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Research article

Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays

  • Received: 08 November 2021 Revised: 13 December 2021 Accepted: 17 December 2021 Published: 28 December 2021
  • MSC : 34K14, 34K20, 92B20

  • We consider the existence and stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. In order to overcome the incompleteness of the space composed of Weyl almost periodic functions, we first obtain the existence of a bounded continuous solution of the system under consideration by using the fixed point theorem, and then prove that the bounded solution is Weyl almost periodic by using a variant of Gronwall inequality. Then we study the global exponential stability of the Weyl almost periodic solution by using the inequality technique. Even when the system we consider degenerates into a real-valued one, our results are new. A numerical example is given to illustrate the feasibility of our results.

    Citation: Yongkun Li, Xiaoli Huang, Xiaohui Wang. Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays[J]. AIMS Mathematics, 2022, 7(4): 4861-4886. doi: 10.3934/math.2022271

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  • We consider the existence and stability of Weyl almost periodic solutions for a class of quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. In order to overcome the incompleteness of the space composed of Weyl almost periodic functions, we first obtain the existence of a bounded continuous solution of the system under consideration by using the fixed point theorem, and then prove that the bounded solution is Weyl almost periodic by using a variant of Gronwall inequality. Then we study the global exponential stability of the Weyl almost periodic solution by using the inequality technique. Even when the system we consider degenerates into a real-valued one, our results are new. A numerical example is given to illustrate the feasibility of our results.



    The boundary value problems (BVPs) for differential equations have important applications in space science and engineering technology. A large number of mathematical models in the fields of engineering, astronomy, mechanics, economics, etc, are often described by differential BVPs [1,2,3]. Except for a few special types, the exact solution of the BVPs is difficult to express in analytical form. It is especially important to find an approximate solution to obtain its numerical solution. In [4], Sinc collocation method provided an exponential convergence rate for two-point BVPs. [5] constructed a simple collocation method by the Haar wavelets for the numerical solution of linear and nonlinear second-order BVPs with periodic boundary conditions. Erge [6] studied the quadratic/linear rational spline collocation method for linear BVPs. In [7], based on B-spline wavelets, the numerical solutions of nonlinear BVPs were derived. Pradip et al. used B-spline to Bratuis problem which is an important nonlinear BVPs in [8,9,10]. [11,12,13,14,15,16] solved BVPs by the reproducing kernel method. Based on the idea of least squares, Xu et al. [17,18,19] gave an effective algorithm in reproducing kernel space for solving fractional differential integral equations and interface problems.

    It is a common technique to use orthogonal polynomials to solve differential equations. In [20,21,22,23], the authors used Chebyshev-Galerkin scheme for the time-fractional diffusion equation. In [24], the authors developed Jacobi rational operational approach for time-fractional sub-diffusion equation on a semi-infinite domain. [25,26,27,28] developed multiscale orthonormal basis to solve BVPs with various boundary conditions, and the stability and convergence order were also discussed. Legendre wavelet is widely used in various fields, such as signal system, because of its good properties. In this paper, a multiscale function is constructed by using Legendre polynomials to solve the approximate solution of differential equations. We use the multiscale fine ability of Legendre wavelet to construct multiwavelet, which has better approximation than single wavelet. In addition, we improve Legendre wavelet for specific problems, and the improved one still has compact support. We know that for functions with compact support, the better the tight support, the more concentrated the energy. Moreover, in the calculation process, the calculation speed can be enhanced, and the error accumulation is low.

    The purpose of this paper is to construct a set of multiscale orthonormal basis with compact support based on Legendre wavelet to find the approximate solution of the boundary value problems:

    {u(x)+p(x)u(x)+q(x)u(x)=F(x,u),x(0,1),a1u(0)+b1u(1)+c1u(0)+d1u(1)=α1,a2u(0)+b2u(1)+c2u(0)+d2u(1)=α2, (1.1)

    where p(x) and q(x) are both smooth. ai,bi,ci,di,i=1,2 are constants. When F is just about the function of x, F(x,u)=f(x), Eq (1.1) is linear boundary value problem. According to [21], the nonlinear boundary value problem can be transformed into a linear boundary value problem by using Quasi-Newton method. So this paper mainly studies the case of F(x,u)=f(x), that is, the linear boundary value problem.

    As we all know, if the basis function has good properties, the approximate solution of the boundary value problem has good convergence, stability and so on. In [25], the orthonormal basis on [0, 1] was constructed by the compact support function to obtain the numerical solution of the boundary value problem. But the basis function is not compactly supported at [0, 1], and the approximating solution is linearly convergent. In this paper, based on the idea of wavelet, a set of orthonormal bases with compact support is constructed by using Legendre polynomials, and the approximate solution of the boundary value problem is obtained by using these bases. Based on the constructed orthonormal basis, the proposed algorithm has convergence and stability, and the convergence order of the algorithm is more than 2 orders.

    The purpose of this work is to deduce the numerical solutions of Eq (1.1). In Section 2, using wavelet theory, a set of multiscale orthonormal basis is presented by Legendre polynomials in W32[0,1]. The constructed basis is compactly supported. It is well known that the compact support performance generates sparse matrices during calculation, thus improving the convergence rate. The numerical method of ε-approximate solution is presented in Section 3. And Section 4 proves the convergence order of ε-approximate solution and stability. In Section 5, the proposed algorithm has been applied to some numerical experiments. Finally, we end with some conclusions in Section 6.

    Wu and Lin introduced the reproducing kernel space W12[0,1] and W32[0,1] [29]. Let

    W32,0[0,1]={u|u(0)=u(0)=u(0)=0,   uW32[0,1]}.

    Clearly, W32,0[0,1] is the closed subspace of W32[0,1].

    Legendre polynomials are mathematically important functions. This section constructs the orthonormal basis in W32[0,1] by Legendre polynomials. Legendre polynomials are known to be orthogonal on L2[1,1]. For convenience, we first compress Legendre's polynomials to [0,1], and get the following four functions:

    φ0(x)=1;  φ1(x)=3(1+2x);φ2(x)=5(16x+6x2);φ3(x)=7(1+12x30x2+20x3).

    By translating and weighting the above four functions, we can construct

    ψl(x)=3j=0(aljφj(2x)+bljφj(2x1)),l=0,1,2,3. (2.1)

    In application, we hope ψl(x) has good properties, for example, as many coefficients as zero and orthogonality, so ψl(x) needs to meet the following conditions

    10xjψl(x)dx=0,j=0,1,2,,l+3, (2.2)
    10ψi(x)ψj(x)dx=δij,i,j=0,1,2,3. (2.3)

    The coefficients alj,blj can be get by Eqs (2.2) and (2.3), immediately ψl(x) is as follows:

    ψ0(x)=1517{3+56x216x2+224x3,x[0,12],61296x+456x2224x3,x[12,1]. (2.4)
    ψ1(x)=121{11+270x1320x2+1680x3, x[0,12],619+2670x3720x2+1680x3,x[12,1]. (2.5)
    ψ2(x)=3517{1+30x174x2+256x3,x[0,12],111450x+594x2256x3,x[12,1]. (2.6)
    ψ3(x)=521{136x+246x2420x3,x[0,12],209804x+1014x2420x3,x[12,1]. (2.7)

    Through the ideas of the wavelet, scale transformation of the functions ψl(x) gets Legendre wavelet

    ψlik(x)=2i12ψl(2ixk),l=0,1,2,3;i=1,2,;k=0,1,,2i11.

    Clearly, ψlik(x) has compactly support in [k2i1,k+12i1]. Let

    Wi=span{ψlik(x)}3l=0,i=1,2,;k=0,1,,2i11.

    Then,

    L2[0,1]=V0i=1Wi,

    where

    V0={φ0(x),φ1(x),φ2(x),φ3(x)}.

    According to the above analysis, we can get the following theorem.

    Theorem 2.1.

    {ρj(x)}j=1={φ0(x),φ1(x),φ2(x),φ3(x),ψ010(x),ψ110(x),ψ210(x),ψ310(x),,ψ0ik(x),ψ1ik(x),ψ2ik(x),ψ3ik(x),}

    is the orthonormal basis in L2[0,1].

    Now we generate the orthonormal basis in W32,0[0,1] from the basis in L2[0,1]. Note

    J3u(x)=12x0(xt)2u(t)dt. (2.8)

    Theorem 2.2. {J3ρj(x)}j=1 is the orthonormal basis in W32,0[0,1].

    Proof. Only need to prove completeness and orthogonality. For uW32,0[0,1], if

    <u,J3ρj>W32,0=0,

    you can deduce u0, then {J3ρj(x)}j=1 are complete. In fact,

    <u,J3ρj>W32,0=<u,ρj>L2=10uρjdx=0. (2.9)

    From Theorem 2.1, u0. Due to uW32,0[0,1], u(0)=u(0)=u(0)=0, then, u0.

    According to Theorem 2.1 and Eq (2.9), orthonormal is obvious.

    Because of W32,0[0,1]W32[0,1] and three more conditions for W32[0,1] than W32,0[0,1]. So the orthonormal basis for W32[0,1] as follows:

    Theorem 2.3.

    {J3gj(x)}j=1={1,x,x22}{J3ρj(x)}j=1

    are the orthonormal basis in W32[0,1].

    Put L: W32[0,1]L2[0,1],

    Lu=u(x)+p(x)u(x)+q(x)u(x).

    L is a linear bounded operator in [27]. Let Bi: W32[0,1]R, and

    Biu=aiu(0)+biu(1)+ciu(0)+diu(1),i=1,2.

    The {Quasi-Newton} method is used to transform Eq (1.1) into a linear boundary value problem, and its operator equation is as follows:

    {Lu=f(x),B1u=α1,  B2u=α2. (3.1)

    Definition 3.1. uε is named ε-approximate solution for Eq (3.1), ε>0, if

    Luεf2L2+2i=1(Biuεαi)2<ε2.

    In [27], it is shown that ε-approximate solution for Eq (3.1) exists by the following theorem.

    Theorem 3.1. Equation (3.1) exists ε-approximate solution

    uεn(x)=nk=1ckJ3gk(x),

    where n is a natural number determined by ε, and ci satisfies

    nk=1ckLJ3gkLu2L2+2l=1(nk=1ckJ3gkBlu)2=minck{nk=1ckLJ3gkLu2L2+2l=1(nk=1ckJ3gkBlu)2}.

    To seek the ε-approximate solution, we just need ck. Let G be quadratic form about

    c=(c1,,cn)T,
    G(c1,,cn)=nk=1ckLJ3gkLu2L2+2l=1(nk=1ckJ3gkBlu)2. (3.2)

    From Theorem 3.1,

    c=(c1,,cn)T

    is the minimum point of G(c1,,cn). If L is reversible, the minimum point of G exists and is unique.

    In fact, the partial derivative of G(c1,,cn) with respect to cj:

    Gcj=2nk=1ckLJ3gk,LJ3gjL22LJ3gj,LuL2+22l=1(nk=1ckJ3gkJ3gjJ3gjBlu).

    Let

    cjG(c1,,cn)=0,

    so

    nk=1ckLJ3gk,LJ3gjL2+2nk=1ckJ3gkJ3gj=LJ3gj,LuL2+2l=1J3gjBlu. (3.3)

    Let An be the n-order matrix and bn be the n-dimensional vector, i.e.,

    An=(LJ3gk,LJ3gjL2+2J3gkJ3gj)n×n,bn=(LJ3gk,LuL2+2l=1J3gjBlu)n.

    Then Eq (3.3) changes to

    Anc=bn. (3.4)

    If L is invertible, Eq (3.4) has only one solution c, and c is minimum point of G. Equation (3.4) has an unique solution is proved as follows.

    Theorem 3.2. If L is invertible, Eq (3.3) has only one solution.

    Proof. The homogeneous linear equation of Eq (3.4) is

    nk=1ckLJ3gk,LJ3gjL2+2nk=1ckJ3gkJ3gj=0.

    Just prove that the above equation has an unique solution. Let cj(j=1,2,,n) multiply to both sides of the equation, and add all equations together so that

    nk=1ckLJ3gk,nj=1cjLJ3gjL2+2nk=1ckJ3gknj=1cjJ3gj=0.

    That is

    nk=1ckLJ3gkL2+2(nk=1ckJ3gk)2=0.

    Clearly,

    nk=1ckLJ3gk2L2=0,(nk=1ckJ3gk)2=0.

    Because J3gk is orthonormal basis, if L is invertible, ck=0. So Eq (3.3) has only one solution.

    Convergence and stability are important properties of algorithms. This section deals with the convergence and stability.

    In order to discuss the convergence, Theorem 4.1 is given as follows:

    Theorem 4.1. J3ψlik(x) is compactly supported in [k2i1,k+12i1].

    Proof. When

    x<k2i1,   J3ψlik(x)=0.

    When x>k+12i1, because of ψlik(x) with compact support, then,

    J3ψlik(x)=12x0(xt)2ψlik(t)dt=12k+12i1k2i1(xt)2ψlik(t)dt=2i32k+12i1k2i1(xt)2ψl(2i1tk)dt=25(i1)210(s2i1xk)2ψl(s)ds, s=2i1tk. (4.1)

    According to Eq (2.2), J3ψlik(x)=0. So J3ψlik(x) has compactly support in [k2i1,k+12i1].

    Note

    (J3ψli,k(x))=J2ψli,k(x), (J3ψli,k(x))=J1ψli,k(x).

    By referring to the proof of Theorem 4.1, J1ψlik(x) and J2ψlik(x) are compactly supported in [k2i1,k+12i1].

    The order of convergence will proceed below. Assume

    u(x)=2j=0cjxjj!+3j=0djJ3φj(x)+i=12i11k=03l=0(c(l)i,kJ3ψli,k), (4.2)

    where

    cj=<u,xjj!>W32,   dj=<u,φj(x)>W32,

    and

    c(l)i,k=<u,   J3ψli,k(x)>W32.

    And

    un(x)=2j=0cjxjj!+3j=0djJ3φj(x)+ni=12i11k=03l=0(c(l)i,kJ3ψli,k).

    Theorem 4.2. Assume uεn(x) is the ε-approximate solution of Eq (3.1). If u(m)(x) is bounded in [0,1], mN,3m7, then,

    |u(x)uεn(x)|2(m2)nM,

    here M is a constant.

    Proof. From Definition 3.1 and Theorem 3.1, we get

    |u(x)uεn(x)|M0uuεnW32M0L1L(uuεn)L2M0L1(L(uuεn)L2+|B1(uuεn)|+|B2(uuεn)|)M0L1(L(uun)L2+B1(uun)+B2(uun)).

    Obviously,

    B1(uun)=0,B2(uun)=0.

    That is

    |u(x)uεn(x)|M0L1L(uun)L2M0L1(10(L(uun))2dx)12M0L1(maxx[0,1]{|L(uun)|2})123M0L1M1maxx[0,1]{|uun|,|uun|,|uun|},

    where

    M1=maxx[0,1]{1,|p(x)|,|q(x)|}.

    We know

    |uun|=|i=n+12i11k=03l=0c(l)i,kJ3ψli,k(x)|i=n+12i11k=03l=0|c(l)i,k||J3ψli,k(x)|.

    By the compactly support of Jpψlik(x),p=1,2,3, fixed i, then Jpψlik(x)0 only in [k2i1,k+12i1],

    |uun|i=n+13l=0|c(l)i,k||J3ψli,k(x)|.

    Similarly,

    |uun|i=n+13l=0|c(l)i,k||J2ψli,k(x)|

    and

    |uun|i=n+13l=0|c(l)i,k||J1ψli,k(x)|.

    Through J1ψli,k(x),J2ψli,k(x) and J3ψli,k(x), you can get

    |u(x)uεn(x)|3M0M1L1|uun|.

    As |uun|, |c(l)i,k| and |J1ψli,k(x)| will be discussed below. We can get that |c(l)i,k| is related to u(m)(x). In fact,

    |c(l)i,k|=|<u,J3ψli,k(x)>W32|=|10(u(x))ψli,k(x)dx|=|k+12i1k2i1u(x)ψli,k(x)dx|. (4.3)

    Taylor's expansion of u(x) at k2i1 is

    u(x)=m1j=3u(j)(k2i1)(j3)!(xk2i1)j3+u(m)(ξ)(m3)!(xk2i1)m3, ξ[k2i1,k+12i1].

    Equation (4.3) is changed to

    |c(l)i,k|=|k+12i1k2i1(m1j=3u(j)(k2i1)(j3)!(xk2i1)j3+u(m)(ξ)(m3)!(xk2i1)m3)ψli,k(x)dx|=|m1j=3u(j)(k2i1)(j3)!k+12i1k2i1(xk2i1)j3ψlik(x)dx|+|k+12i1k2i1u(m)(ξ)(m3)!(xk2i1)m3)ψli,k(x)dx|,

    where

    k+12i1k2i1(xk2i1)j3ψlik(x)dx=2i12k+12i1k2i1(xk2i1)j3ψl(2i1xk)dxt=2i1xk=2(32j)(i1)210(t)j3ψl(t)dt.

    According to Eq (2.2),

    m1j=3u(j)(k2i1)(j3)!k+12i1k2i1(xk2i1)j3ψlik(x)dx=0,

    so

    |c(l)i,k|=|k+12i1k2i1u(m)(ξ)(m3)!(xk2i1)m3ψli,k(x)dx||u(m)(ξ)(m3)!|k+12i1k2i1|xk2i1|m3|ψli,k(x)|dx|u(m)(ξ)(m3)!|2(m3)(i1)k+12i1k2i1|ψli,k(x)|dx|u(m)(ξ)(m3)!|2(m3)(i1)2i12.

    Because u(m)(x) is bounded,

    |u(m)(ξ)(m3)!|M3,

    then,

    |c(l)i,k|2(2m5)(i1)2M3. (4.4)

    By the compactly support of J1ψlik(x),

    |J1ψli,k(x)|=|x0ψli,k(t)dt|k+12i1k2i1|ψli,k(t)|dt2i12.

    According to the above analysis,

    |uun|i=n+14M32(2m5)(i1)22(i1)2=4M32(m2)n.

    That is

    |u(x)uεn(x)|2(m2)nM,

    where M is a constant.

    Stability analysis is conducted below. According to the third section, the stability of the algorithm is related to the stability of Eq (3.4). By the following Property 4.1, the stability of the algorithm can be discussed by the number of conditions of the matrix A.

    Property 4.1. If the matrix A is symmetric and reversible, then

    cond(A)=|λmaxλmin|,

    where λmax and λmin are the largest and smallest eigenvalues of A respectively.

    In this paper,

    An=(aij)n×n=(LJ3gi,LJ3gjL2+2J3giJ3gj)n×n.

    Clearly, An is symmetric. From Theorem 3.2, An is reversible. In order to discuss the stability of the algorithm, only the eigenvalues of matrix An need to be discussed.

    Theorem 4.3. Assume uW32 and uW32=1. If L is an invertible differential operator, then,

    LuL21L1.

    Proof. Since L is an invertible, assume Lu=v, then u=L1v. Moreover

    1=uW32=L1vW32L1vL2.

    Then,

    vL21L1.

    That is,

    LuL21L1.

    Theorem 4.4. Let λ {be} the eigenvalues of matrix A of Eq (3.4), x=(x1,,xn)T is related eigenvalue of λ and x=1, then,

    λL2+2.

    Proof. By Ax=λx,

    λxi=nj=1aijxj=nj=1(LJ3gi,LJ3gjL2+2J3giJ3gj)xj=LJ3gi,nj=1xjLJ3gjL2+2J3ginj=1(J3gjxj),i=1,,n. (4.5)

    Let xi multiply to both sides of Eq (4.5), and then add the equations from j=1 to j=n together so that

    λ=λx2i=ni=1xiLJ3gi,nj=1xjLJ3gjL2+2ni=1(J3gixi)nj=1(J3gjxj)=ni=1xiLJ3gi2L2+2(ni=1(J3gixi))2L2ni=1x2i+2ni=1x2i=(L2+2)x. (4.6)

    Since

    x=1,   λL2+2.

    From Theorem 4.3 and Eq (4.6), we can get

    λni=1xiLJ3giL2=L(ni=1xiJ3gi)L21L1.

    Then,

    cond(A)=|λmaxλmin|L2+21L1=(L2+2)L1.

    That is the condition number of A is bounded, so the presented method is stable.

    This section discusses numerical examples to reveal the accuracy of the proposed algorithm. Examples 5.1 and 5.3 are linear and nonlinear BVPs respectively. Example 5.2 shows that our method also applies to Eq (1.1) with other linear boundary value conditions. In this paper, N is the number of bases, and

    N=7+4(2n1),n=1,2,.

    eN(x) is the absolute errors. C.R. and cond represent the convergence order and the condition number respectively. For convenience, we denote

    eN(x)=|u(x)uN(x)|

    and

    C.R.=log2max|eN(x)|max|eN+1(x)|.

    Example 5.1. Consider the test problem suggested in [28,30]

    {u=u+2u+4x2ex,x(0,1),u(0)=2,u(1)=e1,

    where the exact solution is u(x)=ex2x+1. The numerical results are shown in Table 1. It is clear from Table 1 that the present method produces a converging solution for different values. In addition, the results of the proposed algorithm in Table 1 are compared with those in [28,30]. Obviously, the proposed algorithm is better. Table 2 shows eN(x), C.R., cond and CPU time. The unit of CPU time is second, expressed as s.

    Table 1.  eN(x) of Example 5.1.
    x eN(x) of [30] e66(x) of [28] e35(x) e67(x)
    0 0 5.67e-9 9.94e-14 8.88e-16
    0.1 1.19e-5 3.35e-9 2.04e-13 2.22e-16
    0.2 4.18e-5 3.93e-10 2.16e-13 1.55e-15
    0.3 4.96e-5 1.33e-9 1.42e-13 8.88e-16
    0.4 6.04e-5 1.40e-9 1.68e-14 1.33e-15
    0.5 6.33e-5 1.82e-9 1.82e-13 4.44e-16
    0.6 6.23e-5 5.96e-9 1.52e-13 2.66e-15
    0.7 5.76e-5 1.14e-8 1.66e-13 8.88e-16
    0.8 4.23e-5 1.52e-8 4.36e-13 4.44e-15
    0.9 2.15e-5 1.66e-8 4.93e-13 4.44e-16
    1.0 0 1.90e-8 2.67e-13 4.44e-15

     | Show Table
    DownLoad: CSV
    Table 2.  eN(x), C.R. and cond of Example 5.1.
    n N maxeN(x) C.R. cond CPU(s)
    1 11 1.66e-8 274.262 2.57
    2 19 6.85e-11 7.92 274.262 7.89
    3 35 6.55e-13 6.71 274.262 24.42
    4 67 7.93e-15 6.40 274.262 82.73

     | Show Table
    DownLoad: CSV

    Example 5.2. Consider the problem suggested in [25,28].

    {u+u+xu=f(x),x(0,1),u(0)=2,u(1)+u(12)=sin12+sin1.

    The exact solution is u(x)=sinx, and f(x)=cosxsinx+xsinx. This problem is the boundary value problem with the multipoint boundary value conditions. Table 3 shows maximum absolute error MEn, C.R. and cond., which compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective. The numerical errors are provided in Figures 1 and 2, also show a good accuracy.

    Table 3.  MEn, C.R. and cond of Example 5.2.
    The present method [25] [28]
    n MEn C.R. cond n MEn C.R. cond n MEn C.R. cond
    11 3.73e-9 195.05 10 6.34e-6 3.96 182.06 11 1.88e-4
    19 2.97e-11 6.97 195.05 18 4.04e-7 3.98 182.06 19 5.99e-5 1.65 1.49×106
    35 2.13e-13 7.12 195.05 34 2.54e-8 4.06 182.06 35 1.84e-5 1.70 3.74×108
    67 2.77e-15 6.27 195.05 66 1.59e-9 3.95 182.06 67 4.62e-6 1.99 8.87×1010

     | Show Table
    DownLoad: CSV
    Figure 1.  eN(x) of Example 5.3 (n = 35).
    Figure 2.  eN(x) of Example 5.3 (n = 67).

    Example 5.3. Consider a nonlinear problem suggested in [7,9]

    {u+λeu=0,x(0,1)u(0)=0,u(1)=0,

    where

    u(x)=2ln(cosh((x12)(θ/2))/cosh(θ/4)),

    and θ satisfies

    θ2λcosh(θ/4)=0.

    This is the second-order nonlinear Bratu problem. { Bratu equation is widely used in engineering fields, such as spark discharge, semiconductor manufacturing, etc. In the field of physics, the Bratu equation is used to describe the physical properties of microcrystalline silica gel solar energy. In the biological field, the Bratu equation is used to describe the kinetic model of some biochemical reactions in living organisms.} To this problem, taking u0(x)=x(1x),k=3, where k is the number of iterations of the algorithm mentioned in [27]. when λ=1,λ=2, eN(x) are listed in Tables 4 and 5, respectively.

    Table 4.  eN(x) of Example 5.3 (λ=1).
    x eN(x) of [8] eN(x) of [9] eN(x)
    0 0 0 4.4959e11
    0.2 1.4958e9 2.4390e5 4.1096e11
    0.4 2.7218e9 4.2096e5 7.1502e12
    0.6 2.7218e9 4.2096e5 7.1483e12
    0.8 1.4958e9 2.4390e5 4.1104e11

     | Show Table
    DownLoad: CSV
    Table 5.  eN(x) of Example 5.3 (λ=2).
    x eN(x) of [7] eN(x) of [9] eN(x)
    0 5.8988e26 0 1.1801e12
    0.2 1.3070e7 6.9297e5 2.5646e10
    0.4 1.4681e7 1.0775e4 1.6666e9
    0.6 1.4681e7 1.0775e4 1.6666e9
    0.8 1.3070e7 6.9297e5 2.5646e10

     | Show Table
    DownLoad: CSV

    In this paper, based on Legendre's polynomials, we construct orthonormal basis in L2[0,1] and W32[0,1], respectively. It proves that this group of bases is orthonormal and compactly supported. According to the orthogonality of the basis, we present an algorithm to obtain the approximate solution of the boundary value problems. Using the compact support of the basis, we prove that the convergence order of the presented method related to the boundedness of u(m)(x). Finally, three numerical examples show that the absolute error and convergence order of the algorithm are better than other methods.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This study was supported by National Natural Science Funds of China by Grant number (12101164), Characteristic Innovative Scientific Research Project of Guangdong Province (2023KTSCX181, 2023KTSCX183) and Basic and Applied Basic Research Project Zhuhai City (ZH24017003200026PWC).

    The authors have no conflicts of interest to declare.



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