Fractional heat conduction models and their applications
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Jan Terpak
Abstract
This contribution deals with fractional heat conduction models and their applications. A brief historical overview of the authors who have dealt with the heat conduction equation is given in the introduction. The one-dimensional heat conduction models using integer- and fractional-order derivatives are listed. Numerical methods of solution of the heat conduction models using integer- and fractional-order derivatives for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions are described. In the case of numerical methods we deal with the finite-difference method using the Grünwald-Letnikov definition for the fractional time derivative. Implementation of these individual methods was realized in Matlab. A library of m-functions for the fractional heat conduction model has been created, namely the Time Fractional-Order Diffusion-Wave Equation Toolbox. The simulation examples using this toolbox are listed. At the end of the contribution applications are presented such as experimental verification of the methods for determining thermal diffusivity using the half-order derivative of the temperature by time.
Abstract
This contribution deals with fractional heat conduction models and their applications. A brief historical overview of the authors who have dealt with the heat conduction equation is given in the introduction. The one-dimensional heat conduction models using integer- and fractional-order derivatives are listed. Numerical methods of solution of the heat conduction models using integer- and fractional-order derivatives for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions are described. In the case of numerical methods we deal with the finite-difference method using the Grünwald-Letnikov definition for the fractional time derivative. Implementation of these individual methods was realized in Matlab. A library of m-functions for the fractional heat conduction model has been created, namely the Time Fractional-Order Diffusion-Wave Equation Toolbox. The simulation examples using this toolbox are listed. At the end of the contribution applications are presented such as experimental verification of the methods for determining thermal diffusivity using the half-order derivative of the temperature by time.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Fractional differential equations with bio-medical applications 1
- Fractional-order modeling of electro-impedance spectroscopy information 21
- Numerical solutions of singular time-fractional PDEs 43
- A multi-scale model of nociception pathways and pain mechanisms 55
- Variable-order derivatives and bone remodeling in the presence of metastases 69
- Skeletal muscle modeling by fractional multi-models: analysis of length effect 95
- Fractional calculus for modeling unconfined groundwater 119
- Fractional calculus models in dynamic problems of viscoelasticity 139
- Fractional calculus in structural mechanics 159
- Anomalous solute transport in complex media 193
- Application of variable-order fractional calculus in solid mechanics 207
- Fractional heat conduction models and their applications 225
- Index 247
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Fractional differential equations with bio-medical applications 1
- Fractional-order modeling of electro-impedance spectroscopy information 21
- Numerical solutions of singular time-fractional PDEs 43
- A multi-scale model of nociception pathways and pain mechanisms 55
- Variable-order derivatives and bone remodeling in the presence of metastases 69
- Skeletal muscle modeling by fractional multi-models: analysis of length effect 95
- Fractional calculus for modeling unconfined groundwater 119
- Fractional calculus models in dynamic problems of viscoelasticity 139
- Fractional calculus in structural mechanics 159
- Anomalous solute transport in complex media 193
- Application of variable-order fractional calculus in solid mechanics 207
- Fractional heat conduction models and their applications 225
- Index 247
