# An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

## Abstract

Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.

- Authors:

- Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of)
- (Iran, Islamic Republic of)
- Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (Italy)

- Publication Date:

- OSTI Identifier:
- 22382188

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 283; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; BROWNIAN MOVEMENT; INTEGRAL EQUATIONS; INTEGRALS; MATRICES; MECHANICAL VIBRATIONS; NONLINEAR PROBLEMS; NUMERICAL SOLUTION; OSCILLATIONS; STOCHASTIC PROCESSES; TIME MEASUREMENT

### Citation Formats

```
Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Cattani, C., E-mail: ccattani@unisa.it, Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir, and The Laboratory of Quantum Information Processing, Yazd University, Yazd.
```*An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.11.042.

```
Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Cattani, C., E-mail: ccattani@unisa.it, Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir, & The Laboratory of Quantum Information Processing, Yazd University, Yazd.
```*An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics*. United States. doi:10.1016/J.JCP.2014.11.042.

```
Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir, The Laboratory of Quantum Information Processing, Yazd University, Yazd, Cattani, C., E-mail: ccattani@unisa.it, Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir, and The Laboratory of Quantum Information Processing, Yazd University, Yazd. Sun .
"An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics". United States.
doi:10.1016/J.JCP.2014.11.042.
```

```
@article{osti_22382188,
```

title = {An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics},

author = {Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir and The Laboratory of Quantum Information Processing, Yazd University, Yazd and Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir and The Laboratory of Quantum Information Processing, Yazd University, Yazd and Cattani, C., E-mail: ccattani@unisa.it and Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir and The Laboratory of Quantum Information Processing, Yazd University, Yazd},

abstractNote = {Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.},

doi = {10.1016/J.JCP.2014.11.042},

journal = {Journal of Computational Physics},

number = ,

volume = 283,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}