Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions
We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the d...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2010/649486 |
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| author | Bashir Ahmad Juan J. Nieto |
| author_facet | Bashir Ahmad Juan J. Nieto |
| author_sort | Bashir Ahmad |
| collection | DOAJ |
| description | We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space. |
| format | Article |
| id | doaj-art-7e1bead91b374aaebd5d583fbb95abec |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-7e1bead91b374aaebd5d583fbb95abec2025-02-03T01:02:38ZengWileyInternational Journal of Differential Equations1687-96431687-96512010-01-01201010.1155/2010/649486649486Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary ConditionsBashir Ahmad0Juan J. Nieto1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, SpainWe study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.http://dx.doi.org/10.1155/2010/649486 |
| spellingShingle | Bashir Ahmad Juan J. Nieto Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions International Journal of Differential Equations |
| title | Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions |
| title_full | Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions |
| title_fullStr | Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions |
| title_full_unstemmed | Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions |
| title_short | Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions |
| title_sort | solvability of nonlinear langevin equation involving two fractional orders with dirichlet boundary conditions |
| url | http://dx.doi.org/10.1155/2010/649486 |
| work_keys_str_mv | AT bashirahmad solvabilityofnonlinearlangevinequationinvolvingtwofractionalorderswithdirichletboundaryconditions AT juanjnieto solvabilityofnonlinearlangevinequationinvolvingtwofractionalorderswithdirichletboundaryconditions |