# A new computational approach for the solutions of generalized pantograph-delay differential equations

## Abstract

In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods.

- Authors:

- Ningbo University, Department of Mathematics, Faculty of Science (China)

- Publication Date:

- OSTI Identifier:
- 22769333

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; APPROXIMATIONS; COMPARATIVE EVALUATIONS; DIFFERENTIAL EQUATIONS; ERRORS; FUNCTIONS; LEAST SQUARE FIT

### Citation Formats

```
Xie, Lie-jun, Zhou, Cai-lian, and Xu, Song.
```*A new computational approach for the solutions of generalized pantograph-delay differential equations*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0418-0.

```
Xie, Lie-jun, Zhou, Cai-lian, & Xu, Song.
```*A new computational approach for the solutions of generalized pantograph-delay differential equations*. United States. doi:10.1007/S40314-017-0418-0.

```
Xie, Lie-jun, Zhou, Cai-lian, and Xu, Song. Tue .
"A new computational approach for the solutions of generalized pantograph-delay differential equations". United States. doi:10.1007/S40314-017-0418-0.
```

```
@article{osti_22769333,
```

title = {A new computational approach for the solutions of generalized pantograph-delay differential equations},

author = {Xie, Lie-jun and Zhou, Cai-lian and Xu, Song},

abstractNote = {In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods.},

doi = {10.1007/S40314-017-0418-0},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}