Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments Nov. 6th - 8th 2013, BCAM, Bilbao, Spain Fractional Spectral and Spectral Element Methods (Based on PhD thesis of Mohsen Zayernouri) George Em Karniadakis Division of Applied Mathematics, Brown University & Department of Mechanical Engineering, MIT & Pacific Northwest National Laboratory, CM4/DoE The CRUNCH group: www.cfm.brown.edu/crunch
Outline Fractional Sturm-Liouville Problem (FSLP): Singular & Regular, each of kind-i and kind-ii Fractional Ordinary Differential Equations (FODEs): PG Spectral Methods & Discontinuous Spectral/hp Element Method Discontinuous Petrov-Galerkin (DPG) for Fractional PDEs: Notion of exponentially-accurate time-integration Fractional Collocation Method (Linear/Nonlinear/Multi-Term FPDEs): A new class of fractional Lagrange interpolants A Unified Spectral Method for FPDEs: Hyperbolic, Parabolic, and Elliptic FPDEs in (1+d)-dimensions Conclusion
Singular Sturm-Liouville Problem (Integer-Order): Jacques François Sturm (1803-1855) Joseph Liouville (1809-1882) Local Operator Local Boundary Conditions, (Jacobi polynomials), (Quadratic)
Singular Fractional Sturm-Liouville problem: Global Operator Non-local Boundary Conditions i =1: SFSLP of Kind-I i =2: SFSLP of Kind-II
Singular Fractional Sturm-Liouville problem: Jacobi Polyfractonomials : Theorem: The exact eigenfunctions of SFSLP-I (i=1) and SFSLP-II (i=2) are: and correspondingly, the exact eigenvalues are
Orthogonality Properties: Recurrence relations Fractional derivatives
Recall: SFSLP-I: SFSLP-II: What if? Then, FSLP is no longer singular and it becomes Regular
Regular Fractional Sturm-Liouville problem (RFSLP): Regular Fractional Operator: for RFSLP-I for RFSLP-II
Theorem: The exact eigenfunctions of RFSLP-I (i=1) and RFSLP-II (i=2) are: Regular Fractional Sturm-Liouville problem (RFSLP): Jacobi Polyfractonomials : and correspondingly the exact eigenvalues are:
Regular Eigenvalues
Eigen-solutions of RFSLP-I Eigen-solutions of RFSLP-II The same number of zeros Sharp gradient near Dirichlet end
Approximation Properties of the Polyfractonomials
Fractional Ordinary Differential Equations (FODEs) Model Problem 1: Fractional Initial-Value Problem (RL derivative) Model Problem 2: Fractional Delay Equation (RL derivative)
Petrov-Galerkin Spectral Method (FODEs) BASIS functions: (eigenfunctions of FSLP -I) Taking ( ) Property: TEST functions: (eigenfunctions of FSLP -II) Property:
Petrov-Galerkin Spectral Method (FODEs) Model Problem 1: Fractional Integration-by-Parts Taking Diagonal Stiffness Matrix!
Exponential Decay of the L2-Error
Fractional Ordinary Differential Equations (FODEs) What if the solution is Not Smooth, or, it is Piecewise smooth? How to treat Long-Time Integration?
Discontinuous Spectral/hp Element Method (FODEs) Partitioning the time-domain into non-overlapping elements: Time BASIS: TEST: How to construct them?
Discontinuous Spectral/hp Element Method (FODEs) BASIS functions: TEST functions:
Discontinuous Spectral/hp Element Method (FODEs) Consider the general case: Taking
Discontinuous Spectral/hp Element Method (FODEs) Fractional Integration-by-Parts : History Load Diagonal stiffness Mass and the delay Mass matrices are exact
A=B=0, Non-Smooth Solution: h-ref More Effective than p-ref
CPU Time: PG Methods vs. FDM A=B=0
To fade or not to fade?
Fractional Spectral Collocation Method How to do even better?! Thinking of Employing Nodal rather than previously used Modal expansions Avoiding Quadrature costs (in the construction of the matrices) Efficient treatment of Multi-Term fractional FPDEs Efficient treatment of Nonlinear FPDEs
Fractional Spectral Collocation Method Nodal Expansion: : Fractional Lagrange Interpolants at some arbitrary interpolation/collocation points: We note the difference between and Polynomial/Fourier interpolants!
Fractional Spectral Collocation Method Differentiation Matrices, : Linear Systems
II) Space-Fractional Nonlinear Burgers Equation:
A Unified Spectral Method for FPDEs A Unified Spectral Method, which treats any (1+d)-dimensional FPDE (i.e., hyperbolic, parabolic, and elliptic) with the same ease. Time-Space Hypercube and To also develop a Unified Fast Solver for this general scheme!
A Unified Spectral Method for FPDEs Basis Space: Based on Polyfractonomials of FSLP-I
A Unified Spectral Method for FPDEs Test Space: Based on Polyfractonomials of FSLP-II
A Unified Spectral Method for FPDEs General Lyapunov System
A Unified Spectral Method for FPDEs This choice of basis/test functions leads to Diagonal Temporal Stiffness Diagonal Spatial Stiffness Tridiagonal Spatial Stiffness when when Symmetric Temporal/Spatial Mass Matrices and
A Unified Spectral Method for FPDEs Theorem. A closed-form solution for the multi-dimensional matrix of unknown coefficient matrix is explicitly given as Remark. The fractional order is usually the same in all dimensions, so the eigenvalue problem is usually solved only once for all!
Hyperbolic FPDEs: A Unified Spectral Method for FPDEs
Parabolic FPDEs: A Unified Spectral Method for FPDEs
Concluding Remarks Exact non-polynomial eigenfunctions of the singular/regular fractional Sturm-Liouville problems were obtained and introduced as new BASIS and TEST functions. A Petrov-Galerkin spectral method and a discontinuous spectral element method were developed for FODEs (FIVPs and Delay Problems) An exponentially accurate DPG method for FPDES, also for time-integration method was proposed for first-order in time operators was developed. A fractional spectral collocation method was developed for efficient treatment of multi-term and nonlinear FPDEs A unified spectral method along with a general fast linear solver was developed for FPDEs including hyperbolic, parabolic and elliptic problems Less memory storage and CPU time in the proposed schemes Exponential decay of error was obtained in several test-cases for FODEs and FPDEs.
Special Issue of Journal of Computational Physics on FPDEs 20 papers Submission: April 30 th, 2014 Production: August 30 th, 2014
Publications 1) M. Zayernouri and G.E. Karniadakis, Fractional Sturm-Liouville Eigen-Problems: Theory and Numerical Approximation, J. Comput. Phys. vol. 252, (2013), Pages 495 517 2) M. Zayernouri and G.E. Karniadakis, Exponentially Accurate Spectral and Spectral Element Methods for Fractional ODEs, J. Comput. Phys. vol. 257-Part A (2014), Pages 460-480. 3) M. Zayernouri and G.E. Karniadakis, Fractional Spectral Collocation Method, SIAM J. Scientific Computing, (2013), Accepted. 4) M. Zayernouri, W. Cao, and G.E. Karniadakis, Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations, Submitted to SIAM J. Scientific Computing, (2013). 5) M. Zayernouri, I. Alvey, and G.E. Karniadakis, Discontinuous Petrov-Galerkin (DPG) Methods for Time- & Space-Fractional Advection Equation, Submitted to SIAM J. Scientific Computing, (2013). 6) M. Zayernouri, M. Ainsworth, and G.E. Karniadakis, A Unified Petrov-Galerkin Spectral Method for Fractional PDEs, to be Submitted to CMAME, (2013).
I) Time- and Space-Fractional Multi-Term FPDEs : Lyapunov equation: