Galerkin Projections and Finite Elements for Fractional Order Derivatives
Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) co...
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Gespeichert in:
| Autor*in: |
Singh, Satwinder Jit [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2006 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science + Business Media, Inc. 2006 |
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| Übergeordnetes Werk: |
Enthalten in: Nonlinear dynamics - Kluwer Academic Publishers, 1990, 45(2006), 1-2 vom: 21. Apr., Seite 183-206 |
|---|---|
| Übergeordnetes Werk: |
volume:45 ; year:2006 ; number:1-2 ; day:21 ; month:04 ; pages:183-206 |
| Links: |
|---|
| DOI / URN: |
10.1007/s11071-005-9002-z |
|---|
| Katalog-ID: |
OLC2051081018 |
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| 520 | |a Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. | ||
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10.1007/s11071-005-9002-z doi (DE-627)OLC2051081018 (DE-He213)s11071-005-9002-z-p DE-627 ger DE-627 rakwb eng 510 VZ 11 ssgn Singh, Satwinder Jit verfasserin aut Galerkin Projections and Finite Elements for Fractional Order Derivatives 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Fractional derivative Galerkin projection finite dimensional approximation Chatterjee, Anindya aut Enthalten in Nonlinear dynamics Kluwer Academic Publishers, 1990 45(2006), 1-2 vom: 21. Apr., Seite 183-206 (DE-627)130936782 (DE-600)1058624-6 (DE-576)034188126 0924-090X nnns volume:45 year:2006 number:1-2 day:21 month:04 pages:183-206 https://doi.org/10.1007/s11071-005-9002-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_23 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_4307 AR 45 2006 1-2 21 04 183-206 |
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10.1007/s11071-005-9002-z doi (DE-627)OLC2051081018 (DE-He213)s11071-005-9002-z-p DE-627 ger DE-627 rakwb eng 510 VZ 11 ssgn Singh, Satwinder Jit verfasserin aut Galerkin Projections and Finite Elements for Fractional Order Derivatives 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Fractional derivative Galerkin projection finite dimensional approximation Chatterjee, Anindya aut Enthalten in Nonlinear dynamics Kluwer Academic Publishers, 1990 45(2006), 1-2 vom: 21. Apr., Seite 183-206 (DE-627)130936782 (DE-600)1058624-6 (DE-576)034188126 0924-090X nnns volume:45 year:2006 number:1-2 day:21 month:04 pages:183-206 https://doi.org/10.1007/s11071-005-9002-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_23 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_4307 AR 45 2006 1-2 21 04 183-206 |
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10.1007/s11071-005-9002-z doi (DE-627)OLC2051081018 (DE-He213)s11071-005-9002-z-p DE-627 ger DE-627 rakwb eng 510 VZ 11 ssgn Singh, Satwinder Jit verfasserin aut Galerkin Projections and Finite Elements for Fractional Order Derivatives 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Fractional derivative Galerkin projection finite dimensional approximation Chatterjee, Anindya aut Enthalten in Nonlinear dynamics Kluwer Academic Publishers, 1990 45(2006), 1-2 vom: 21. Apr., Seite 183-206 (DE-627)130936782 (DE-600)1058624-6 (DE-576)034188126 0924-090X nnns volume:45 year:2006 number:1-2 day:21 month:04 pages:183-206 https://doi.org/10.1007/s11071-005-9002-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_23 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_4307 AR 45 2006 1-2 21 04 183-206 |
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10.1007/s11071-005-9002-z doi (DE-627)OLC2051081018 (DE-He213)s11071-005-9002-z-p DE-627 ger DE-627 rakwb eng 510 VZ 11 ssgn Singh, Satwinder Jit verfasserin aut Galerkin Projections and Finite Elements for Fractional Order Derivatives 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Fractional derivative Galerkin projection finite dimensional approximation Chatterjee, Anindya aut Enthalten in Nonlinear dynamics Kluwer Academic Publishers, 1990 45(2006), 1-2 vom: 21. Apr., Seite 183-206 (DE-627)130936782 (DE-600)1058624-6 (DE-576)034188126 0924-090X nnns volume:45 year:2006 number:1-2 day:21 month:04 pages:183-206 https://doi.org/10.1007/s11071-005-9002-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_23 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_4307 AR 45 2006 1-2 21 04 183-206 |
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10.1007/s11071-005-9002-z doi (DE-627)OLC2051081018 (DE-He213)s11071-005-9002-z-p DE-627 ger DE-627 rakwb eng 510 VZ 11 ssgn Singh, Satwinder Jit verfasserin aut Galerkin Projections and Finite Elements for Fractional Order Derivatives 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. Fractional derivative Galerkin projection finite dimensional approximation Chatterjee, Anindya aut Enthalten in Nonlinear dynamics Kluwer Academic Publishers, 1990 45(2006), 1-2 vom: 21. Apr., Seite 183-206 (DE-627)130936782 (DE-600)1058624-6 (DE-576)034188126 0924-090X nnns volume:45 year:2006 number:1-2 day:21 month:04 pages:183-206 https://doi.org/10.1007/s11071-005-9002-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-CHE SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_23 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2006 GBV_ILN_4307 AR 45 2006 1-2 21 04 183-206 |
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Galerkin Projections and Finite Elements for Fractional Order Derivatives |
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Singh, Satwinder Jit |
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Nonlinear dynamics |
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Nonlinear dynamics |
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eng |
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500 - Science |
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2006 |
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Singh, Satwinder Jit Chatterjee, Anindya |
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Singh, Satwinder Jit |
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10.1007/s11071-005-9002-z |
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510 |
| title_sort |
galerkin projections and finite elements for fractional order derivatives |
| title_auth |
Galerkin Projections and Finite Elements for Fractional Order Derivatives |
| abstract |
Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. © Springer Science + Business Media, Inc. 2006 |
| abstractGer |
Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. © Springer Science + Business Media, Inc. 2006 |
| abstract_unstemmed |
Abstract Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method. © Springer Science + Business Media, Inc. 2006 |
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Galerkin Projections and Finite Elements for Fractional Order Derivatives |
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Chatterjee, Anindya |
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