 2 ( I ) , there hold:

(3.3) ∫ I ( 1 + ζ ) ℒ m σ m ℒ m ς m d ζ = ∫ I ( 1 + ζ ) σ m ″ ς m ″ d ζ + ( 2 m 2 + 1 ) ∫ I 1 1 + ζ σ m ′ ς m ′ d ζ + ( m 4 − 4 m 2 ) ∫ I 1 ( 1 + ζ ) 3 σ m ς m d ζ + σ m ′ ( 1 ) ς m ′ , ( 1 )

with ∣ m ∣ ≠ 1 , and

(3.4) ∫ I ( 1 + ζ ) ℒ m σ m ℒ m ς m d ζ = ( 1 + ∣ m ∣ ) σ m ′ ( 1 ) ς m ′ ( 1 ) + ∫ I ( 1 + ζ ) σ m ′ − ∣ m ∣ 1 ζ + 1 σ m ′ ς m ′ − ∣ m ∣ 1 ζ + 1 σ m ′ d ζ + ∫ I ( 1 + ∣ m ∣ ) 2 1 + ζ σ m ′ − ∣ m ∣ 1 1 + ζ σ m v m ′ − ∣ m ∣ 1 1 + ζ ς m d ζ ,

with ∣ m ∣ = 1 .

Proof

When ∣ m ∣ ≠ 1 , we derive from (2.12) that

(3.5) ∫ I σ m ″ ς m ′ + σ m ′ ς m ″ d ζ = σ m ′ ( 1 ) ς m ′ ( 1 ) ,

(3.6) ∫ I 1 ( 1 + ζ ) 2 ( σ m ′ ς m + σ m ς m ′ ) d ζ = 2 ∫ I 1 ( 1 + ζ ) 3 σ m ς m d ζ ,

(3.7) ∫ I 1 1 + ζ ( σ m ″ ς m + σ m ς m ″ ) d v = 2 ∫ I 1 ( 1 + ζ ) 3 σ m ς m d ζ − 2 ∫ I 1 ( 1 + ζ ) σ m ′ ς m ′ d ζ .

We derive from (5.8)–(3.7) that

∫ I ( 1 + ζ ) ℒ m σ m ℒ m ς m d ζ = ∫ I ( 1 + ζ ) σ m ″ + σ m ′ − m 2 1 + ζ σ m ς m ″ + ς m ′ ζ + 1 − m 2 ( 1 + ζ ) 2 ς m d ζ = ∫ I ( 1 + ζ ) σ m ″ ς m ″ d ζ + ∫ I σ m ″ ς m ′ + σ m ′ ς m ″ d ζ − ∫ I m 2 ( ζ + 1 ) 2 ( σ m ′ ς m + σ m ς m ′ ) d ζ − ∫ I m 2 ζ + 1 ( σ m ″ ς m + σ m ς m ″ ) d ζ + ∫ I m 4 ( 1 + ζ ) 3 σ m ς m d ζ + ∫ I 1 1 + ζ σ m ′ ς m ′ d ζ = ∫ I ( 1 + ζ ) σ m ″ ς m ″ d t + ( 2 m 2 + 1 ) ∫ I 1 1 + ζ σ m ′ ς m ′ d ζ + σ m ′ ( 1 ) ς m ′ ( 1 ) + ( m 4 − 4 m 2 ) ∫ I 1 ( 1 + ζ ) 3 σ m ς m d ζ .

Then, (3.3) follows. When ∣ m ∣ = 1 , in view of pole condition (2.11), we derive that

∫ I ( 1 + ζ ) ℒ m σ m ℒ m ς m d ζ = ∫ I ( 1 + ζ ) σ m ″ + σ m ′ 1 + ζ − m 2 ( 1 + ζ ) 2 σ m ς m ″ + ς m ′ 1 + ζ − m 2 ( 1 + ζ ) 2 ς m d ζ = ∫ I ( 1 + ζ ) σ m ′ − ∣ m ∣ 1 + ζ σ m ′ ς m ′ − ∣ m ∣ 1 + ζ ς m ′ d ζ + ∫ I ( 1 + ∣ m ∣ ) 2 ζ + 1 σ m ′ − ∣ m ∣ 1 + ζ σ m ς m ′ − ∣ m ∣ 1 + ζ ς m d ζ + ∫ I ( 1 + ∣ m ∣ ) σ m ′ − ∣ m ∣ 1 + ζ σ m ς m ′ − ∣ m ∣ 1 + ζ ς m ′ d ζ = ∫ I ( 1 + ζ ) σ m ′ − ∣ m ∣ 1 + ζ σ m ′ ς m ′ − ∣ m ∣ 1 + ζ ς m ′ d ζ + ∫ I ( 1 + ∣ m ∣ ) 2 1 + ζ σ m ′ − ∣ m ∣ 1 1 + ζ σ m ς m ′ − ∣ m ∣ 1 1 + ζ ς m d ζ + ( 1 + ∣ m ∣ ) σ m ′ ( 1 ) ς m ′ ( 1 ) .

Then, (3.4) follows.□

Lemma 2

For any σ m ∈ H m 2 ( I ) , there hold

(3.8) ∫ I ( 1 + ζ ) σ m 2 d ζ ≲ ∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + ζ σ m 2 d ζ ,

(3.9) ∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + ζ σ m 2 d ζ ≲ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ .

Proof

By the boundary condition σ m ( 1 ) = 0 and the Cauchy-Schwarz inequality, we have

∫ − 1 1 ( 1 + ζ ) σ m 2 d ζ = ∫ − 1 1 ∫ ζ 1 1 s + 1 s + 1 σ m ′ ( s ) d s 2 ( 1 + ζ ) d ζ ≤ ∫ − 1 1 ∫ ζ 1 ( 1 + s ) [ σ m ′ ( s ) ] 2 d s [ ln 2 − ln ( 1 + ζ ) ] ( 1 + ζ ) d ζ ≤ ∫ − 1 1 ( 1 + s ) [ σ m ′ ( s ) ] 2 d s ∫ − 1 1 [ ( 1 + ζ ) ln 2 − ( 1 + ζ ) ln ( 1 + ζ ) ] d ζ = ∫ − 1 1 ( 1 + s ) [ σ m ′ ( s ) ] 2 d s ≤ ∫ − 1 1 ( 1 + ζ ) ( σ m ′ ) 2 d t + m 2 1 + ζ σ m 2 d ζ .

Then, (3.8) follows. Next, we prove Inequality (3.9). When ∣ m ∣ = 1 , we derive from (3.4) that

∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + ζ σ m 2 d ζ = ∫ I ( 1 + ζ ) σ m ′ − ∣ m ∣ 1 + ζ σ m 2 d ζ ≲ ∫ I 1 1 + ζ σ m ′ − ∣ m ∣ 1 + ζ σ m 2 d ζ ≲ ∫ I ( 1 + ζ ) σ m ′ − ∣ m ∣ ζ + 1 σ m ′ 2 + ∫ I 1 1 + ζ σ m ′ − ∣ m ∣ 1 + ζ σ m 2 d ζ + ( 1 + ∣ m ∣ ) ( σ m ′ ) 2 ≲ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ .

When ∣ m ∣ ≠ 1 , with the help of Hardy inequality (cf. B8.2 in [29]), we have

(3.10) ∫ I 1 ( 1 + ζ ) 3 σ m 2 d ζ ≲ ∫ I 1 1 + ζ ( ∂ ζ σ m ) 2 d ζ .

Then, we derive from (3.3) and (3.10) that

∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + t σ m 2 d ζ ≤ ∫ I ( 1 + ζ ) ( σ m ″ ) 2 d ζ + 4 ∫ I 1 1 + ζ ( σ m ′ ) 2 d ζ + 4 m 2 ∫ I 1 ( 1 + ζ ) 3 σ m 2 d ζ ≲ ∫ I ( 1 + ζ ) ( σ m ″ ) 2 d ζ + ( 2 m 2 + 1 ) ∫ I 1 1 + ζ ( σ m ′ ) 2 d ζ ≲ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ .

Then, (3.9) follows.□

Lemma 3

A m ( σ m , ς m ) is a continuous and coercive bilinear form on H m 2 ( I ) × H m 2 ( I ) , i.e.,

∣ A m ( σ m , ς m ) ∣ ≲ ‖ σ m ‖ 2 , ω , m ‖ ς m ‖ 2 , ω , m , A m ( σ m , σ m ) ≳ ‖ σ m ‖ 2 , ω , m 2 .

Proof

We derive from Lemmas 1–2 and Cauchy-Schwarz inequality that

∣ A m ( σ m , ς m ) ∣ ≲ ∫ I ( 1 + ζ ) ∣ ℒ m σ m ℒ m ς m ∣ d ζ + ∫ I ( 1 + ζ ) ∣ σ m ′ ς m ′ ∣ + m 2 1 + ζ ∣ σ m ς m ∣ d ζ + ∫ I 1 1 + ζ ∣ σ m ς m ∣ d ζ ≲ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ + ∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + ζ σ m 2 d ζ + ∫ I 1 1 + ζ σ m 2 d ζ 1 2 × ∫ I ( 1 + ζ ) ( ℒ m ς m ) 2 d ζ + ∫ I ( 1 + ζ ) ( ς m ′ ) 2 + m 2 1 + ζ ς m 2 d ζ + ∫ I 1 1 + t ς m 2 d ζ 1 2 ≲ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 1 2 ∫ I ( 1 + ζ ) ( ℒ m ς m ) 2 1 2 = ‖ σ m ‖ 2 , ω , m ‖ ς m ‖ 2 , ω , m .

On the other hand, we can derive that

□ A m ( σ m , σ m ) = ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ + α R 2 4 ∫ I ( 1 + ζ ) ( σ m ′ ) 2 + m 2 1 + ζ σ m 2 d ζ + β R 2 4 ∫ I 1 1 + ζ ( σ m ) 2 d ζ ≥ ∫ I ( 1 + ζ ) ( ℒ m σ m ) 2 d ζ = ‖ σ m ‖ 2 , ω , m 2 .

Theorem 1

If f m ( ζ ) ∈ L ω 2 ( I ) , then equations (3.1) and (3.2) exist a unique weak solution σ m and approximate solution σ m N , respectively.

Proof

Note that if f m ( ζ ) ∈ L ω 2 ( I ) , then ℱ m ( ς m ) is a continuous linear functions on H m 2 ( I ) , i.e.,

∣ ℱ m ( ς ) ∣ ≲ ‖ ς m ‖ 2 , ω , m .

In fact, from the Cauchy-Schwarz inequality and Lemma 2, we derive that

∣ ℱ m ( ς m ) ∣ = R 4 16 ∫ I ( ζ + 1 ) f m ς m d ζ ≤ ∫ I ( ζ + 1 ) ∣ f m ∣ 2 d ζ 1 2 ∫ I ( ζ + 1 ) ∣ ς m ∣ 2 d ζ 1 2 ≲ ∫ I ( ζ + 1 ) ∣ ℒ m ς m ∣ 2 d ζ 1 2 = ‖ ς m ‖ 2 , ω , m .

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 σ m and σ m N be the solutions of equations (3.1) and (3.2), respectively. Then, the following inequality holds

‖ σ m − σ m N ‖ 2 , ω , m ≲ inf ς m N ∈ X N ( m ) ‖ ∂ ζ 2 ( σ m − ς m N ) ‖ .

Proof

It follows from equations (3.1) and (3.2) that

A m ( σ m , ς m N ) = ℱ m ( ς m N ) ∀ ς m N ∈ X N ( m ) , A m ( σ m N , ς m N ) = ℱ m ( ς m N ) ∀ ς m N ∈ X N ( m ) .

Thus, we have

(4.1) A m ( σ m − σ m N , ς m N ) = 0 ∀ ς m N ∈ X N ( m ) .

By Lemma 3 and (4.1), we derive that

‖ σ m − σ m N ‖ 2 , ω , m 2 ≲ A m ( σ m − σ m N , σ m − σ m N ) = A m ( σ m − σ m N , σ m − ς m N + ς m N − σ m N ) = A m ( σ m − σ m N , σ m − ς m N ) + A m ( σ m − σ m N , ς m N − σ m N ) ≲ ‖ σ m − σ m N ‖ 2 , ω , m ‖ σ m − ς m N ‖ 2 , ω , m .

Namely,

‖ σ m − σ m N ‖ 2 , ω , m ≲ ‖ σ m − ς m N ‖ 2 , ω , m , ∀ ς m N ∈ X N ( m ) .

Note that

(4.2) ‖ σ m − ς m N ‖ 2 , ω , m ≲ ‖ ∂ t 2 ( σ m − ς m N ) ‖ , ∀ ς m N ∈ X N ( m ) .

In fact, using Hardy inequality (cf. B8.6 in [29]), we derive that

(4.3) ∫ I 1 ( 1 + ζ ) 3 σ m 2 d ζ ≲ ∫ I 1 1 + ζ ( ∂ ζ σ m ) 2 d ζ ,

(4.4) ∫ I 1 ( 1 + ζ ) 2 σ m 2 d ζ ≲ ∫ I ( ∂ ζ σ m ) 2 d ζ ,

with σ m ( − 1 ) = 0 . When ∣ m ∣ ≠ 1 , by (2.12) and Cauchy-Schwarz inequality, we derive that

‖ σ m − ς m N ‖ 2 , ω , m 2 = ∫ I ( 1 + ζ ) [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ + ∫ I [ ∂ ζ 2 ( σ m − ς m N ) d ζ ] 2 + ( 2 m 2 + 1 ) ∫ I 1 1 + ζ [ ∂ ζ ( σ m − ς m N ) ] 2 d ζ + ( m 4 − 4 m 2 ) ∫ I 1 ( 1 + ζ ) 3 ( σ m − ς m N ) 2 d ζ ≲ ∫ I ( 1 + ζ ) [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ + ∫ I [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ + ∫ I 1 1 + ζ [ ∂ ζ ( σ m − ς m N ) ] 2 d ζ ≲ ∫ I [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ + ∫ I 1 ( 1 + ζ ) 2 [ ∂ ζ ( σ m − ς m N ) ] 2 d ζ ≲ ∫ I [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ .

When ∣ m ∣ = 1 , in view of equations (4.3) and (2.11), we derive that

∫ I 1 ( 1 + ζ ) 3 [ ( σ m − ς m N ) − ( 1 + ζ ) ∂ ζ ( σ m − ς m N ) ] 2 d ζ ≲ ∫ I ( 1 + ζ ) [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ .

On the other hand, we derive that

(4.5) ∣ ∂ ζ ( σ m − ς m N ) ∣ ζ = 1 ∣ 2 = 1 4 ∫ I ∂ ζ ( 1 + ζ ) ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) d ζ 2 ≲ ∫ I ( 1 + ζ ) ∂ ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ + ∫ I 1 1 + ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ .

Then, from Cauchy-Schwarz inequality and (4.5), we derive that

‖ σ m − ς m N ‖ 2 , ω , m 2 = ∫ I ( 1 + ζ ) ∂ ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ + ∫ I 4 1 + ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ + 2 ∣ ∂ ζ ( σ m − ς m N ) ∣ ζ = 1 ∣ 2 ≲ ∫ I ( 1 + ζ ) ∂ ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ + ∫ I 1 1 + ζ ∂ ζ ( σ m − ς m N ) − 1 1 + ζ ( σ m − ς m N ) 2 d ζ ≲ ∫ I ( 1 + ζ ) [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ + ∫ I 1 ( 1 + ζ ) 3 [ ( ζ + 1 ) ∂ ζ ( σ m − ς m N ) − ( σ m − ς m N ) ] 2 d ζ ≲ ∫ I ( 1 + ζ ) [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ ≲ ∫ I [ ∂ ζ 2 ( σ m − ς m N ) ] 2 d ζ .

Then, (4.2) follows. This finishes our proof.□

Let ω α , β ( ζ ) = ( 1 − ζ ) α ( 1 + ζ ) β be a Jacobi weight function. We introduce a class of non-uniformly weighted Sobolev spaces [30]:

H ω α , β , * s ( I ) ≔ { u : ∂ ζ k u ∈ L ω α + k , β + k 2 , 0 ≤ k ≤ s } ,

and the corresponding inner product and norm are as follows:

( u , v ) s , ω α , β , * = ∑ k = 0 s ( ∂ ζ k u , ∂ ζ k v ) ω α + k , β + k , ‖ u ‖ s , ω α , β , * = [ ( u , u ) s , ω α , β , * ] 1 2 .

Define

H ω − 2 , − 2 , m s ( I ) ≔ { σ m ∈ H m 2 ( I ) : ∂ ζ k σ m ∈ L ω − 2 + k , − 2 + k 2 , 2 ≤ k ≤ s }

with the following inner product and norm:

( σ m , ς m ) s , ω − 2 , − 2 , m = ( σ m , ς m ) 2 , ω , m + ∑ k = 2 s ( ∂ ζ k σ m , ∂ ζ k ς m ) ω − 2 + k , − 2 + k , ‖ σ m ‖ s , ω − 2 , − 2 , m = [ ( σ m , σ m ) s , ω − 2 , − 2 , m ] 1 2 .

Let

G N − 2 , − 2 = { q N ∈ P N : q N ( ± 1 ) = q N ′ ( ± 1 ) = 0 } .

Define an orthogonal projection operator: Π N − 2 : L ω − 2 , − 2 2 ( I ) ↦ G N − 2 , − 2 by:

( u − Π N − 2 u , v N ) ω − 2 , − 2 = 0 , ∀ v N ∈ G N − 2 , − 2 .

According to Theorem 1.8.2 in [30], we have the following lemma:

Lemma 5

For any u ∈ H ω − 2 , − 2 , * s ( I ) , there holds:

‖ ∂ ζ 2 ( Π N − 2 u − u ) ‖ ≲ N 2 − s ‖ ∂ ζ s u ‖ ω − 2 + s , − 2 + s .

Theorem 2

There exists an operator Π N 2 , m : H m 2 ( I ) ↦ P N 0 with ∣ m ∣ ≠ 1 such that Π N 2 , m σ m ( ± 1 ) = σ m ( ± 1 ) = 0 , ∂ ζ Π N 2 , m σ m ( − 1 ) = ∂ ζ σ m ( − 1 ) = 0 and ∂ ζ Π N 2 , m σ m ( 1 ) = ∂ ζ σ m ( 1 ) , and for any σ m ∈ H ω − 2 , − 2 , m s ( I ) with s ≥ 2 , it holds

‖ ∂ ζ 2 ( Π N 2 , m σ m − σ m ) ‖ ≲ N 2 − s ( ‖ ∂ ζ s σ m ‖ ω − 2 + s , − 2 + s + ‖ ∂ ζ 2 σ m ‖ ) ,

where P N 0 = { p N ∈ P N : p N ( ± 1 ) = p N ′ ( − 1 ) = 0 } .

Proof

Let σ m * ( ζ ) = − 1 4 ( 1 − ζ 2 ) ( 1 + ζ ) ∂ ζ σ m ( 1 ) , ∀ σ m ∈ H m 2 ( I ) . Then, for any σ m ∈ H ω − 2 , − 2 , m s ( I ) , we have ∂ ζ k ( σ m − σ m * ) ( ± 1 ) = 0 , ( k = 0 , 1 ) , namely, σ m − σ m * ∈ H ω − 2 , − 2 , * s ( I ) . Actually, according to the Hardy inequality (cf. B8.8 in [29]), we derive that

∫ I ω − 2 , − 2 ( σ m − σ m * ) 2 d ζ ≲ ∫ I [ ∂ ζ ( σ m − σ m * ) ] 2 d ζ , ∫ I ω − 2 , − 2 [ ∂ ζ ( σ m − σ m * ) ] 2 d ζ ≲ ∫ I [ ∂ ζ 2 ( σ m − σ m * ) ] 2 d ζ .

Then, it follows that

∫ I ω − 2 , − 2 ( σ m − σ m * ) 2 d ζ ≲ ∫ I ω − 1 , − 1 [ ∂ ζ ( σ m − σ m * ) ] 2 d ζ ≲ ∫ I [ ∂ ζ 2 ( σ m − σ m * ) ] 2 d ζ .

Note that

∫ I [ ∂ ζ 2 σ m * ] 2 d ζ = 2 [ ∂ ζ σ m ( 1 ) ] 2 = 2 ∫ I ∂ ζ 2 σ m d ζ 2 ≲ ∫ I ( ∂ ζ 2 σ m ) 2 d ζ .

Namely,

(4.6) ∫ I [ ∂ ζ 2 ( σ m − σ m * ) ] 2 d ζ ≲ ∫ I ( ∂ ζ 2 σ m ) 2 d ζ + ∫ I ( ∂ ζ 2 σ m * ) 2 d ζ ≲ ∫ I ( ∂ ζ 2 σ m ) 2 d ζ .

Similarly, we can derive that

(4.7) ∫ I ω 1 , 1 [ ∂ ζ 3 ( σ m − σ m * ) ] 2 d ζ ≲ ∫ I ω 1 , 1 ( ∂ ζ 3 σ m ) 2 d ζ + ∫ I ( ∂ ζ 2 σ m ) 2 d ζ .

For k > 3 , we have

(4.8) ∫ I ω − 2 + k , − 2 + k [ ∂ ζ k ( σ m − σ m * ) ] 2 d ζ = ∫ I ω − 2 + k , − 2 + k ( ∂ ζ k σ m ) 2 d ζ ,

which means σ m − σ m * ∈ H ω − 2 , − 2 , * s ( I ) . Define

Π N 2 , m σ m = Π N − 2 ( σ m − σ m * ) + σ m * ∈ P N 0 , ∀ σ m ∈ H ω − 2 , − 2 , m s ( I ) .

By Lemma 5, we have

‖ ∂ ζ 2 ( Π N 2 , m σ m − σ m ) ‖ = ‖ ∂ ζ 2 ( Π N − 2 ( σ m − σ m * ) − ( σ m − σ m * ) ) ‖ ≲ N 2 − s ‖ ∂ ζ s ( σ m − σ m * ) ‖ ω − 2 + s , − 2 + s .

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 Π N 2 , 1 : H m 2 ( I ) ↦ P N 1 with ∣ m ∣ = 1 such that Π N 2 , 1 σ m ( ± 1 ) = σ m ( ± 1 ) = 0 , ∂ ζ Π N 2 , 1 σ m ( ± 1 ) = ∂ ζ σ m ( ± 1 ) , and for any σ m ∈ H ω − 2 , − 2 , m s ( I ) with s ≥ 2 , there holds

‖ ∂ ζ 2 ( Π N 2 , 1 σ m − σ m ) ‖ ≲ N 2 − s ( ‖ ∂ ζ s σ m ‖ ω − 2 + s , − 2 + s + ‖ ∂ ζ 2 σ m ‖ ) ,

where P N 1 = { p N ∈ P N : p N ( ± 1 ) = 0 } .

We now give the main theorem as follows:

Theorem 4

Let σ m and σ m N be the solutions of (3.1) and (3.2), respectively. Then, for any σ m ∈ H ω − 2 , − 2 , m s ( I ) with s ≥ 2 , it holds

‖ σ m − σ m N ‖ 2 , ω , m ≲ N 2 − s ( ‖ ∂ ζ s σ m ‖ ω − 2 + s , − 2 + s + ‖ ∂ ζ 2 σ m ‖ ) .

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 L i ( ζ ) a i -degree Legendre polynomial. Let

(5.1) φ i ( ζ ) = ( 1 − ζ 2 ) [ L i ( ζ ) − L i + 2 ( ζ ) ] , ζ ∈ I , i = 0 , … , N − 4 .

We can check that

(5.2) X N ( m ) = span { φ 0 , … , φ N − 4 } ⊕ span { φ N − 3 } , ( ∣ m ∣ ≠ 1 ) ;

(5.3) X N ( m ) = span { φ ˜ 0 , … , φ ˜ N − 4 } ⊕ span { φ ˜ N − 3 , φ ˜ N − 2 } , ( ∣ m ∣ = 1 ) ,

where

(5.4) φ N − 3 = 1 4 ( ζ − 1 ) ( ζ + 1 ) 2 , φ ˜ N − 3 = ζ 2 − 1 ,

(5.5) φ ˜ N − 2 = ζ ( ζ 2 − 1 ) , φ ˜ j = φ i , ( 0 ≤ j ≤ N − 4 ) .

Setting

a i j = ∫ I ( ζ + 1 ) φ j ″ φ i ″ d ζ , b i j = ∫ I 1 ζ + 1 φ j ′ ϕ i ′ d ζ , c i j = ∫ I ( ζ + 1 ) φ i φ j d ζ , d i j = ∫ I ( ζ + 1 ) φ i ′ φ j ′ d ζ , e i j = ∫ I 1 ( ζ + 1 ) 3 φ j φ i d ζ , g i j = ∫ I 1 ζ + 1 φ j φ i d ζ , h i j = ∫ I ( ζ + 1 ) ℒ 1 φ ˜ j ℒ 1 φ ˜ i d ζ , k i j = ∫ I ( ζ + 1 ) φ ˜ j φ ˜ i d ζ , p i j = φ j ′ ( 1 ) φ i ′ ( 1 ) , w i j = ∫ I ( ζ + 1 ) φ ˜ j ′ φ ˜ i ′ d ζ + ∫ I 1 ζ + 1 φ ˜ j φ ˜ i d ζ , f i m = ∫ I ( ζ + 1 ) f m φ i d ζ , ( ∣ m ∣ ≠ 1 ) f i m = ∫ I ( ζ + 1 ) f m φ ˜ i d ζ , ( ∣ m ∣ = 1 ) .

We shall look for

(5.6) σ m N = ∑ i = 0 N − 3 σ i m φ i , ( ∣ m ∣ ≠ 1 ) ;

(5.7) σ m N = ∑ i = 0 N − 2 σ i m φ ˜ i , ( ∣ m ∣ = 1 ) .

Plugging equations (5.6) and (5.7) into equation (3.2) and taking ς m N through all the basis functions in X N ( m ) , we can derive the following matrix form:

(5.8) H + α R 2 4 W + β R 2 4 K σ m = R 4 16 F m , ( ∣ m ∣ = 1 ) ,

(5.9) A + ( 2 m 2 + 1 ) B + α R 2 4 ( D + m 2 G ) + P + ( m 4 − 4 m 2 ) E + β R 2 4 C σ m = R 4 16 F m , ( ∣ m ∣ ≠ 1 ) ,

where

A = ( a i j ) , B = ( b i j ) , C = ( c i j ) , D m = ( d i j ) , H = ( h i j ) , E = ( e i j ) , P = ( p i j ) , G = ( g i j ) , K = ( k i j ) , W = ( w i j ) , σ m = ( σ 0 m , … , σ N − 2 m ) T , F m = ( f 0 m , … , f N − 2 m ) T , ( ∣ m ∣ = 1 ) , σ m = ( σ 0 m , … , σ N − 3 m ) T , F m = ( f 0 m , … , f N − 3 m ) T , ( ∣ m ∣ ≠ 1 ) .

Solving equations (5.8) and (5.9), one can obtain the expansion coefficients σ m . Then, plugging σ m into equations (5.6) and (5.7) can obtain the approximation solutions.

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 φ i with ∣ m ∣ ≠ 1 and φ ˜ i with ∣ m ∣ = 1 .

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 σ M N ( x ) be an approximation solution of the exact solution σ ( x ) . Using the following variable substitution:

x 1 = ζ + 1 2 R cos θ and x 2 = ζ + 1 2 R sin θ , ( − 1 ≤ ζ ≤ 1 , 0 ≤ θ < 2 π ) ,

we have

σ ( x ) = σ ˆ ( ζ , θ ) = ∑ ∣ m ∣ = 0 ∞ σ m ( ζ ) e i m θ , σ M N ( x ) = σ ˆ M N ( ζ , θ ) = ∑ ∣ m ∣ = 0 M σ m N ( ζ ) e i m θ .

Define an error between σ ( x ) and σ M N ( x ) in the sense of L ∞ norm as follows:

e ( σ ( x ) , σ M N ( x ) ) = ‖ σ ( x ) − σ M N ( x ) ‖ L ∞ ( Ω ) = ‖ σ ˆ ( ζ , θ ) − σ ˆ M N ( ζ , θ ) ‖ L ∞ ( D ) .

Example 1. We take α = 2 , β = 3 , R = 1 , and σ = ( x 2 + y 2 − 1 ) 3 ( x 2 + y 2 ) e 2 x + 3 y . By plugging σ into (2.1), we can obtain g . Based on the algorithm proposed in Section 5, the error e ( σ ( x ) , σ M N ( x ) ) is listed in Table 1 for different M and N . In addition, the images of exact solution and its numerical solution and the error figures between them are also given in Figures 1 and 2, respectively.

Table 1

Error e ( σ ( x ) and σ M N ( x ) ) with different N and M

N	M = 4	M = 8	M = 12	M = 16
20	0.0059	6.1163 × 1 0 ‒ 6	1.5288 × 1 0 ‒ 5	0.0015
25	0.0060	6.3064 × 1 0 ‒ 6	3.0742 × 1 0 ‒ 9	1.3503 × 1 0 ‒ 6
30	0.0060	6.1912 × 1 0 ‒ 6	1.6203 × 1 0 ‒ 9	1.2977 × 1 0 ‒ 10
35	0.0060	6.3041 × 1 0 ‒ 6	1.6539 × 1 0 ‒ 9	1.6381 × 1 0 ‒ 13
40	0.0060	6.2756 × 1 0 ‒ 6	1.6592 × 1 0 ‒ 9	1.6107 × 1 0 ‒ 13

$Figure 1 Images of σ ( x ) \sigma \left({\bf{x}}) (left) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) (right) with N = 25 N=25 and M = 16 M=16 .$

Figure 1

Images of σ ( x ) (left) and σ M N ( x ) (right) with N = 25 and M = 16 .

$Figure 2 Error between σ ( x ) \sigma \left({\bf{x}}) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) with N = 40 N=40 and M = 20 M=20 (left) and N = 45 N=45 and M = 25 M=25 (right).$

Figure 2

Error between σ ( x ) and σ M N ( x ) with N = 40 and M = 20 (left) and N = 45 and M = 25 (right).

It can be seen from Table 1 that the σ M N ( x ) arrive at an accuracy of about 1 0 − 13 with N ≥ 35 and M = 16 . In addition, the images 1–2 also show the convergence and spectral accuracy of our algorithm.

Example 2. We take α = β = 1 , R = 1 , and σ = ( x 2 + y 2 − 1 ) 3 ( x 2 + y 2 ) cos ( 3 x + 5 y ) . Likewise, g can be obtained by plugging σ into equation (2.1). The error e ( σ ( x ) , σ M N ( x ) ) is listed in Table 2 for different M and N . In addition, the images of exact solution and its numerical solution and the error figures between them are also given in Figures 3 and 4, respectively.

Table 2

Error e ( σ ( x ) and σ M N ( x ) ) with different N and M

N	M = 4	M = 8	M = 12	M = 16
20	0.0078	5.3994 × 1 0 ‒ 5	2.7637 × 1 0 ‒ 4	0.0019
25	0.0078	4.9361 × 1 0 ‒ 5	2.3091 × 1 0 ‒ 7	1.6346 × 1 0 ‒ 5
30	0.0078	4.8993 × 1 0 ‒ 5	7.7776 × 1 0 ‒ 8	5.0354 × 1 0 ‒ 8
35	0.0078	4.9522 × 1 0 ‒ 5	7.9075 × 1 0 ‒ 8	4.8346 × 1 0 ‒ 11
40	0.0078	4.9061 × 1 0 ‒ 5	7.9169 × 1 0 ‒ 8	4.6403 × 1 0 ‒ 11

$Figure 3 Images of σ ( x ) \sigma \left({\bf{x}}) (left) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) (right) with N = 35 N=35 and M = 15 M=15 .$

Figure 3

Images of σ ( x ) (left) and σ M N ( x ) (right) with N = 35 and M = 15 .

$Figure 4 Error between σ ( x ) \sigma \left({\bf{x}}) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) with N = 46 N=46 and M = 22 M=22 (left) and N = 52 N=52 and M = 26 M=26 (right).$

Figure 4

Error between σ ( x ) and σ M N ( x ) with N = 46 and M = 22 (left) and N = 52 and M = 26 (right).

It can be seen from Table 2 that the σ M N ( x ) arrive at an accuracy of about 1 0 − 11 with N ≥ 35 and M = 16 . In addition, the images 3–4 also show the convergence and spectral accuracy of our algorithm.

Example 3. We take α = β = 0 , R = 1 , and σ = ( x 2 + y 2 − 1 ) 3 ( x 2 + y 2 ) ( cos 2 x + y ) . Again, g can be obtained by plugging σ into (2.1).

Similarly, the error e ( σ ( x ) , σ M N ( x ) ) is listed in Table 3 for different M and N . In addition, the images of exact solution and its numerical solution and the error figures between them are also given in Figures 5 and 6, respectively.

Table 3

Error e ( σ ( x ) and σ M N ( x ) ) with N and different M

N	M = 4	M = 8	M = 12	M = 16
20	2.1023 × 1 0 ‒ 5	1.6421 × 1 0 ‒ 9	7.6897 × 1 0 ‒ 8	6.5268 × 1 0 ‒ 5
25	2.1571 × 1 0 ‒ 5	1.6932 × 1 0 ‒ 9	9.7465 × 1 0 ‒ 14	5.3320 × 1 0 ‒ 10
30	2.1617 × 1 0 ‒ 5	1.6722 × 1 0 ‒ 9	3.4060 × 1 0 ‒ 14	2.3441 × 1 0 ‒ 14
35	2.1403 × 1 0 ‒ 5	1.6889 × 1 0 ‒ 9	3.4205 × 1 0 ‒ 14	6.1062 × 1 0 ‒ 16
40	2.1627 × 1 0 ‒ 5	1.6898 × 1 0 ‒ 9	3.4615 × 1 0 ‒ 14	2.6368 × 1 0 ‒ 16

$Figure 5 Images of σ ( x ) \sigma \left({\bf{x}}) (left) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) (right) with N = 35 N=35 and M = 15 M=15 .$

Figure 5

Images of σ ( x ) (left) and σ M N ( x ) (right) with N = 35 and M = 15 .

$Figure 6 Error between σ ( x ) \sigma \left({\bf{x}}) and σ M N ( x ) {\sigma }_{MN}\left({\bf{x}}) with N = 39 N=39 and M = 16 M=16 (left) and N = 48 N=48 and M = 25 M=25 (right).$

Figure 6

Error between σ ( x ) and σ M N ( x ) with N = 39 and M = 16 (left) and N = 48 and M = 25 (right).

It can be seen from Table 3 that the σ M N ( x ) arrive at an accuracy of about 1 0 − 15 with N ≥ 35 and M = 16 . In addition, the images 5–6 also show the convergence and spectral accuracy of our algorithm.

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.

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).
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.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: The data are available from the corresponding author on reasonable request.

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Received: 2023-04-20

Revised: 2023-08-13

Accepted: 2023-09-13

Published Online: 2023-12-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0128

Keywords for this article

fourth-order equation; singular potential; SSP boundary condition; Legendre-Galerkin approximation; error estimation

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