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CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series

CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series. Tomas Oppenheim MESA (Mechatronics, Embedded Systems and Automation) Lab School of Engineering, University of California, Merced E : toppenheim@ucmerced.edu Phone: 310-853-9234 Lab : CAS Eng 820 ( T : 228-4398).

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CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series

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  1. CTRW MATLAB ToolboxApplied Fractional Calculus Workshop Series Tomas Oppenheim MESA (Mechatronics, Embedded Systems and Automation)Lab School of Engineering, University of California, Merced E: toppenheim@ucmerced.eduPhone: 310-853-9234 Lab: CAS Eng 820 (T: 228-4398) June 16, 2014. Monday 4:00-6:00 PM Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

  2. Outline • Continuous Time Random Walk (CTRW) Background • CTRW MATLAB Basic Equations • 1D Forward Modeling: Generating Temporal/Spatial Profiles • 1D Inverse Problem: Fitting CTRW Solutions to Data Sets • Conclusions and Future Work AFC Workshop Series @ MESALAB @ UCMerced

  3. CTRW Background Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008). “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” PHYSICAL REVIEW E 84, 051139 (2011) AFC Workshop Series @ MESALAB @ UCMerced

  4. CTRW Background Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008). “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” Probability of the particle being in point x at instant t. AFC Workshop Series @ MESALAB @ UCMerced

  5. CTRW Background Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Porous Media Dye Flowing Through Porous Media Different velocities due to heterogeneities in Porous media Use CTRW to model flow of non-Fickian (anomalous) and Fickian (Gaussian) transport http://www.newcastle.edu.au/research-and-innovation/centre/cgmm/research/thermal-transport-in-composites-and-porous-media AFC Workshop Series @ MESALAB @ UCMerced

  6. CTRW Background Based on paper: Berkowitz et. al. Modeling Non-Fickoan Transport in Geoligical Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Gaussian or Anomalous Transport (PDF)? Hu Sheng, YangQuan Chen, TianShuangQiu. Fractional Processes and Fractional-Order Signal Processing, Techniques and Applications. Springer (2012) AFC Workshop Series @ MESALAB @ UCMerced

  7. CTRW Background Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Generalized Master Equation (GME): Continuous transitions of tracer transport Mean, ensemble-averaged, normalized concentration (e.g. of dye) Defined later AFC Workshop Series @ MESALAB @ UCMerced

  8. CTRW Background Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). CTRW Transport Equations: Using Laplace Transform, GME = CTRW Probability/time for a walker to just arrive at site s at time t Probability/time for a displacement s with a difference of arrival times of t AFC Workshop Series @ MESALAB @ UCMerced

  9. CTRW Background Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). CTRW Transport Equations: Using Laplace Transform, GME = CTRW Basic PDF. Determines nature of transport (Fickian or Non-Fickian) Correspondence between GME and LT of GME AFC Workshop Series @ MESALAB @ UCMerced

  10. CTRW MATLAB Basic Equations Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Laplace Transform of CTRW “heart” of CTRW formulation Transport velocity Generalized Diffusion Coefficient AFC Workshop Series @ MESALAB @ UCMerced

  11. CTRW MATLAB Basic Equations Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  12. 1D Forward Modeling: Generating Temporal Profiles The Options Variable: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  13. 1D Forward Modeling: Generating Temporal Profiles The Options Variable: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  14. 1D Forward Modeling: Generating Temporal Profiles The Options Variable: ? Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  15. 1D Forward Modeling: Generating Temporal Profiles The Options Variable: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  16. 1D Forward Modeling: Generating Temporal Profiles “A concentration measurement in the field or laboratory will be most closely related to either the resident or the flux concentration. The former describes all solute particles present at some point while the latter describessolute particles that have moved irrevocably past some point.” Zhang et al. Geophysical Research Letters. (33) 2006. AFC Workshop Series @ MESALAB @ UCMerced

  17. 1D Forward Modeling: Generating Temporal Profiles Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Assume: Porous Column, use ADE model Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  18. 1D Forward Modeling: Generating Temporal Profiles Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Matlab Commands: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  19. 1D Forward Modeling: Generating Temporal Profiles Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Matlab Commands: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  20. 1D Forward Modeling: Generating Temporal Profiles Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Resulting Temporal Profile: Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  21. Outline • Continuous Time Random Walk (CTRW) Background • CTRW MATLAB Basic Equations • 1D Forward Modeling: Generating Temporal/ Spatial Profiles • 1D Inverse Problem: Fitting CTRW Solutions to Data Sets • Conclusions and Future Work AFC Workshop Series @ MESALAB @ UCMerced

  22. Conclusions CTRW is good model for Fickian and Non-Fickian transport – “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” MATLAB toolbox available with detailed instruction manual to generate temporal/spatial concentration profiles and model data using either TPM, ADE, or ETA model Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  23. Future Work Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

  24. Future Work Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). AFC Workshop Series @ MESALAB @ UCMerced

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