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# The numerical solution of partial differential-algebraic equations

- Muhammed Yigider
^{1}Email author and - Ercan Çelik
^{2}

**2013**:8

https://doi.org/10.1186/1687-1847-2013-8

© Yigider and Çelik; licensee Springer. 2013

**Received:**1 October 2012**Accepted:**20 December 2012**Published:**10 January 2013

## Abstract

In this paper, a numerical solution of partial differential-algebraic equations (PDAEs) is considered by multivariate Padé approximations. We applied this method to an example. First, PDAE has been converted to power series by two-dimensional differential transformation, and then the numerical solution of the equation was put into a multivariate Padé series form. Thus, we obtained the numerical solution of PDAEs.

## Keywords

- partial differential-algebraic equation (PDAE)
- two-dimensional differential transformation
- multivariate Padé approximation

## 1 Introduction

where $t\in (0,{t}_{e})$ and $x\in (-l,l)\subset R$, $A,B,C\in {R}^{n,xn}$ are constant matrices, $u,f:[0,{t}_{e}]\times [-l,l]\to {R}^{n}$. We are interested in cases where at least one of the matrices, *A* or *B*, is singular. The two special cases $A=0$ or $B=0$ lead to ordinary differential equations or DAEs which are not considered here. Therefore, in this paper we assume that none of the matrices *A* or *B* is the zero matrix [1–3]. Many important mathematical models can be expressed in terms of PDAEs. Such models arise in many areas of mathematics, engineering, the physical sciences and population growth. In recent years, much research has been focused on the numerical solution of PDAEs [4, 5]. Some numerical methods have been developed using Runge-Kutta methods [6, 7]. The purpose of this paper is to consider the numerical solution of PDAEs by using multivariate Padé approximations.

## 2 Two-dimensional differential transformation

*T*-function and lower case and upper case letters represent the original and transformed functions respectively. The differential inverse transform of $W(k,h)$ is defined as

## 3 Multivariate Padé approximants

*j*th row in $p(x)$ and $q(x)$ by ${x}^{j+m-1}$ ($j=2,\dots ,n+1$) and afterwards divide the

*j*th column in $p(x)$ and $q(x)$ by ${x}^{j-1}$ ($j=2,\dots ,n+1$). This results in a multiplication of numerator and denominator by ${x}^{mn}$. Having done so, we get

if ($D=det{D}_{m,n}\ne 0$).

This quotient of determinants can also immediately be written down for a bivariate function $f(x,y)$. The sum ${\sum}_{i=0}^{k}{c}_{i}{x}^{i}$ will be replaced by the *k* th partial sum of the Taylor series development of $f(x,y)$ and the expression ${c}_{k}{x}^{k}$ by an expression that contains all the terms of degree *k* in $f(x,y)$. Here a bivariate term ${c}_{ij}{x}^{i}{y}^{j}$ is said to be of degree $i+j$.

## 4 Numerical example

## 5 Conclusions

**Comparison of the numerical solution of**
${\mathit{u}}_{\mathbf{1}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$
**with exact solutions (**
$\mathit{t}\mathbf{=}\mathbf{0.01}$
**)**

x | ${\mathit{u}}_{\mathbf{1}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | ${\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | $\mathbf{|}{\mathit{u}}_{\mathbf{1}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{-}{\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{|}$ |
---|---|---|---|

−0.5 | 0.2475124584 | 0.2475124584 | 0 |

−0.4 | 0.1584079734 | 0.1584079734 | 0 |

−0.3 | 0.08910448503 | 0.08910448502 | 1.10 |

−0.2 | 0.03960199335 | 0.03960199334 | 1.10 |

−0.1 | 0.009900498337 | 0.009900498336 | 1.10 |

0.1 | 0.009900498337 | 0.009900498336 | 1.10 |

0.2 | 0.03960199335 | 0.03960199334 | 1.10 |

0.3 | 0.08910448503 | 0.08910448502 | 1.10 |

0.4 | 0.1584079734 | 0.1584079734 | 0 |

0.5 | 0.2475124584 | 0.2475124584 | 0 |

**Comparison of the numerical solution of**
${\mathit{u}}_{\mathbf{2}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$
**with exact solutions (**
$\mathit{t}\mathbf{=}\mathbf{0.01}$
**)**

x | ${\mathit{u}}_{\mathbf{2}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | ${\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | $\mathbf{|}{\mathit{u}}_{\mathbf{2}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{-}{\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{|}$ |
---|---|---|---|

−0.5 | 0.2487531198 | 0.2487531198 | 0 |

−0.4 | 0.1592019967 | 0.1592019967 | 0 |

−0.3 | 0.08955112313 | 0.08955112314 | 1.10 |

−0.2 | 0.03980049917 | 0.03980049917 | 0 |

−0.1 | 0.009950124792 | 0.009950124793 | 1.10 |

0.1 | 0.009950124792 | 0.009950124793 | 1.10 |

0.2 | 0.03980049917 | 0.03980049917 | 0 |

0.3 | 0.08955112313 | 0.08955112314 | 1.10 |

0.4 | 0.1592019967 | 0.1592019967 | 0 |

0.5 | 0.2487531198 | 0.2487531198 | 0 |

**Comparison of the numerical solution of**
${\mathit{u}}_{\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$
**with exact solutions (**
$\mathit{t}\mathbf{=}\mathbf{0.01}$
**)**

x | ${\mathit{u}}_{\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | ${\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$ | $\mathbf{|}{\mathit{u}}_{\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{-}{\mathit{r}}_{\mathbf{4}\mathbf{,}\mathbf{3}}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{|}$ |
---|---|---|---|

−0.5 | 0.002499958334 | 0.002499958333 | 1.10 |

−0.4 | 0.001599973333 | 0.001599973333 | 0 |

−0.3 | 0.0008999850001 | 0.0008999850000 | 1.10 |

−0.2 | 0.0003999933334 | 0.0003999933333 | 1.10 |

−0.1 | 0.00009999833334 | 0.00009999833333 | 1.10 |

0.1 | 0.00009999833334 | 0.00009999833333 | 1.10 |

0.2 | 0.0003999933334 | 0.0003999933333 | 1.10 |

0.3 | 0.0008999850001 | 0.0008999850000 | 1.10 |

0.4 | 0.001599973333 | 0.001599973333 | 0 |

0.5 | 0.002499958334 | 0.002499958333 | 1.10 |

## Declarations

### Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript. This study was supported by The Scientific Research Projects of Atatürk University.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.