Multiscale method for solving high-order BVPs

: This paper presents a numerical algorithm for solving high-order BVPs. We in-troduce the construction method of multiscale orthonormal basis in W m 2 [0 , 1] by multiscale orthonormal basis in W 12 [0 , 1]. We deﬁne ε − approximate solution, and obtain the ε − approximate solution of high-order BVPs by using the approximate theory. Moreover, the convergence and stability of the algorithm are improved. At last, several numerical experiments show the feasibility of the proposed method.

Higher-order BVPs are important mathematical models in the field of electromagnetics, fluid mechanics and material science. For example, the common Cahn-Hilliard equations and Molecular Beam Expital equations are higher-order models [1]. It is difficult to find the analytic solutions of higher-order BVPs because of the complexity of the systems, many numerical algorithms for high-order BVPs have been proposed in recent years. The multistage integration method is an important method to solve the numerical solution of higher-order models by reducing the order gradually [2][3][4][5]. Cao [6] solved a class of higher-order fractional ordinary differential equations by the quadratic interpolation function method. The collocation method proposed by Toutounian [7] and orthonormal Bernstein polynomials method proposed by Mirzaee [8] can solve higher-order linear complex differential equations effectively. Raslan et al. [9][10][11] proposed a variety of numerical algorithms for solving higher-order integro differential equations. Many scholars have also proposed many methods in the field of numerical solution of higher-order partial differential equations, such as Galerkin finite element method [12] and Diethelms method [13].
In this paper, we construct a set of multiscale orthonormal basis in the reproducing kernel space W m 2 . Multiscale is an effective theory for numerical analysis, which is based on the idea of wavelet. As a numerical algorithm for solving boundary value problems, multiscale theory has attracted more and more attention. Zhang [14] constructed an algorithm based on multiscale orthonormal basis for solving second-order BVPs. Multiscale theory combined with wavelet method or reproducing kernel method can also be used to solve BVPs [15,16]. Pezza [17] and Aminikhah [18] proposed a multiscale numerical algorithm for fractional BVPs. In recent years, multiscale finite element method has also been applied to solve the numerical solutions of partial differential equations [19,20]. Reproducing kernel space is an important banach space, which has been used in the field of numerical analysis. Wu and Lin [21] discussed the theory of reproducing kernel in detail, and designed a polynomial reproducing kernel method for solving various operator equations. Niu [22] proposed a new reproducing kernel algorithm for solving nonlinear singular BVPs. Mei and Shen [23][24][25] have proposed numerical algorithms for solving impulsive differential equations. The reproducing kernel methods are used in the numerical solutions of high-order models, singular BVPs and interface problems [26][27][28][29][30].

Multiscale orthonormal basis
In this section, the reproducing kernel space is defined and a set of multiscale orthonormal basis is constructed. These knowledge is very useful in the following article.
In order to express the algorithm for solving Eq. (1.1) , this paper constructs a set of orthonormal basis in W m Proof We just prove the orthonormality and completeness. First, orthonormality. obviously, Orthonormality is true.
If the basis in W m 2 [0, 1] is constructed from the basis in W m 2,0 [0, 1], we need to look for m + 1 functions g k (x) ∈ W m 2 , k = 0, 1, 2, · · · , m, such that . By definition of the inner product, g k (x) satisfies Eq. (2.2)-Eq. (2.4), and from Eq. (2.3), we can obtain a = 1 k! . Based on the above analysis, we get the the orthonormal basis in W m 2 .
Proof According to Th. 2.1 and Eq. (2.2)-Eq. (2.4), it's clear that Next, we just need to prove completeness. That is , if u, ρ j W m 2 = 0, then u ≡ 0. In fact,
where M is a positive constant. Then Eq. (1.1) is equivalent to the following equation Zhang [14] proposed the ε−approximate theory of second-order differential equations, now we define the ε−approximate solution of Eq. (3.4) based on the idea.
Put J is quadratic form about c = (c 1 , · · · , c n ), c * k is the minimum point of J. In fact, in order to find the minimum value of J, that is, Then Eq. (3.7) changes to According to [14], the unique solution of Eq. (3.8) is the minimum point of J.

Convergence and stability analysis
In this section, the properties of the algorithm are introduced, such as uniform convergence and stability.

Convergence analysis
Theorem 4.1 Assume u is the exact solution of Eq. (1.1), u n is the ε−approximation of (1.1).
That is We can obtain (c i,k ) 2 ≤ 1 2 3i C 1 . In fact, According to else.
where C is a constant.
where C is a constant, K(x, y) is the reproducing kernel of W m 2 . From theorem 4.1, u n uniformly convergence to u, which is the following theorem

Stability analysis
It is well known that if A is a reversible symmetrical matrix, then the condition number of A is where λ 1 and λ n are the maximum and minimum eigenvalues of A respectively. Obviously, A of Eq. (3.8) is invertible symmetric matrix. Therefore, in order to prove the stability of the algorithm, we can first prove the boundedness of the eigenvalues. lemma 4.1 Suppose λx = Ax, x = 1, where x = (x 1 , · · · , x n ) T is related eigenvector of λ, then Proof According to λx = Ax, Multiply both sides of (4.2) by x i (i = 1, 2, · · · , n) and add up, and it get Without loss of generality, put u W m 2,0 = 1. According to inverse operator theorem [21], To sum up From lemma 4.1, we get That is, the presented method is stable.

Numerical experiments
In this section, we give several numerical experiments to verify the effectiveness of the proposed algorithm. We denote by u j the approximation to the exact solution u(x j ) obtained by the numerical schemes in the present work, and we measure the errors in the following sense: where n is the number of bases. C.R. represents the convergence order.

Conclusion
In summary, this study used a set of multi-scale orthonormal basis to find the ε−approximate solutions of higher-order BVPs. This paper not only demonstrates the convergence and stability in theory, but also demonstrates the feasibility of the method through numerical experiments. Through theoretical analysis and numerical experiments, this method can be extended to solve general linear models, such as linear integral equations, differential equations, fractional differential equations.