# Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Sinc-Approximation Formula

**Play-Wiener**class, denoted by $\mathbf{W}(\pi /h)$ which is the family of all analytic functions that are in $\mathbb{C}$, and satisfy some decaying conditions. For a function f and its derivative ${f}^{\prime}$ to be in the class $\mathbf{W}(\pi /h)$, the $kth$ derivative of the function f can be approximated by

## 3. The Sinc–Galerkin Method: Balance Laws

**Theorem**

**1**

**.**Let the matrix U be as defined in (15). Then for the number of points in both space and time ${N}_{x},{N}_{t}$ to be bigger than some constant that depends on $d,\alpha $, we have

**Theorem**

**2**

**.**For any positive constant R, we can find another positive constant ${T}_{0}$ so that whenever $\parallel {U}_{1}-{U}_{0}\parallel <\frac{R}{2}$, then the solution of (15) is unique. Also, the iteration system

**Definition**

**1.**

**Theorem**

**3.**

**Proof.**

#### Treatment of Non-Zero Boundary Conditions

## 4. The Adomian Decomposition Method (ADM)

#### Convergence of the ADM Approximation

**Definition**

**2.**

## 5. Applications: Riemann Type

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{N}}_{\mathit{x}}={\mathit{N}}_{\mathit{t}}$ | ${\Vert \mathit{u}\Vert}_{\mathbf{\infty}}$ | ${\Vert \mathit{v}\Vert}_{\mathbf{\infty}}$ |
---|---|---|

4 | $8.5450\times {10}^{-03}$ | $4.1314\times {10}^{-02}$ |

8 | $5.7183\times {10}^{-03}$ | $1.6575\times {10}^{-02}$ |

16 | $1.3617\times {10}^{-03}$ | $3.5789\times {10}^{-03}$ |

32 | $8.8632\times {10}^{-05}$ | $2.4625\times {10}^{-04}$ |

64 | $5.3823\times {10}^{-06}$ | $9.6498\times {10}^{-05}$ |

Iteration No. | $\mathit{u}(\mathit{x},\mathit{t})$ | $\mathit{v}(\mathit{x},\mathit{t})$ |
---|---|---|

First | $0.999753$ | $0.289434$ |

Second | $0.937269$ | $0.284561$ |

Third | $0.941174$ | $0.281119$ |

Fourth | $0.904930$ | $0.273621$ |

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Al-Khaled, K.; Rababah, N.M.
Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions. *Symmetry* **2020**, *12*, 757.
https://doi.org/10.3390/sym12050757

**AMA Style**

Al-Khaled K, Rababah NM.
Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions. *Symmetry*. 2020; 12(5):757.
https://doi.org/10.3390/sym12050757

**Chicago/Turabian Style**

Al-Khaled, Kamel, and Nid’a M. Rababah.
2020. "Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions" *Symmetry* 12, no. 5: 757.
https://doi.org/10.3390/sym12050757