Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

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Introduction
In the past decades, various numerical methods are designed for solving kinds of differential equations, such as finite element method [1][2][3], spectral method [4][5][6], shifted Legendre approximation [7,8], and differential transformation method [9,10].To avoid the constraints and workload of region divisions, a new family of computational methods has emerged.The so-called meshless or mesh-free methods have been investigated and used by many researchers.The advantage of meshless methods reads that the interpolation accuracy is not significantly affected by the nodal distribution.And hence meshless methods attract great attentions in various disciplines for treating a large variety of engineering problems.In fact, the MPS is originally proposed with the radial basis functions for solving various kinds of differential equations.Recently, the MPS has been continuously employed to solve various interesting models and proven to be an effective method in numerical simulations.For more details about this numerical scheme, please refer to [11][12][13] and the references cited therein.
To the best of our current knowledge, the meshless schemes, including Kansa method [14], method of fundamental solutions [15], method of particular solutions [16,17], element-free Galerkin method [18], local point interpolation [19], and boundary knot method [20], are widely used to approximate a large class of partial differential equations in science and engineering fields.As reported in the literatures, the MPS has been applied to solve the Navier-Stokes problem [21], wave propagation problem [22], and time-fractional diffusion problem [23].Despite the effectiveness of the MPS, there are some disadvantages such as the ill-conditioned collocation matrix, the uncertainty of the shape parameters, and difficulties in deriving the closed-form particular solutions for general differential operators, and for more details, please refer to [12,17,[24][25][26] and the references cited therein.
In order to overcome these disadvantages, lots of works have been done on efficient numerical schemes for the MPS.And many basis functions have been designed to discretize partial differential equations.Chebyshev polynomials [11,27], polynomials basis functions [16,18,28], and trigonometric functions [29] were employed with their closedform particular solutions to approximate kinds of models.However, few results about error estimates of the MPS are illustrated in the current literatures.
In this paper, Legendre polynomials are used to design the particular solutions for the MPS.Specially, boundary conditions are naturally imposed, and the corresponding discretized scheme is constructed in a collocation scheme.
The closed-form particular solutions for given differential operators with variable coefficients are derived via recursive relationships of Legendre polynomials.Compared with the radial basis functions for the MPS, our proposed scheme provides a simple approach to effectively solve a kind of differential equations with variable coefficients.
Meanwhile, with an orthogonal projector and the Aubin-Nitsche duality argument, we provide rigorous studies on two a-priori error estimates for this numerical method.For sufficiently smooth solutions, the a-priori error estimations show that asymptotic super-exponential convergence orders of the MPS approximations are readily achieved in H 1 -and L ∞ -norms.
The remainder of this paper is organized as follows.Some preliminaries and a brief review of the MPS are presented in Section 2. The numerical procedures of the MPS for solving differential equations with variable coefficients are proposed in Section 3. In Section 4, two a-priori error estimates are given in different norms with rigorous proofs.And three numerical examples are provided with numerical errors and convergence orders to demonstrate the effectiveness of the proposed methods in Section 5. Furthermore, some conclusions and discussions are listed in Section 6.And in the last part, an appendix is given to sketch a rigorous proof for the recalled lemma.

Preliminaries
Let us introduce some basic notations which will be used in the sequel.Hereafter, we select a unit interval I = ð−1, 1Þ to show the sketch of the MPS approximations and a-priori error estimates and adopt the standard notation W m,q ðIÞ for Sobolev space on I. Setting W m,q 0 ðIÞ = fv ∈ W m,q ðIÞ: ðd k v/dx k Þð±1Þ = 0, 0 ≤ k ≤ m − 1g, we denote H m 0 ðIÞ = W m,2 and L 2 ðIÞ, respectively.We use C and c to denote different constants in different formulae.For simplicity, we omit subscripts if m = 0. Particularly, if m = 1, we set Thereby, the scalar product in L 2 ðIÞ and bilinear form in H 1 ðIÞ are defined as We define the following polynomial sets: 2.1.Legendre Polynomials.We denote by L i ðxÞ the i-th degree Legendre polynomial with x ∈ I. Three-term recurrence relationship for Legendre polynomials reads and L 0 ðxÞ = 1, L 1 ðxÞ = x.We recall that fL i ðxÞg i≥1 satisfy and hence there holds Also, there is an orthogonality And for i ≤ N, it is obvious that L ′ i ðxÞ ∈ PN−1 and 2.2.The Method of Particular Solutions.In this subsection, we consider the second-order differential equation with homogeneous Dirichlet boundary condition: and the constraint on ωðxÞ will be stated in the sequel.By (3), we obtain the equivalent weak formulation of (10) reads: In view of (9), we design the corresponding particular solutions for (10) as which guarantee ψ i ð±1Þ = 0.
And then we define P N as where ψ i ðxÞ satisfies the homogeneous Dirichlet boundary conditions in (10).For more details about the completeness of P N in ( 13), please refer to [30].

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According to (13), the MPS approximation of uðxÞ can be stated as where fc j g N−1 i=1 are the coefficients to be determined.For the sake of convenience, we use fx k g M k=1 to denote the collocations in the interval.And then the corresponding equivalent collocation scheme for (10) reads: where v k = δðx − x k Þ denotes the Dirac delta distribution on x k .For readers interested in the collocation approximations, please refer to [31].

The Model Problem and Its Approximation Scheme
3.1.The Model Problem with ωðxÞ = 1 − x 2 .In the following parts, we focus on ωðxÞ = 1 − x 2 .Since there does not exist any positive constant c satisfying ωðxÞ ≥ c in I, we miss the sufficient conditions for the uniqueness of (10).And hence we have to restate the uniqueness of the solution for (11) with some novel techniques.
Proof.For any v, w ∈ H 1 ðIÞ, there holds where we used ωðxÞ j j≤ 1.One directly states the continuation of the bilinear form and also the existence of solutions.
Now we are at the point to investigate the uniqueness of the solution for (10).Obviously, the bilinear form is not elliptic.We have to prove the uniqueness with new techniques.Assuming there exist two solutions u 1 and u 2 satisfying (10), one readily gets that for all x ∈ I, there almost holds which means where UðxÞ = u 1 ðxÞ − u 2 ðxÞ.Now, we turn to prove that the solution of boundary value problem ( 18) is zero.And hence we employ integrations by parts to get the unique solution Considering the boundary conditions and properties of function ln ðxÞ at x = ±1, we easily declare that c 1 = 0 and c 2 = 0, which means UðxÞ = 0, a:e:x ∈ I.Then, we readily depict that u 1 ðxÞ = u 2 ðxÞ, a:e:x ∈ I, which directly verifies the uniqueness of solution of (11).

3.2.
The MPS with Legendre Polynomials.Noticing that, one of the challenges of the MPS is how to derive closedform particular solutions for given differential operators.Although the particular solutions are not unique, it is always a complicated task to find appropriate particular solutions for given differential operators.In general, finding or designing closed-form particular solutions are nontrivial (for more details on this topic, please refer to [32] and the references therein).
It is well-known that the size of globally dense matrices in the MPS grows with the increase of collocation points and will cause bigger condition numbers of resultant matrices.Hence, the crucial task of the MPS is to choose pertinent PN such that the basis functions are as simple as possible.According to the recursive relationships of Legendre polynomials, we derive efficient basis functions for corresponding particular solutions bit by bit.
With (9), it is direct to state that And then we have Hence, the basis functions for the approximations of the right hand term can be set as which satisfy the following identity One readily gets that the discretized formulation of (11) reads: The details about the equivalent weak formulation can be found in [31].Meanwhile, the existence and uniqueness of the numerical solution in P N of ( 24) can be readily proved by the same techniques given in Theorem 1.

The A-Priori Error Estimates
In this section, we study a-priori error estimates of the MPS approximations by an orthogonal projector.For any w ∈ H 1 0 ðIÞ, there holds In view of orthogonal properties of Legendre polynomials and (20), we get the following identities: We recall the first derivative orthogonal projector Here, an error estimate for this first derivative orthogonal projector is shown in the following lemma.
Lemma 3 (See [34,35]).For bounded interval I and F ∈ H −1 ðIÞ, we set y F as the unique solution of the following homogeneous boundary value problem where h•, • i stands for the dual product on H −1 ðIÞ × H 1 0 ðIÞ.Then, y F ∈ H 1 ðIÞ, and there holds By the above results, we derive the following a-priori error estimate.
Theorem 4. Let u and u N be the solutions of ( 11) and (15), respectively.Then for all u ∈ H 1 0 ðIÞ ∩ H m ðIÞ, it holds that Proof.It follows that Then, the a-priori error estimation in (31) is yielded.

The A-Priori
Error Estimate in L ∞ -Norm.In this subsection, we give the corresponding error estimate in L ∞ -norm with a rigorous relationship during L ∞ ðIÞ and H 1 ðIÞ.
Lemma 5.For all v ∈ H 1 ðIÞ, there holds the following estimate Proof.Since the interval is bounded, one gets that W m,2 ðIÞ ⊂ W m,1 ðIÞ.By the embedding theorems (refer to Chapter 12 in [36]), we know that W 1,1 ðIÞ is embedded in L ∞ ðIÞ.Furthermore, H 1 ðIÞ is a subset of W 1,1 ðIÞ due to the bounded interval I. Hence, H 1 ðIÞ is embedded in L ∞ ðIÞ.About the constants within the above estimate, please refer to the Theorem 1.9 in [37] for further details.And a theoretical proof is listed in the appendix, which improves the proof given in [38].
Theorem 6.Let u and u N be the solutions of ( 11) and ( 15), respectively.Then, for all u ∈ H 1 0 ðIÞ ∩ H m ðIÞ, there holds Journal of Function Spaces where C I denotes the constant within the Poincaré inequality.One readily gets that the desired result listed in (34) holds.
The above two a-priori error estimations, which are given in H 1 -and L ∞ -norms, show that an asymptotic super-exponential convergence order for the MPS approximations can be achieved for any sufficiently smooth solution.

Numerical Results
In the following different kinds of numerical examples, we show the approximation data in tables and figures, which illustrate the efficiency of the MPS for (10).For simplicity, we evenly select distributed nodes as the collocation points.
Example 7. Setting the boundary value problem (10) with we get the analytic solution Obviously, this analytic solution is sufficient smooth on I.The numerical data listed in Table 1 show error esti-mates of numerical approximations and the first derivatives of numerical solutions versus N, respectively.And two a-priori error estimations with L ∞ -and L 2 -norms verify our theoretical analyses.Hence, by the numerical data in the first five columns, we obtain the high accuracy property of the MPS approximations.

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The last column depicts the convergent orders, which will validate the high efficiency of the MPS.Since the errors arrive at the machine accuracy, the convergence order, 1.0401 in the last column, has no essential significance.Here, the convergence order is calculated by where the subscripts denote corresponding i-th and ði + 1Þth information.It is obvious that for any sufficiently smooth analytic solution, the convergence orders of the MPS can be sharply enhanced by the increased N.
For the given right-hand side function f in (36), the analytic solutions and the MPS approximations of N = 2 and N = 4 are pointwise delineated in Figure 1.
And numerical results of kðu − u N Þ ′ k are shown by the semi-logarithmic scale in Figure 2. By the Poincaré inequality, we know that the approximation errors in H 1 -norm are naturally consistent with our proposed a-priori error estimates.These figures show the efficiency of the MPS approximations for this example.
Following the above numerical data shown in Table 1 and Figures 1 and 2, it is clear that the numerical errors decrease exponentially with increased N.And hence, the convergence and high accuracy of our proposed numerical scheme are demonstrated.
Example 8. We consider the boundary value problem (10) with and the corresponding analytic solution reads By our proposed MPS schemes, corresponding numerical errors are listed in Table 2. Also convergence orders are given, which depict the finite algebraic convergence properties.Since numerical data of N = 32 and N = 64 approach the machine accuracy, which lead to that the last convergence order 0:9125 is unworthy of consideration.And the curves of numerical solution and analytic solution are shown in Figure 3.
Considering the above results and figures, we readily know the sharply approximation properties of the MPS.
Example 9.In this example, we consider the boundary value problem (10) with Since the third derivative of this solution is singular at the boundary points x = ±1, the convergence order is not exponential.By the MPS schemes, numerical errors of our proposed approximations are listed in Table 3.And the convergence orders are shown in the last column, which depict the finite algebraic convergence properties.
Furthermore, considering accumulations of round-off errors and convergence orders, we show that the MPS approximations perform well for this kind of second-order differential equations.
Here, we present Figure 4 to show the errors of numerical solutions against various N by semi-logarithmic scale.The error curve of kðu − u N Þ ′ k is around the reference line, whose slope reads k = −1:5.This indicates that our error estimates uniformly predict the numerical errors of the MPS, which is consistent with the regularity of the given solution.
In the light of kinds of classical solutions with different smoothness, we demonstrate that our a-priori error estimates uniformly predict the errors of the MPS approximations.Furthermore, considering accumulations of round-off errors, the current section verifies our theoretical results for the model problems with the proposed MPS approximations.

Conclusions
The highlight of this work is that we skillfully employed Legendre polynomials to solve second-order differential equations by the MPS.To investigate the efficiency and accuracy of proposed numerical schemes, we study the errors of corresponding numerical approximations.By orthogonal projector and Aubin-Nitsche duality argument, we obtain the a-priori error estimate in H 1 -norm with rigorous analyses.Meanwhile, with the help of relationships between L ∞ -and H 1 -norms on any bounded interval, we readily get corresponding a-priori error estimate in L ∞ -norm.In the numerical examples, three analytic solutions with different regularity are selected: One is with finite smoothness and others are with infinite regularity.Furthermore, convergence orders and numerical errors are listed to confirm our theoretical results, which also validate the efficiency and high accuracy of the MPS.
The success of dealing with this typical model problem by the MPS will pave the way for solving other more challenging models in science and engineering applications.In our ongoing researches, corresponding further discussions have been listed for the MPS in high dimensional domains, such as how to design the basis functions and corresponding particular solutions based on orthogonal polynomials and how to select collocation points for singular domains.Fortunately, the tensor product of orthogonal polynomials will help us to reformulate the particular solutions and corresponding discretizations.We believe that this method will be applicable for a large amount of partial differential equations and is an efficient numerical scheme in applications.

Appendix A. The Proof of Lemma 4.3
This appendix follows the proof of both Theorem 7.10 in [39] and (3.9) in [38] and gives a rigorous proof for Lemma 5 on any bounded interval ða, bÞ.
Firstly, we proceed from ∀v ∈ C ∞ ½a, b.By the first mean value theorem of integrals, we know that there exists a σ ∈ ða, bÞ satisfying   Meanwhile, by the Newton-Leibniz integration formula, we have ðA:3Þ Hence, in view of C ∞ ½a, b is dense in H 1 ða, bÞ, then for any v ∈ H 1 ða, bÞ, there exists Now it is obvious that ½ ,∀k, l: ðA:5Þ By (A.3), one arrives at ðA:6Þ Then for ∀x ∈ ½a, b and k, l ⟶ ∞, there holds Secondly, we identify the relationship between v and ṽ.By Minkowski's inequality and Lebesgue integration, we have , ðA:10Þ then for k ⟶ ∞, there holds Finally, by (A.3), we know that for all v k , there holds ðA:14Þ Then, ðA:15Þ With the help of (A.13), we directly get

(a) N = 2 - 1 = 4 Figure 1 :
Figure 1: Pointwise curve of u and u N at two different N.

Figure 2 :
Figure 2: Errors of kðu − u N Þ′k versus N in the semi-logarithmic scale.

Figure 3 :
Figure 3: u and u N at two different N.

Figure 4 :
Figure 4: Errors of kðu − u N Þ′k versus N in semi-logarithmic scale and reference convergence line.
:7Þ which means that fv k ðxÞg is a Cauchy sequence in C½a, b.Meanwhile, in the light of the completeness of C½a, b, we know that there exists ṽ ∈ C½a, b such thatv k ⟶ C a,b ½ ṽ, k ⟶ ∞,ðA:8Þ i.e., max x∈ a,b ½ ṽ x ð Þ, a:e:x ∈ a, b ½ : ðA:13Þ , a:e:x ∈ a, b v k k 2 + 2 b − a j j• v′ 2 : ðA:17Þ This is the desired result in Lemma 5. 8 Journal of Function Spaces

Table 1 :
Errors of u − u N and orders of convergence for Example 7.

Table 2 :
Errors of u − u N and orders of convergence for Example 8.

Table 3 :
Errors of u − u N and orders of convergence for Example 9.