On some ψ Caputo fractional Čebyšev like inequalities for functions of two and three variables

1.   Introduction

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:

where $p(x)$ and $q(x)$ are both smooth. $a_i, b_i, c_i, d_i, \; i = 1, 2$ 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 $W_2^3[0, 1]$. 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 ${\varepsilon}$-approximate solution is presented in Section 3. And Section 4 proves the convergence order of ${\varepsilon}$-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.

2.   Basis functions in $W_2^3[0, 1]$

Wu and Lin introduced the reproducing kernel space $W_2^1[0, 1]$ and $W_2^3[0, 1]$ ^[29]. Let

Clearly, $W_{2, 0}^3[0, 1]$ is the closed subspace of $W_2^3[0, 1]$.

Legendre polynomials are mathematically important functions. This section constructs the orthonormal basis in $W_2^3[0, 1]$ by Legendre polynomials. Legendre polynomials are known to be orthogonal on $L^2[-1, 1]$. For convenience, we first compress Legendre's polynomials to $[0, 1]$, and get the following four functions:

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

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

The coefficients $a_{lj}, b_{lj}$ can be get by Eqs (2.2) and (2.3), immediately $\psi^l(x)$ is as follows:

Through the ideas of the wavelet, scale transformation of the functions $\psi^l(x)$ gets Legendre wavelet

Clearly, $\psi_{ik}^{l}(x)$ has compactly support in $[\frac{k}{2^{i-1}}, \frac{k+1}{2^{i-1}}]$. Let

Then,

where

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

Theorem 2.1.

is the orthonormal basis in $L^2[0, 1]$.

Now we generate the orthonormal basis in $W_{2, 0}^3[0, 1]$ from the basis in $L^2[0, 1]$. Note

Theorem 2.2. $\{J^3\rho_j(x)\}_{j = 1}^{\infty}$ is the orthonormal basis in $W_{2, 0}^3[0, 1]$.

Proof. Only need to prove completeness and orthogonality. For $u \in W_{2, 0}^3[0, 1]$, if

you can deduce $u \equiv 0$, then $\{J^3\rho_j(x)\}_{j = 1}^{\infty}$ are complete. In fact,

From Theorem 2.1, $u^{\prime\prime\prime}\equiv 0$. Due to $u \in W_{2, 0}^3[0, 1]$, $u(0) = u^{\prime}(0) = u^{\prime\prime}(0) = 0$, then, $u\equiv 0$.

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

Because of $W^3_{2, 0}[0, 1]\subset W^3_2[0, 1]$ and three more conditions for $W_{2}^3[0, 1]$ than $W_{2, 0}^3[0, 1]$. So the orthonormal basis for $W_{2}^3[0, 1]$ as follows:

Theorem 2.3.

are the orthonormal basis in $W_{2}^3[0, 1]$.

3.   A multiscale algorithm for Eq (1.1)

Put $L$: $W_2^3[0, 1]\to L^2[0, 1]$,

$L$ is a linear bounded operator in ^[27]. Let $B_i$: $W_2^3[0, 1]\to \mathbb{R}$, and

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

Definition 3.1. $u^{{\varepsilon}}$ is named ${\varepsilon}$-approximate solution for Eq (3.1), $\forall {\varepsilon} > 0$, if

In ^[27], it is shown that ${\varepsilon}$-approximate solution for Eq (3.1) exists by the following theorem.

Theorem 3.1. Equation (3.1) exists ${\varepsilon}$-approximate solution

where $n$ is a natural number determined by ${\varepsilon}$, and $c^{*}_i$ satisfies

To seek the ${\varepsilon}$-approximate solution, we just need $c_k^*$. Let $G$ be quadratic form about

From Theorem 3.1,

is the minimum point of $G(c_1, \cdots, c_n)$. If $L$ is reversible, the minimum point of $G$ exists and is unique.

In fact, the partial derivative of $G(c_1, \cdots, c_n)$ with respect to $c_j$:

Let

so

Let $\mathit{\boldsymbol{A_{n}}}$ be the n-order matrix and $\mathit{\boldsymbol{b_{n}}}$ be the n-dimensional vector, i.e.,

Then Eq (3.3) changes to

If $L$ is invertible, Eq (3.4) has only one solution $\mathit{\boldsymbol{c}}^*$, and $\mathit{\boldsymbol{c}}^*$ 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

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

That is

Clearly,

Because $J^3g_k$ is orthonormal basis, if $L$ is invertible, $c_k = 0$. So Eq (3.3) has only one solution. □

4.   Analysis of convergence and stability

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

4.1.   Analysis of convergence

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

Theorem 4.1. $J^3\psi_{ik}^{l}(x)$ is compactly supported in $[\frac{k}{2^{i-1}}, \frac{k+1}{2^{i-1}}]$.

Proof. When

When $x > \frac{k+1}{2^{i-1}}$, because of $\psi_{ik}^{l}(x)$ with compact support, then,

According to Eq (2.2), $J^3\psi_{ik}^{l}(x) = 0$. So $J^3\psi_{ik}^{l}(x)$ has compactly support in $[\frac{k}{2^{i-1}}, \frac{k+1}{2^{i-1}}]$. □

Note

By referring to the proof of Theorem 4.1, $J^1\psi_{ik}^{l}(x)$ and $J^2\psi_{ik}^{l}(x)$ are compactly supported in $[\frac{k}{2^{i-1}}, \frac{k+1}{2^{i-1}}]$.

The order of convergence will proceed below. Assume

where

and

And

Theorem 4.2. Assume $u_n^{{\varepsilon}}(x)$ is the ${\varepsilon}$-approximate solution of Eq (3.1). If $u^{(m)}(x)$ is bounded in $[0, 1]$, $m \in \mathbb{N}, 3\le m\le7$, then,

here $M$ is a constant.

Proof. From Definition 3.1 and Theorem 3.1, we get

Obviously,

That is

where

We know

By the compactly support of $J^p\psi_{ik}^{l}(x), \; p = 1, 2, 3$, fixed $i$, then $J^p\psi_{ik}^{l}(x)\neq 0$ only in $[\frac{k}{2^{i-1}}, \frac{k+1}{2^{i-1}}]$,

Similarly,

and

Through $J^1\psi_{i, k}^{l}(x), J^2\psi_{i, k}^{l}(x)$ and $J^3\psi_{i, k}^{l}(x)$, you can get

As $|u^{\prime\prime}-u_n^{\prime\prime}|$, $|c_{i, k}^{(l)}|$ and $|J^1\psi_{i, k}^{l}(x)|$ will be discussed below. We can get that $|c_{i, k}^{(l)}|$ is related to $u^{(m)}(x)$. In fact,

Taylor's expansion of $u^{\prime\prime\prime}(x)$ at $\frac{k}{2^{i-1}}$ is

Equation (4.3) is changed to

where

According to Eq (2.2),

so

Because $u^{(m)}(x)$ is bounded,

then,

By the compactly support of $J^1\psi_{ik}^{l}(x)$,

According to the above analysis,

That is

where $M$ is a constant. □

4.2.   Analysis of stability

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 $\mathit{\boldsymbol{A}}$.

Property 4.1. If the matrix $\mathit{\boldsymbol{A}}$ is symmetric and reversible, then

where ${\lambda_{max}}$ and ${\lambda_{min}}$ are the largest and smallest eigenvalues of $\mathit{\boldsymbol{A}}$ respectively.

In this paper,

Clearly, $\mathit{\boldsymbol{A_{n}}}$ is symmetric. From Theorem 3.2, $\mathit{\boldsymbol{A_{n}}}$ is reversible. In order to discuss the stability of the algorithm, only the eigenvalues of matrix $\mathit{\boldsymbol{A_{n}}}$ need to be discussed.

Theorem 4.3. Assume $u\in W_2^3$ and $\|u\|_{W_2^3} = 1$. If $L$ is an invertible differential operator, then,

Proof. Since $L$ is an invertible, assume $Lu = v$, then $u = L^{-1}v$. Moreover

Then,

That is,

□

Theorem 4.4. Let ${\lambda}$ {be} the eigenvalues of matrix $\mathit{\boldsymbol{A}}$ of Eq (3.4), $\mathit{\boldsymbol{x}} = (x_1, \cdots, x_n)^{T}$ is related eigenvalue of $\lambda$ and $\|\mathit{\boldsymbol{x}}\| = 1$, then,

Proof. By $\mathit{\boldsymbol{Ax}} = \lambda \mathit{\boldsymbol{x}}$,

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

Since

□

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

Then,

That is the condition number of $\mathit{\boldsymbol{A}}$ is bounded, so the presented method is stable.

5.   Numerical examples

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

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

and

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

where the exact solution is $u(x) = e^x-2x+1$. 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 $e_N(x)$, $C.R.$, $cond$ and CPU time. The unit of CPU time is second, expressed as $s$.

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

The exact solution is $u(x) = \sin{x}$, and $f(x) = \cos{x}-\sin{x}+x\sin{x}$. This problem is the boundary value problem with the multipoint boundary value conditions. Table 3 shows maximum absolute error $ME_n$, $C.R.$ 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.

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

where

and $\theta$ satisfies

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 $u_0(x) = x(1-x), \; k = 3$, where $k$ is the number of iterations of the algorithm mentioned in ^[27]. when $\lambda = 1, \lambda = 2$, $e_N(x)$ are listed in Tables 4 and 5, respectively.

6.   Conclusions

In this paper, based on Legendre's polynomials, we construct orthonormal basis in $L^2[0, 1]$ and $W_2^3[0, 1]$, 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.

Use of AI tools declaration

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

Acknowledgments

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).

Conflict of interest

The authors have no conflicts of interest to declare.