# Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Inverse Nodal Problem

**Theorem**

**1.**

**Proof.**

**Algorithm**

**1.**

- (1)
- For each fixed$x\in [0,1]$, choose a sequence$\left\{{x}_{{n}_{k}}^{{j}_{{n}_{k}}}\right\}\subseteq {X}_{0}$such that${lim}_{\left|{n}_{k}\right|\to \infty}{x}_{{n}_{k}}^{{j}_{{n}_{k}}}=x$;
- (2)
- (3)
- (4)
- (5)

**Theorem**

**2.**

## 3. Numerical Solution of Inverse Nodal Problems

**Numerical solution:**for sufficiently large n, given the nodal points ${x}_{n}^{j},j=\overline{1,n}$ and constants h, a, and ${\int}_{0}^{1}q\left(t\right)\mathrm{d}t$, reconstruct the potential function $q\left(x\right)$. The solution $\phi (x,\rho )$ of (1) can be written as follows (see [10]):

**Theorem**

**3.**

- (1)
- Choose $k,M$. Set $n={2}^{k-1}M$.
- (2)
- Calculate the unknown vector
**C**by the following linear equation:$$\mathbf{A}\mathbf{C}=\mathbf{B},$$$$\mathbf{A}=\left(\begin{array}{ccccccccccc}{a}_{1,0}^{1}& {a}_{1,1}^{1}& \dots & {a}_{1,M-1}^{1}& {a}_{2,0}^{1}& \dots & {a}_{2,M-1}^{1}& \dots & {a}_{{2}^{k-1},0}^{1}& \dots & {a}_{{2}^{k-1},M-1}^{1}\\ {a}_{1,0}^{2}& {a}_{1,1}^{2}& \dots & {a}_{1,M-1}^{1}& {a}_{20}^{2}& \dots & {a}_{2,M-1}^{2}& \dots & {a}_{{2}^{k-1},0}^{2}& \dots & {a}_{{2}^{k-1},M-1}^{2}\\ \vdots & & & & \vdots & & & & & \vdots \\ {a}_{1,0}^{{2}^{k-1}M}& {a}_{1,1}^{{2}^{k-1}M}& \dots & {a}_{1,M-1}^{{2}^{k-1}M}& {a}_{2,0}^{{2}^{k-1}M}& \dots & {a}_{2,M-1}^{{2}^{k-1}M}& \dots & {a}_{{2}^{k-1},0}^{{2}^{k-1}M}& \dots & {a}_{{2}^{k-1},M-1}^{{2}^{k-1}M}\end{array}\right),$$$$\begin{array}{cc}& {a}_{l,m}^{j}={\int}_{0}^{{x}_{n}^{j}}{\psi}_{l,m}\left(t\right){cos}^{2}\left({\rho}_{n}^{0}\right)t\mathrm{d}t,\hfill \\ & l=\overline{1,{2}^{k-1}},\phantom{\rule{1.em}{0ex}}m=\overline{0,M-1},\phantom{\rule{1.em}{0ex}}n={2}^{k-1}M,\phantom{\rule{1.em}{0ex}}j=\overline{1,n},\hfill \end{array}$$$$\begin{array}{c}\mathbf{B}=\left(\begin{array}{c}-{\rho}_{n}^{0}cot\left({\rho}_{n}^{0}{x}_{n}^{1}\right)-h\\ -{\rho}_{n}^{0}cot\left({\rho}_{n}^{0}{x}_{n}^{2}\right)-h\\ \xb7\\ \xb7\\ \xb7\\ -{\rho}_{n}^{0}cot\left({\rho}_{n}^{0}{x}_{n}^{n}\right)-h\end{array}\right),\end{array}$$ - (3)
- To approximate $q\left({t}_{i}\right),i=\overline{1,{2}^{k-1}M}$, we use the following formula:$$\left[q\left({t}_{i}\right)\right]={\mathbf{C}}^{\mathrm{T}}\mathsf{\Phi},$$$${t}_{i}=\frac{2i-1}{{2}^{k}M},\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,{2}^{k-1}M,$$$$\mathsf{\Phi}=\left[\mathsf{\Psi}\left(\frac{1}{{2}^{k}M}\right),\mathsf{\Psi}\left(\frac{3}{{2}^{k}M}\right),\dots ,\mathsf{\Psi}\left(\frac{{2}^{k}M-1}{{2}^{k}M}\right)\right].$$

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

- (1)
- If $k=4$ and $M=5,7,9$ and ${x}_{n}^{j}$ satisfy the Formula (7), we find three approximation solutions of the potential function $q\left(x\right)$;
- (2)
- If $k=5,7$, $M=5$ and $k=6$, $M=10$ and ${x}_{n}^{j}$ satisfy the Formula (7), we find three approximation solutions of the potential function $q\left(x\right)$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numerical values of $q\left(x\right)$ and absolute errors between the approximation and exact solutions of $q\left(x\right)$ with k = 4 and M = 5, 7, 9 in Example 1.

**Figure 2.**Numerical values of $q\left(x\right)$ and absolute errors between the approximation and exact solutions of $q\left(x\right)$ with k = 4 and M = 5, 7, 9 in Example 2.

**Figure 3.**Numerical values of $q\left(x\right)$ and absolute errors between the approximation and exact solutions of $q\left(x\right)$ with $k=5,7$, $M=5$, and $k=6$, $M=10$ in Example 2.

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**MDPI and ACS Style**

Tang, Y.; Ni, H.; Song, F.; Wang, Y.
Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem. *Mathematics* **2022**, *10*, 4204.
https://doi.org/10.3390/math10224204

**AMA Style**

Tang Y, Ni H, Song F, Wang Y.
Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem. *Mathematics*. 2022; 10(22):4204.
https://doi.org/10.3390/math10224204

**Chicago/Turabian Style**

Tang, Yong, Haoze Ni, Fei Song, and Yuping Wang.
2022. "Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem" *Mathematics* 10, no. 22: 4204.
https://doi.org/10.3390/math10224204