# Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

## Abstract

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

- Authors:

- Cairo University, Department of Engineering Mathematics, Faculty of Engineering (Egypt)

- Publication Date:

- OSTI Identifier:
- 22769310

- 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; ANALYTICAL SOLUTION; INTEGRO-DIFFERENTIAL EQUATIONS; ITERATIVE METHODS; VARIATIONAL METHODS; VOLTERRA INTEGRAL EQUATIONS

### Citation Formats

```
Deif, Sarah A., E-mail: sarah-deif@hotmail.com, E-mail: sdeif@cu.edu.eg, and Grace, Said R.
```*Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0464-7.

```
Deif, Sarah A., E-mail: sarah-deif@hotmail.com, E-mail: sdeif@cu.edu.eg, & Grace, Said R.
```*Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations*. United States. doi:10.1007/S40314-017-0464-7.

```
Deif, Sarah A., E-mail: sarah-deif@hotmail.com, E-mail: sdeif@cu.edu.eg, and Grace, Said R. Tue .
"Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations". United States. doi:10.1007/S40314-017-0464-7.
```

```
@article{osti_22769310,
```

title = {Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations},

author = {Deif, Sarah A., E-mail: sarah-deif@hotmail.com, E-mail: sdeif@cu.edu.eg and Grace, Said R.},

abstractNote = {The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.},

doi = {10.1007/S40314-017-0464-7},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}