A new application of the Legendre reproducing kernel method

1.   Introduction

In this work, we take into consideration the following nonlinear system of ordinary differential equation ^[21]:

subject to the boundary conditions

where $S, \, P_1, \, P_2, \, \delta, \, \beta$ and $\gamma$ are real finite constants.

We can see these problems in paper production, polymer extraction, aerodynamics, reaction-diffusion processes, fluid dynamics, biology and rheometry domains. These problems show up mainly due to the suction and injection effects on the unsteady magneto-hydrodynamic flow ^[24].

Many methods have been improved for the analytical and approximate solution of nonlinear ordinary differential systems. These techniques contain finite-difference methods ^[5,31,32,33], Adams-Bashforth method ^[20,23], B-spline approximation method ^[8], Chebyshev finite difference method ^[28], finite element method ^[6], He's homotopy perturbation method ^[27], $G'/G$- method ^[22], multi-step methods ^[14].

In recent years, much attempt has been done to the newly developed methods to introduce an analytic and approximate solution of nonlinear boundary value problems ^{[10,11,12,13,15,16,17,18,19,25]}. For more details see ^{[1,2,3,26,34,35,36,37]}. In this work, we present an approximate-analytical technique for solving a coupled system of second and fourth order boundary value problems.

The rest of this paper is organized as follows. In Section 2, an overview of shifted Legendre polynomials and their relevant properties required henceforward are presented. Also in this section, we will recall a brief review of the reproducing kernel spaces. In Section 3, we construct an orthogonal basis in the Legendre reproducing kernel space and construct a reproducing kernel space which includes boundary conditions. In Section 4, our method to approximate the solution of nonlinear system via shifted Legendre reproducing kernel basis function is considered. We present the convergence analysis and error estimation in Section 5. We demonstrate the numerical results in Section 6. We give the conclusion in the last section.

2.   Legendre reproducing kernel functions

In this section, we will recall some basic polynomial functionals and define some new reproducing kernel functions. The well-known shifted Legendre polynomials are described on [0, 1] and can be obtained by the following iterative formula

The polynomials $P_n(x)$ are orthogonal on $[0, 1]$ with $\rho(x) = 1$, in the sense that

where

We use shifted Jacobi basis functions which provide the homogeneous boundary conditions as:

Lemma 2.1. Let $\alpha, \beta\geq 1$ and $\alpha, \beta \in \mathbb{Z}$. We have $\{\textrm{a}_j\}$ such that

where $P_j(x)$ are the shifted Legender polynomial of degree $j$ and $J_n^{-\alpha, -\beta}(x)$ is the shifted Jacobi polynomial on $[0, 1]$. Then, we have

Proof. For the proof of Lemma 2.1 (see ^[30], Lemma 1.4.3).

Now, by utilizing the shifted Jacobi basis function and shifted Legendre functions, we will introduce a reproducing kernel Hilbert space method.

Since $P_n(1) = 1$ and $P_n(0) = (-1)^n$, we have

Therefore, we describe

and

Definition 2.2. ^[10] For a nonempty set $E$, let $H$ be a Hilbert space of real value functions on some set $E$. A function $K: E \times E\longrightarrow \mathbb{R}$ is said to be the reproducing kernel function of $H$ if and only if:

(i) For every $y\in E$, $K(\cdot, y) \in H$.

(ii) For every $y\in E$ and $f\in H, \quad \langle f(\cdot), K(\cdot, y)\rangle = f(y).$

Also, a Hilbert space of function $H$ that possesses a reproducing kernel $K$ is a reproducing kernel Hilbert space; we represent the reproducing kernel Hilbert space and it's kernel by $H_{K}(E)$ and $K_{y}$ respectively.

Theorem 2.3. ^[7] Let $H$ be $n$-dimensional Hilbert space, $\{w_i \}_{i = 1}^{n}$ is an orthonormal basis of $H$, then the reproducing kernel of H as:

Theorem 2.4. (^[29] Theorem 1.24) For the orthonormal system $\{w_{n}\}_{n = 1}^{\infty}$, formula $(2.7)$ yields the Christoffel-Darboux formula:

Where, $k_{n} > 0$ is the coefficient of $x^{n}$ in $w_{n}(x)$. We get

Definition 2.5. Let $H_{\omega, K_{1, n}}[0, 1]$ be the weighted inner product space of Jacobi functions described as (2.5) on $[0, 1]$ with degree less than or equal to $n$. The inner product and norm are given respectively by

where $\omega(x) = (1-x)^{-1}(1+x)^{-1}$ and $K_{1, n}(x, y)$ the reproducing kernel of $H_{\omega, K_{1, n}}[0, 1]$ is constructed using (2.8) with $w_{n}(x): = u_{n}(x)$. From definition

for any fixed $n$, $H_{\omega, K_{1, n}}[0, 1]$ is a subspace of $L_{\omega}^2[0, 1]$ and

From Definition 2.5, $H_{\omega, K_{1, n}}[0, 1]$ is a finite dimensional inner product space. Every finite dimensional inner product space is a Hilbert space. Therefore, from this result and Theorem 2.3, $H_{\omega, K_{1, n}}[0, 1]$ is a reproducing kernel Hilbert space.

Definition 2.6. Let $H_{K_{2, n}}[0, 1]$ be the inner product space of Legendre functions described as (2.6) on $[0, 1]$ with degree less than or equal to $n$. The inner product and norm are given respectively by

where the reproducing kernel $K_{2, n}(x, y)$ of $H_{K_{2, n}}[0, 1]$ is constructed using (2.8) with $w_{n}(x): = v_{n}(x)$. From definition

for any fixed $n$, $H_{K_{2, n}}[0, 1]$ is a subspace of $L^2[0, 1]$ and

3.   Construction of reproducing kernel spaces

$H_{\omega, R_1}[0, 1]$ and $H_{R_2}[0, 1]$ are described by:

Clearly, $H_{\omega, R_1}[0, 1]$ and $H_{R_2}[0, 1]$ are closed subspaces of $H_{\omega, K_{1, n}}[0, 1]$ and $H_{K_{2, n}}[0, 1]$, respectively.

Let $T_1f = f^{\prime}(0)-\beta f^{\prime\prime}(0)$ and $T_2f = f^{\prime}(1)+\beta f^{\prime\prime}(1)$ be the boundary condition of function $f(x)$. Put

and

where, the symbol $T_{1, x}$ shows that the operator $T_1$ implements to the function of $x$.

Theorem 3.1. If $T_{1, x}T_{1, y}K_{1, n}(x, y)\neq 0$ and $T_{2, x}T_{2, y}R_{1, 1}(x, y)\neq 0$, then $R_{1}(x, y)$ given by $(3.2)$ satisfies the boundary conditions $T_1f = 0$ and $T_2f = 0$ exactly.

Proof. By applying the operator $T_{1, x}$ to $R_{1, 1}(x, y)$ in Eq (3.1), we get

Furthermore, by applying the operator $T_{1, x}T_{2, y}$ to $R_{1, 1}(x, y)$, we have

Then, by applying the operator $T_{1, x}$ to $R_{1}(x, y)$ in Eq (3.2) and using Eqs (3.3) and (3.4), we get

Also, by applying the operator $T_{2, x}$ to $R_{1}(x, y)$ in Eq (3.2), we have

Theorem 3.2. ^[9] If $T_{1, x}T_{1, y}K_{1, n}(x, y)\neq 0$ and $T_{2, x}T_{2, y}R_{1, 1}(x, y)\neq 0$, then, we obtain

Let $T_3\theta = \theta(0)-\gamma \theta^{\prime}$ and $T_4\theta = \theta(1)+\gamma \theta^{\prime}(1) = 0$ be the boundary condition of function $\theta(x)$. Put

and

Theorem 3.3. If $T_{3, x}T_{3, y}K_{2, n}(x, y)\neq 0$ and $T_{4, x}T_{4, y}R_{2, 2}(x, y)\neq 0$, then $R_{2}(x, y)$ given by $(3.6)$ satisfies the boundary conditions $T_3\theta = 0$ and $T_4\theta = 0$ exactly.

Proof. The proof of this theorem is similar to the proof of Theorem 3.1.

Theorem 3.4. If $T_{3, x}T_{3, y}K_{2, n}(x, y)\neq 0$ and $T_{4, x}T_{4, y}R_{2, 2}(x, y)\neq 0$, then $H_{R_2}[0, 1]$ is a reproducing kernel space and its reproducing kernel is

Note that $R_x(y) = R(x, y)$, $R_{1, x}(y) = R_1(x, y)$ and $R_{2, x}(y) = R_2(x, y)$. Henceforth and not to conflict unless stated otherwise, we denote $H[0, 1] = H_{\omega, R_1}\oplus H_{R_2}$, $L[0, 1] = L_{\omega}^2[0, 1]\oplus L^2[0, 1]$ and $R_x(y) = (R_{1, x}(y), R_{2, x}(y))^T$.

Definition 3.5. (a) The Hilbert space $H[0, 1]$ is described by:

The inner product in $H[0, 1]$ is building as

and the norm is $\|z\|_{H} = \|z_1\|_{H_{\omega, K_n}}+\|z_2\|_{H_{K_n}}$ where $z, w\in H[0, 1].$

(b) The Hilbert space $L[0, 1]$ is described by:

The inner product in $L[0, 1]$ is building as

and the norm is $\|z\|_{L} = \|z_1\|_{L_{\omega}^2}+\|z_2\|_{L^2}$ where $z, w\in L[0, 1]$.

4.   Representation of approximate solutions

We assume

and

where

and ${\gamma}_1 = \frac{1}{2\gamma+1}$, then Eqs (1.1) and (1.2) changes to the following problem:

and the boundary conditions changes to the following conditions:

where

Put

then, the coupled differential systems of Eqs (1.1) and (1.2) can be written as follows:

with boundary conditions:

where $g = (g_1, g_2)^T$, $N = (N_1, N_2)^T$, $\Phi\in H[0, 1]$, $g-N\in L[0, 1]$, $e_1 = (1, 0)^T$, $e_2 = (0, 1)^T$ and $\mathbb{L}:H[0, 1]\rightarrow L[0, 1]$.

Here, $\big(e_1^T\Phi(i)\big)e_1 = (F(i), 0)^T$ and $\big(e_2^T\Phi(i)\big)e_2 = (0, \Theta(i))^T, \, \, i = 0, 1$.

Lemma 4.1. (^[10], Lemma 4.1) The operators ${\mathbb{L}}_{22}:H_{R_2}[0, 1]\rightarrow L^2[0, 1]$ and ${\mathbb{L}}_{i1}[0, 1]:H_{\omega, R_1}\rightarrow L_{\omega}^2[0, 1], \, \, i = 1, 2,$ are linear bounded operators.

Theorem 4.2. The operator $\mathbb{L}:H[0, 1]\rightarrow L[0, 1]$ is bounded linear operator.

Proof. For each $\Phi\in H[0, 1]$, using Definition 3.5, we have

The boundedness of ${\mathbb{L}}_{22}$ and ${\mathbb{L}}_{i1}$ for $i = 1, 2,$ shows that ${\mathbb{L}}$ is bounded. The proof is complete.

Let $D = \{x_i\}_{i = 1}^{\infty}$ is countable dense subset in the domain $[0, 1]$, then for any fixed $x_i\in [0, 1]$, we have

where ${\mathbb{L}}^{*} = \left(\begin{array}{cc} \mathbb{L}_{11}^{*} & \mathbb{L}_{21}^{*} \\ 0 & \mathbb{L}_{22}^{*} \\ \end{array} \right)$ is the adjoint operator of ${\mathbb{L}}$ and the subscript $y$ in the operator ${\mathbb{L}_y}$ indicates that the operator ${\mathbb{L}}$ applied to the function $y$. For any fixed $x_i\in (0, 1)$, $\Psi_{ij}(x)\in H[0, 1]$.

Theorem 4.3. If $\{x_i\}_{i = 1}^{\infty}$ is distinct points dense on $[0, 1]$ and ${\mathbb{L}}^{-1}$ is existent, then

Proof. Clearly $\psi_{ij}(x)\in H_{\omega, R_1}([0, 1])\oplus H_{R_2}([0, 1])$. For any $\Phi \in (Im {\mathbb{L}}^{\ast})^{\perp}$, since $\psi_{ij}(x) = {\mathbb{L}}^{\ast}R_{x_i}(x)e_j$, we have

On the other hand,

Thus, by Eq (4.8), we get

Note that $\{x_i\}_{i = 1}^{\infty}$ is dense on $[0, 1]$. Hence $(\mathbb{L}\Phi)(x) = 0$. So from the existence $\mathbb{L}^{-1}$, we have $\Phi(x) = 0$. That is $(Im \mathbb{L}^{\ast})^{\perp} = 0$. Therefore $Im \mathbb{L}^{\ast} = H_{\omega, R_1}([0, 1])\oplus H_{R_2}([0, 1])$. Similarly, we can show $(Ker {\mathbb{L}}^{\ast})^{\perp} = L_{\omega}^2([0, 1])\oplus L^2([0, 1])$.

Corollary 4.4. For Eqs $(1.1)–(1.4)$, if $\{x_i\}_{i = 1}^{\infty}$ is distinct points dense on $[0, 1]$ and ${\mathbb{L}}^{-1}$ is existent, then $\{\psi_{ij}(x)\}^{(\infty, 2)}_{(i, j) = (1, 1)}$ is the complete function system of the space $H([0, 1])$.

By Gram-Schmidt process, we acquire an orthogonal basis $\{\overline{\psi}_{ij}(x)\}_{(i, j) = (1, 1)}^{(\infty, 2)}$ of $H([0, 1])$, such that

where $\alpha_{lk}^{ij}$ represents orthogonal coefficients, which are given by the following relations ^[4]:

such that $a_{lk}^{ij} = \sqrt{\| \psi_{lk}\|_{H}^2-\sum_{s = i}^{l-1}({c_{lk}^{sj}})^2}$ and $c_{lk}^{sj} = \langle \psi_{lk}, \overline{\psi}_{sk} \rangle_{_{H}}^2.$

Lemma 4.5. Let $\{\overline{\psi}_{ij}(x)\}_{(i, j) = (1, 1)}^{(\infty, 2)}$ be an orthonormal basis of $H$ then we have

Proof. Let $g\in H$, then

Theorem 4.6. If $\{x_{i} \}_{i = 1}^{\infty}$ is dense on $[0, 1]$ and ${\mathbb{L}}^{-1}$ is existent, then the solution of Eq $(4.6)$ satisfies the form

where

Proof. Since $\{\overline{\psi}_{ij}(x)\}_{(i, j) = (1, 1)}^{(\infty, 2)}$ is orthonormal system, $\Phi(x)$ is expressed as

where ${r}_{lk}(y) = R_{x_l}(y)e_k$.This completes the proof.

Now, let

Define $H[0, 1]-$orthogonal projection $\mathcal{T}_N:H[0, 1]\rightarrow H_N[0, 1]$ such that for $\Phi\in H[0, 1],$

or equivalently,

Then, we get the approximate solution as:

Here, $\mathcal{T}_0\Phi(x)$ is any fixed function in $H([0, 1])$.

5.   Convergence and error estimation

Theorem 5.1. Assume that the problem $(4.6)$ has a unique solution. If $\{x_{i} \}_{i = 1}^{\infty}$ is dense on $[0, 1]$, then $\Phi_N(x)$ in $(4.10)$ is convergence to the $\Phi(x)$ and for any fixed $\Phi_0(x)\in H([0, 1]), \, \, \Phi_N(x)$ is also represented by

Proof. We have

where

Then for $s\leq N$ and $p\leq 2$, we have

Therefore

If $s = 1$, we get

that is,

For $s = 2$, we have

that is,

Hence it can be obtained by induction,

Since $\{x_n\}_{n = 1}^{\infty}$ is dense, for any $x\in [0, 1]$ there exists a subsequence $\{x_{n_i}\}_{i = 1}^{\infty}$ such that $x_{n_i}\rightarrow x$, as $i\rightarrow \infty$. Then, we reach:

Moreover, according to (4.10) we have

So, from Eqs (5.2) and (5.3), we conclude that

Thus, we obtain

Theorem 5.2. Let $\Phi_n(x) = (F_n(x), \Theta_n(x))^T$ be approximate solution that has obtained from the present method in the space $H[0, 1]$ and $\Phi(x) = (F(x), \Theta(x))^T$ be exact solution for the differential equation $(4.6)$ with boundary conditions $(4.7)$. Also, assume that $x_n\rightarrow x\, \, \, (n\rightarrow \infty)$, $\|\Phi_n \|_{_{H}}$ is bounded and $G(t, \Phi(t), \Phi^{\prime}(t), \Phi^{\prime\prime}(t))$ is continuous for $t\in [0, 1]$, then

as $n \rightarrow \infty$.

Proof. For any $x\in [0, 1]$, using the boundedness of $\|\partial_x^iR_1(x, y)\|_{_{H_{{w}, {R_{1}}}}}\, \, \, (i = 0, 1, 2)$ and reproducing property of $R_1(x, y)$, we have

where for $i = 0, 1, 2$, $\alpha_i$ are positive constants.

Similarly, for each $x\in [0, 1]$ and $i = 0, 1$, we get

where for $i = 0, 1, 2$, $\beta_i$ are positive constants.

Furthermore, if $\Phi\in H[0, 1]$, then $\Phi(x) = (F(x), \Theta(x))^T$ where $F(x)\in H_{w, R_1}[0, 1]$ and $\Theta(x)\in H_{R_2}[0, 1]$. Thus for $i = 0, 1, 2$, we have

Note that, since $\Phi_n\in H[0, 1]$, exist a constant $c_1$ such that

Therefore

where $y_1$ lies between $x_n$ and $x$. Now will show that $\Phi_{n-1}^{\prime}(x_n)\rightarrow \Phi^{\prime}(x)$. Since $\Phi_{n}(x)\in H[0, 1]$, exist a constant $c_2$ such that $|\Phi_{n-1}^{\prime\prime}(x)|\leq c_2$, so we get

where $y_2$ lies between $x_n$ and $x$. Similarly, we can write

Now, from the continuation of $G(t, \Phi(t), \Phi^{\prime}(t), \Phi^{\prime\prime}(t))$, it is implies that

6.   Numerical experiment

In this section, some illustrative examples demonstrate the applicability, efficiency and utility of the proposed technique. The computations associated with the examples were performed using Maple$16$ on a personal computer.

Let us consider Eqs (1.1)–(1.4), using the shifted Legendre reproducing kernel Hilbert space method. We apply the technique on this problem with $N = 12$ and

Table 1 demonstrate the obtained solutions of $f^{\prime\prime}(x)$ and at $x = 1$ for various values of $S$, $M$, $\beta$ and compares the results with homotopy analysis method (HAM) presented in ^[21]. Table 2 demonstrates the approximate solutions of velocity $\theta^{\prime}(x)$ at $x = 1$ with $N = 12$ and $P_1 = M = P_2 = 1.0$, $\delta = 0.1$ for different values of $S$, $\beta$, $\gamma$ and compares the result with the HAM presented in ^[21]. In ^[21], there is no analysis about the convergence or error estimate of results, whereas in the current work we discussed about the convergence of method and residual errors. Hence, we can claim from the error analysis that out obtained results are more accurate than ^[21]. For example, the result of $\theta'(1)$ with respect to $S = 4.00$ in ^[21] is 0.281319, but in the present techniqe we get 0.2880499297. It is evident that there is little difference between the obtained results, which the present method gives more accurate results.

The effect of Hartmann number M on the radial velocity $f^{\prime}(x)$ is exhibited in Figure 1. The radial velocity $f^{\prime}(x)$, decreased for higher values of the Hartmann number on $0.24 \leq x\leq 0.76$. The influence of parameter $\beta$ on $f^{\prime}(x)$ is plotted in Figure 2 with $M = S = 1.0$. This Figure suggests that the $f^{\prime}(x)$ show decreasing behavior with an increase in $\beta$. Figures 3 and 4 display the temperature profiles $\theta(x)$ for the various embedded parameters viz thermal slip parameter $\gamma$ and Eckert number $P_2$ on interval $[0, 1]$. It is seen that when $\gamma = 0$, which corresponds to no thermal slip, the temperature of the fluid and that of the disks surfaces is the same, which in this case is $0$ and $1$ for lower and upper disks, respectively.

Since the exact solution of problems (1.1)–(1.4) is not known, we discuss the absolute residual error function which is a measure of how well the approximation satisfies the Eq (4.6) with $S = P_1 = M = P_2 = 1.0$ and $\beta = \gamma = \delta = 0.1$ as

Note that the norm 2 of the residual function on the domain is

and it is employed in this paper to check the accuracy and the convergence of the proposed method. The absolute residual errors are plotted in Figure 5.

7.   Conclusions and perspectives

In this paper, the shifted Legendre reproducing kernel method is employed to compute approximate solutions of a nonlinear system of ordinary differential equation. In this approach, a truncated series based on shifted Legendre reproducing kernel functions with easily computable components. The convergence analysis and error estimation of the approximate solution using the proposed method are investigated. The validity and applicability of the method is demonstrated by solving several numerical examples. The main advantage of the present method lies in the lower computational cost and high accuracy. System of differential equations appear in various branches of science and technology. Results of current study show that the shifted Legendre reproducing kernel method is a reliable technique for the physical models in the system of differential equations form. Moreover, this method could be developed for systems of differential equations with fractional order derivatives or system of integro-differential equations.

Acknowledgments

The authors wish to express their thanks to the referees for comments which improved the paper.

Conflict of interest

The authors declare that they have no conflicts of interest.