A Survey of Numerical Methods in Fractional Calculus
|
|
- Juliet Palmer
- 5 years ago
- Views:
Transcription
1 A Survey of Numerical Methods in Fractional Calculus Fractional Derivatives in Mechanics: State of the Art CNAM Paris, 17 November 2006 Kai Diethelm Gesellschaft für numerische Simulation mbh, Braunschweig Institut Computational Mathematics, TU Braunschweig K. Diethelm, Numerical Methods in Fractional Calculus p. 1/21
2 Outline The basic problem Grünwald-Letnikov methods Lubich s fractional linear multistep methods Methods based on quadrature theory Examples Extensions K. Diethelm, Numerical Methods in Fractional Calculus p. 2/21
3 The Basic Problem Find a numerical solution for the fractional differential equation D α y(x) = f(x, y(x)) with appropriate initial condition(s), where D α is either the Riemann-Liouville derivative or the Caputo derivative. K. Diethelm, Numerical Methods in Fractional Calculus p. 3/21
4 Fundamental Definitions Riemann-Liouville integral: J α f(x) = 1 Γ(α) x 0 (x t) α 1 f(t)dt K. Diethelm, Numerical Methods in Fractional Calculus p. 4/21
5 Fundamental Definitions Riemann-Liouville integral: J α f(x) = 1 Γ(α) x 0 (x t) α 1 f(t)dt Riemann-Liouville derivative: D α f(x) = D n J n α f(x), n= α K. Diethelm, Numerical Methods in Fractional Calculus p. 4/21
6 Fundamental Definitions Riemann-Liouville integral: J α f(x) = 1 Γ(α) x 0 (x t) α 1 f(t)dt Riemann-Liouville derivative: D α f(x) = D n J n α f(x), n= α Caputo derivative: D α f(x) = J n α D n f(x), n= α K. Diethelm, Numerical Methods in Fractional Calculus p. 4/21
7 Fundamental Definitions Riemann-Liouville integral: J α f(x) = 1 Γ(α) x 0 (x t) α 1 f(t)dt Riemann-Liouville derivative: D α f(x) = D n J n α f(x), n= α natural initial condition for DE: D α k y(0) = y (k) 0, k =1, 2,..., n Caputo derivative: D α f(x) = J n α D n f(x), n= α K. Diethelm, Numerical Methods in Fractional Calculus p. 4/21
8 Fundamental Definitions Riemann-Liouville integral: J α f(x) = 1 Γ(α) x 0 (x t) α 1 f(t)dt Riemann-Liouville derivative: D α f(x) = D n J n α f(x), n= α natural initial condition for DE: D α k y(0) = y (k) 0, k =1, 2,..., n Caputo derivative: D α f(x) = J n α D n f(x), n= α natural initial condition for DE: D k y(0) = y (k) 0, k =0, 1,..., n 1 K. Diethelm, Numerical Methods in Fractional Calculus p. 4/21
9 Grünwald-Letnikov Methods Integer-order derivatives: D n 1 y(x) = lim h 0 h n k=0 ( 1) k Γ(n + 1) y(x kh) (n N) Γ(k + 1)Γ(n k + 1) K. Diethelm, Numerical Methods in Fractional Calculus p. 5/21
10 Grünwald-Letnikov Methods Integer-order derivatives: D n 1 y(x) = lim h 0 h n k=0 ( 1) k Γ(n + 1) y(x kh) (n N) Γ(k + 1)Γ(n k + 1) Fractional extension (Liouville 1832, Grünwald 1867, Letnikov 1868): DGLy(x) α 1 ( 1) k Γ(α + 1) = lim y(x kh) (α > 0) h 0 h α Γ(k + 1)Γ(α k + 1) k=0 K. Diethelm, Numerical Methods in Fractional Calculus p. 5/21
11 Grünwald-Letnikov Methods Integer-order derivatives: D n 1 y(x) = lim h 0 h n k=0 ( 1) k Γ(n + 1) y(x kh) (n N) Γ(k + 1)Γ(n k + 1) Fractional extension (Liouville 1832, Grünwald 1867, Letnikov 1868): x/h DGLy(x) α 1 ( 1) k Γ(α + 1) = lim y(x kh) (α > 0) h 0 h α Γ(k + 1)Γ(α k + 1) k=0 y defined only on [0, X]; thus assume y(x) = 0 for x < 0. K. Diethelm, Numerical Methods in Fractional Calculus p. 5/21
12 Grünwald-Letnikov Methods Integer-order derivatives: D n 1 y(x) = lim h 0 h n k=0 ( 1) k Γ(n + 1) y(x kh) (n N) Γ(k + 1)Γ(n k + 1) Fractional extension (Liouville 1832, Grünwald 1867, Letnikov 1868): x/h DGLy(x) α 1 ( 1) k Γ(α + 1) = lim y(x kh) (α > 0) h 0 h α Γ(k + 1)Γ(α k + 1) k=0 y defined only on [0, X]; thus assume y(x) = 0 for x < 0. D α GL y(x) = Dα y(x) if y is smooth. K. Diethelm, Numerical Methods in Fractional Calculus p. 5/21
13 Grünwald-Letnikov Methods Exact analytical expression: DGLy(x) α 1 = lim h 0 h α x/h k=0 ( 1) k Γ(α + 1) y(x kh) Γ(k + 1)Γ(α k + 1) K. Diethelm, Numerical Methods in Fractional Calculus p. 6/21
14 Grünwald-Letnikov Methods Numerical approximation: hd α GLy(x) = 1 h α x/h k=0 ( 1) k Γ(α + 1) y(x kh) Γ(k + 1)Γ(α k + 1) for some h > 0. K. Diethelm, Numerical Methods in Fractional Calculus p. 6/21
15 Grünwald-Letnikov Methods Numerical approximation: hd α GLy(x) = 1 h α x/h k=0 ( 1) k Γ(α + 1) y(x kh) Γ(k + 1)Γ(α k + 1) for some h > 0. Error = O(h) + O(f(0)) if f is smooth, i.e. slow convergence if f(0) = 0, completely unfeasible if f(0) 0. K. Diethelm, Numerical Methods in Fractional Calculus p. 6/21
16 Fractional Linear Multistep Methods Generalization of classical linear multistep methods for first-order ODEs (Lubich ) General form: y m = α 1 j=0 y (j) 0 x j m j! + h α m j=0 ω m j f(x j, y j ) + h α s j=0 w m,j f(x j, y j ) K. Diethelm, Numerical Methods in Fractional Calculus p. 7/21
17 Fractional Linear Multistep Methods Generalization of classical linear multistep methods for first-order ODEs (Lubich ) General form: y m = α 1 j=0 y (j) 0 x j m j! + h α m j=0 ω m j f(x j, y j ) + h α s j=0 w m,j f(x j, y j ) initial condition convolution weights starting weights K. Diethelm, Numerical Methods in Fractional Calculus p. 7/21
18 Fractional Linear Multistep Methods Generalization of classical linear multistep methods for first-order ODEs (Lubich ) General form: y m = α 1 j=0 y (j) 0 x j m j! + h α m j=0 ω m j f(x j, y j ) + h α s j=0 w m,j f(x j, y j ) initial condition convolution weights starting weights Computation of convolution weights directly from coefficients of underlying classical linear multistep method K. Diethelm, Numerical Methods in Fractional Calculus p. 7/21
19 Fractional Linear Multistep Methods Computation of starting weights: Linear system of equations K. Diethelm, Numerical Methods in Fractional Calculus p. 8/21
20 Fractional Linear Multistep Methods Computation of starting weights: Linear system of equations Properties of linear system depend strongly on α: K. Diethelm, Numerical Methods in Fractional Calculus p. 8/21
21 Fractional Linear Multistep Methods Computation of starting weights: Linear system of equations Properties of linear system depend strongly on α: Assume underlying classical method to be of order p, and let A = {γ = j + kα : j, k = 0, 1,..., γ p 1}. Then, error of resulting fractional method = O(h p ), dimension of linear system = cardinality of A, condition of linear system depends on distribution of elements of A. K. Diethelm, Numerical Methods in Fractional Calculus p. 8/21
22 Fractional Linear Multistep Methods Further comments on linear system for starting weights: needs to be solved at every grid point, coefficient matrix always identical, coefficient matrix has generalized Vandermonde structure, right-hand sides change from one grid point to another, accuracy of numerical computation of right-hand side suffers from cancellation of digits. K. Diethelm, Numerical Methods in Fractional Calculus p. 9/21
23 Fractional Linear Multistep Methods A well-behaved example: p = 4, α = 1/3 A = { 0, 1, 2, 1, 4, 5,..., 3} card(a) = 10, highly regular spacing of elements of A (equispaced) Matrix of coefficients can be transformed to usual Vandermonde form mildly ill conditioned system special algorithms can be used for sufficiently accurate solution (Björck & Pereyra 1970) K. Diethelm, Numerical Methods in Fractional Calculus p. 10/21
24 Fractional Linear Multistep Methods A badly behaved example: p = 4, α = 0.33 A = {0, 0.33, 0.66, 0.99, 1, 1.32, 1.33,..., 2.97, 2.98, 2.99, 3} card(a) = 22, spacing of the elements of A is highly irregular (in particular, some elements are very close together) severely ill conditioned system no algorithms for sufficiently accurate solution known (Diethelm, Ford, Ford & Weilbeer 2006) K. Diethelm, Numerical Methods in Fractional Calculus p. 11/21
25 Quadrature-Based Methods Direct approach: Use representation D α 1 x y(x) = (x t) α 1 y(t)dt Γ( α) 0 (finite-part integral) in differential equation Discretize integral by classical quadrature techniques Example: Replace y in integrand by piecewise linear interpolant with step size h (Diethelm 1997) product trapezoidal method; Error = O(h 2 α ) Similar concept applicable to Caputo derivatives K. Diethelm, Numerical Methods in Fractional Calculus p. 12/21
26 Quadrature-Based Methods Indirect approach: (Diethelm, Ford & Freed ) Rewrite Caputo initial value problem as integral equation α 1 y(x) = y (k) x k 0 k! + J α [f(, y( ))](x) k=0 Apply product integration idea to J α Example 1: Product rectangle quadrature fractional Adams-Bashforth method explicit method Error = O(h) K. Diethelm, Numerical Methods in Fractional Calculus p. 13/21
27 Quadrature-Based Methods Indirect approach (continued): Example 2: Product trapezoidal quadrature fractional Adams-Moulton method implicit method Error = O(h 2 ) P(EC) m E scheme: Combine Adams-Bashforth predictor and m Adams-Moulton correctors Error = O(h p ), p = min{2, 1 + mα} Similar concept applicable to Riemann-Liouville derivatives K. Diethelm, Numerical Methods in Fractional Calculus p. 14/21
28 Quadrature-Based Methods Common properties: Coefficients can be computed in accurate and stable way without problems Method is applicable for any type of differential equation Reasonable rate of convergence can be achieved K. Diethelm, Numerical Methods in Fractional Calculus p. 15/21
29 Example 1 D α y(x) = x Γ(5 α) x4 α y(x), y(0) = 1 exact solution on [0, 2]: y(x) = x Error at x=2 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 1e-10 α = K. Diethelm, Numerical Methods in Fractional Calculus p. 16/21
30 Example 1 D α y(x) = x Γ(5 α) x4 α y(x), y(0) = 1 exact solution on [0, 2]: y(x) = x Error at x=2 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 1e-10 α = K. Diethelm, Numerical Methods in Fractional Calculus p. 16/21
31 Example 1 D α y(x) = x Γ(5 α) x4 α y(x), y(0) = 1 exact solution on [0, 2]: y(x) = x Error at x=2 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 1e-10 α = K. Diethelm, Numerical Methods in Fractional Calculus p. 16/21
32 Example 2 D α y(x) = Γ(5 + α/2) Γ(9 α) x8 α 3 Γ(5 α/2) x4 α/2 + 9 Γ(α + 1) 4 + ( 1.5x α/2 x 4) 3 y(x) 3/2, y(0) = 0 exact solution on [0, 1]: y(x) = x 8 3x 4+α/ x α Error at x=1 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 17/21
33 Example 2 D α y(x) = Γ(5 + α/2) Γ(9 α) x8 α 3 Γ(5 α/2) x4 α/2 + 9 Γ(α + 1) 4 + ( 1.5x α/2 x 4) 3 y(x) 3/2, y(0) = 0 exact solution on [0, 1]: y(x) = x 8 3x 4+α/ x α Error at x=1 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 17/21
34 Example 2 D α y(x) = Γ(5 + α/2) Γ(9 α) x8 α 3 Γ(5 α/2) x4 α/2 + 9 Γ(α + 1) 4 + ( 1.5x α/2 x 4) 3 y(x) 3/2, y(0) = 0 exact solution on [0, 1]: y(x) = x 8 3x 4+α/ x α Error at x=1 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 17/21
35 Example 3 D α y(x) = x 1 α E 1,2 α ( x) + exp( 2x) y(x) 2, y(0) = 1 exact solution on [0, 5]: y(x) = exp( x) Error at x=5 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 18/21
36 Example 3 D α y(x) = x 1 α E 1,2 α ( x) + exp( 2x) y(x) 2, y(0) = 1 exact solution on [0, 5]: y(x) = exp( x) Error at x=5 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 18/21
37 Example 3 D α y(x) = x 1 α E 1,2 α ( x) + exp( 2x) y(x) 2, y(0) = 1 exact solution on [0, 5]: y(x) = exp( x) Error at x=5 vs. no. of nodes Grünwald-Letnikov direct quadrature indirect quadrature multistep (BDF4) 0,01 0,0001 1e-06 1e-08 α = e K. Diethelm, Numerical Methods in Fractional Calculus p. 18/21
38 Summary Grünwald-Letnikov: reliable but slowly convergent Linear multistep methods: very bad for some values of α, very good for others Quadrature-based methods: useful compromise, in particular because of simple speed-up possibility All types of basic routines available in GNS Numerical Fractional Calculus Library (presently FORTRAN77) K. Diethelm, Numerical Methods in Fractional Calculus p. 19/21
39 Extensions Application of methods to time-/space-fractional partial differential equations, fractional optimal control problems. Common problem: High arithmetic complexity due to non-locality of fractional operators. Solution for quadrature methods: Nested mesh concept (Ford & Simpson 2001). K. Diethelm, Numerical Methods in Fractional Calculus p. 20/21
40 Merci pour votre attention! contact: K. Diethelm, Numerical Methods in Fractional Calculus p. 21/21
Numerical Fractional Calculus Using Methods Based on Non-Uniform Step Sizes. Kai Diethelm. International Symposium on Fractional PDEs June 3 5, 2013
Numerical Fractional Calculus Using Methods Based on Non-Uniform Step Sizes Gesellschaft für numerische Simulation mbh Braunschweig AG Numerik Institut Computational Mathematics Technische Universität
More informationSMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS. Kai Diethelm. Abstract
SMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS Kai Diethelm Abstract Dedicated to Prof. Michele Caputo on the occasion of his 8th birthday We consider ordinary fractional
More informationFourth Order RK-Method
Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationNumerical solution of the Bagley Torvik equation. Kai Diethelm & Neville J. Ford
ISSN 1360-1725 UMIST Numerical solution of the Bagley Torvik equation Kai Diethelm & Neville J. Ford Numerical Analysis Report No. 378 A report in association with Chester College Manchester Centre for
More informationHigher order numerical methods for solving fractional differential equations
Noname manuscript No. will be inserted by the editor Higher order numerical methods for solving fractional differential equations Yubin Yan Kamal Pal Neville J Ford Received: date / Accepted: date Abstract
More informationSolving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations
Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Outline ODEs and initial conditions. Explicit and implicit Euler methods. Runge-Kutta methods. Multistep
More informationReview Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationarxiv: v2 [math.ca] 13 Oct 2015
arxiv:1510.03285v2 [math.ca] 13 Oct 2015 MONOTONICITY OF FUNCTIONS AND SIGN CHANGES OF THEIR CAPUTO DERIVATIVES Kai Diethelm 1,2 Abstract It is well known that a continuously differentiable function is
More informationInitial-Value Problems for ODEs. Introduction to Linear Multistep Methods
Initial-Value Problems for ODEs Introduction to Linear Multistep Methods Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationAN EFFICIENT PARALLEL ALGORITHM FOR THE NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS. Kai Diethelm 1,2
RESEARCH PAPER AN EFFICIENT PARALLEL ALGORITHM FOR THE NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS Kai Diethelm 1,2 Abstract The numerical solution of differential equations of fractional order
More informationJim Lambers MAT 772 Fall Semester Lecture 21 Notes
Jim Lambers MAT 772 Fall Semester 21-11 Lecture 21 Notes These notes correspond to Sections 12.6, 12.7 and 12.8 in the text. Multistep Methods All of the numerical methods that we have developed for solving
More informationA computationally effective predictor-corrector method for simulating fractional order dynamical control system
ANZIAM J. 47 (EMA25) pp.168 184, 26 168 A computationally effective predictor-corrector method for simulating fractional order dynamical control system. Yang F. Liu (Received 14 October 25; revised 24
More informationComparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations
Comparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations Farhad Farokhi, Mohammad Haeri, and Mohammad Saleh Tavazoei Abstract This paper is a result of comparison of some
More information9.6 Predictor-Corrector Methods
SEC. 9.6 PREDICTOR-CORRECTOR METHODS 505 Adams-Bashforth-Moulton Method 9.6 Predictor-Corrector Methods The methods of Euler, Heun, Taylor, and Runge-Kutta are called single-step methods because they use
More informationModule 4: Numerical Methods for ODE. Michael Bader. Winter 2007/2008
Outlines Module 4: for ODE Part I: Basic Part II: Advanced Lehrstuhl Informatik V Winter 2007/2008 Part I: Basic 1 Direction Fields 2 Euler s Method Outlines Part I: Basic Part II: Advanced 3 Discretized
More informationCHAPTER 5: Linear Multistep Methods
CHAPTER 5: Linear Multistep Methods Multistep: use information from many steps Higher order possible with fewer function evaluations than with RK. Convenient error estimates. Changing stepsize or order
More informationThis work has been submitted to ChesterRep the University of Chester s online research repository.
This work has been submitted to ChesterRep the University of Chester s online research repository http://chesterrep.openrepository.com Author(s): Kai Diethelm; Neville J Ford Title: Volterra integral equations
More informationmultistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):
MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25.
Logistics Week 12: Monday, Apr 18 HW 6 is due at 11:59 tonight. HW 7 is posted, and will be due in class on 4/25. The prelim is graded. An analysis and rubric are on CMS. Problem du jour For implicit methods
More informationCS 450 Numerical Analysis. Chapter 8: Numerical Integration and Differentiation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationWe are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors
We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 4, 6, 2M Open access books available International authors and editors Downloads Our authors are
More informationHigher Order Numerical Methods for Fractional Order Differential Equations
Higher Order Numerical Methods for Fractional Order Differential Equations Item Type Thesis or dissertation Authors Pal, Kamal Citation Pal, K. K., (5. Higher order numerical methods for fractional order
More informationIntegration, differentiation, and root finding. Phys 420/580 Lecture 7
Integration, differentiation, and root finding Phys 420/580 Lecture 7 Numerical integration Compute an approximation to the definite integral I = b Find area under the curve in the interval Trapezoid Rule:
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More information16.7 Multistep, Multivalue, and Predictor-Corrector Methods
740 Chapter 16. Integration of Ordinary Differential Equations 16.7 Multistep, Multivalue, and Predictor-Corrector Methods The terms multistepand multivaluedescribe two different ways of implementing essentially
More informationScientific Computing: Numerical Integration
Scientific Computing: Numerical Integration Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Fall 2015 Nov 5th, 2015 A. Donev (Courant Institute) Lecture
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Multistep Methods Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 1. Runge Kutta methods 2. Embedded RK methods
More informationNumerical Methods - Initial Value Problems for ODEs
Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationLinear Multistep Methods I: Adams and BDF Methods
Linear Multistep Methods I: Adams and BDF Methods Varun Shankar January 1, 016 1 Introduction In our review of 5610 material, we have discussed polynomial interpolation and its application to generating
More informationCOSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods
COSC 336 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods Fall 2005 Repetition from the last lecture (I) Initial value problems: dy = f ( t, y) dt y ( a) = y 0 a t b Goal:
More informationOn the fractional-order logistic equation
Applied Mathematics Letters 20 (2007) 817 823 www.elsevier.com/locate/aml On the fractional-order logistic equation A.M.A. El-Sayed a, A.E.M. El-Mesiry b, H.A.A. El-Saka b, a Faculty of Science, Alexandria
More informationOn the Fractional Signals and Systems
On the Fractional Signals and Systems (Invited paper) Richard Magin 1, Manuel D. Ortigueira (member Eurasip) 2, Igor Podlubny 3, and Juan Trujillo 4 1 University of Illinois at Chicago, Bioengineering
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
More information16.7 Multistep, Multivalue, and Predictor-Corrector Methods
16.7 Multistep, Multivalue, and Predictor-Corrector Methods 747 } free_vector(ysav,1,nv); free_vector(yerr,1,nv); free_vector(x,1,kmaxx); free_vector(err,1,kmaxx); free_matrix(dfdy,1,nv,1,nv); free_vector(dfdx,1,nv);
More informationHIGHER ORDER METHODS. There are two principal means to derive higher order methods. b j f(x n j,y n j )
HIGHER ORDER METHODS There are two principal means to derive higher order methods y n+1 = p j=0 a j y n j + h p j= 1 b j f(x n j,y n j ) (a) Method of Undetermined Coefficients (b) Numerical Integration
More informationMTH 452/552 Homework 3
MTH 452/552 Homework 3 Do either 1 or 2. 1. (40 points) [Use of ode113 and ode45] This problem can be solved by a modifying the m-files odesample.m and odesampletest.m available from the author s webpage.
More informationOn The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions
On The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions Xiong Wang Center of Chaos and Complex Network, Department of Electronic Engineering, City University of
More informationSolving Ordinary Differential equations
Solving Ordinary Differential equations Taylor methods can be used to build explicit methods with higher order of convergence than Euler s method. The main difficult of these methods is the computation
More informationNUMERICAL MATHEMATICS AND COMPUTING
NUMERICAL MATHEMATICS AND COMPUTING Fourth Edition Ward Cheney David Kincaid The University of Texas at Austin 9 Brooks/Cole Publishing Company I(T)P An International Thomson Publishing Company Pacific
More information5.6 Multistep Methods
5.6 Multistep Methods 1 Motivation: Consider IVP: yy = ff(tt, yy), aa tt bb, yy(aa) = αα. To compute solution at tt ii+1, approximate solutions at mesh points tt 0, tt 1, tt 2, tt ii are already obtained.
More informationApplied Numerical Analysis
Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458 Contents Preface xi 1 Foundations 1 1.1 Introductory
More informationECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations
ECE257 Numerical Methods and Scientific Computing Ordinary Differential Equations Today s s class: Stiffness Multistep Methods Stiff Equations Stiffness occurs in a problem where two or more independent
More informationLecture 10: Linear Multistep Methods (LMMs)
Lecture 10: Linear Multistep Methods (LMMs) 2nd-order Adams-Bashforth Method The approximation for the 2nd-order Adams-Bashforth method is given by equation (10.10) in the lecture note for week 10, as
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationPart IB Numerical Analysis
Part IB Numerical Analysis Definitions Based on lectures by G. Moore Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationMultistep Methods for IVPs. t 0 < t < T
Multistep Methods for IVPs We are still considering the IVP dy dt = f(t,y) t 0 < t < T y(t 0 ) = y 0 So far we have looked at Euler s method, which was a first order method and Runge Kutta (RK) methods
More informationChapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods
Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Matematics National Institute of Tecnology Durgapur Durgapur-7109 email: anita.buie@gmail.com 1 . Capter 8 Numerical Solution of Ordinary
More informationOrdinary Differential Equations
CHAPTER 8 Ordinary Differential Equations 8.1. Introduction My section 8.1 will cover the material in sections 8.1 and 8.2 in the book. Read the book sections on your own. I don t like the order of things
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationHigh Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional Differential Equation
International Symposium on Fractional PDEs: Theory, Numerics and Applications June 3-5, 013, Salve Regina University High Order Numerical Methods for the Riesz Derivatives and the Space Riesz Fractional
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. One-step methods (Taylor series, Runge-Kutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. One-step methods (Taylor series, Runge-Kutta) 2. Multistep methods (Predictor-Corrector, Adams methods) Both of these types of methods are
More information8.1 Introduction. Consider the initial value problem (IVP):
8.1 Introduction Consider the initial value problem (IVP): y dy dt = f(t, y), y(t 0)=y 0, t 0 t T. Geometrically: solutions are a one parameter family of curves y = y(t) in(t, y)-plane. Assume solution
More information2 Numerical Methods for Initial Value Problems
Numerical Analysis of Differential Equations 44 2 Numerical Methods for Initial Value Problems Contents 2.1 Some Simple Methods 2.2 One-Step Methods Definition and Properties 2.3 Runge-Kutta-Methods 2.4
More informationOn the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method
Open Math. 215; 13: 547 556 Open Mathematics Open Access Research Article Haci Mehmet Baskonus* and Hasan Bulut On the numerical solutions of some fractional ordinary differential equations by fractional
More informationarxiv: v1 [math.ca] 30 Sep 2014
SOLVING ABEL INTEGRAL EQUATIONS OF FIRST KIND VIA FRATIONAL ALULUS arxiv:149.8446v1 [math.a] 3 Sep 214 S. JAHANSHAHI, E. BABOLIAN, D. F. M. TORRES, AND A. R. VAHIDI Abstract. We give a new method for numerically
More informationNumerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag Leffler Functions
mathematics Article Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag Leffler Functions Marina Popolizio ID Dipartimento di Matematica e Fisica Ennio De Giorgi,
More informationAPPM/MATH Problem Set 6 Solutions
APPM/MATH 460 Problem Set 6 Solutions This assignment is due by 4pm on Wednesday, November 6th You may either turn it in to me in class or in the box outside my office door (ECOT ) Minimal credit will
More informationOn the Designing of Fractional Order FIR Differentiator Using Radial Basis Function and Window
On the Designing of Fractional Order FIR Differentiator Using Radial Basis Function and Window Manjeet Kumar Digital Signal Processing Laboratory, Room No. 135, ECE Division, Netaji Subhas Institute of
More informationInitial value problems for ordinary differential equations
AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential
More informationJim Lambers MAT 460/560 Fall Semester Practice Final Exam
Jim Lambers MAT 460/560 Fall Semester 2009-10 Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding
More information5 Numerical Integration & Dierentiation
5 Numerical Integration & Dierentiation Department of Mathematics & Statistics ASU Outline of Chapter 5 1 The Trapezoidal and Simpson Rules 2 Error Formulas 3 Gaussian Numerical Integration 4 Numerical
More informationCS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationMathematics for Engineers. Numerical mathematics
Mathematics for Engineers Numerical mathematics Integers Determine the largest representable integer with the intmax command. intmax ans = int32 2147483647 2147483647+1 ans = 2.1475e+09 Remark The set
More informationarxiv: v1 [math.na] 15 May 2015
Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory arxiv:1505.03967v1 [math.na] 15 May 2015 Christopher L. MacDonald, Nirupama Bhattacharya,
More informationChapter 5 Exercises. (a) Determine the best possible Lipschitz constant for this function over 2 u <. u (t) = log(u(t)), u(0) = 2.
Chapter 5 Exercises From: Finite Difference Methods for Ordinary and Partial Differential Equations by R. J. LeVeque, SIAM, 2007. http://www.amath.washington.edu/ rjl/fdmbook Exercise 5. (Uniqueness for
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The
More informationDELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science
DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS ( WI3097 TU AESB0 ) Thursday April 6
More informationMathematics for chemical engineers. Numerical solution of ordinary differential equations
Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 2015-2016 Outline 1 Introduction 2 One step methods Euler
More informationdu dθ We wish to approximate the initial value problem for the fractional ordinary differential equation of order β :
Mixed Collocation for Fractional Differential Equations François Dubois and Stéphanie Mengué December 5, 3 This contribution is dedicaced to Claude Brézinski at the occasion of his sixtieth birthday. Abstract
More informationNumerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.
Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley
More informationCLASSICAL AND FRACTIONAL ASPECTS OF TWO COUPLED PENDULUMS
(c) 018 Rom. Rep. Phys. (for accepted papers only) CLASSICAL AND FRACTIONAL ASPECTS OF TWO COUPLED PENDULUMS D. BALEANU 1,, A. JAJARMI 3,, J.H. ASAD 4 1 Department of Mathematics, Faculty of Arts and Sciences,
More informationNumerical Analysis Preliminary Exam 10 am to 1 pm, August 20, 2018
Numerical Analysis Preliminary Exam 1 am to 1 pm, August 2, 218 Instructions. You have three hours to complete this exam. Submit solutions to four (and no more) of the following six problems. Please start
More informationCh. 03 Numerical Quadrature. Andrea Mignone Physics Department, University of Torino AA
Ch. 03 Numerical Quadrature Andrea Mignone Physics Department, University of Torino AA 2017-2018 Numerical Quadrature In numerical analysis quadrature refers to the computation of definite integrals. y
More informationCS 257: Numerical Methods
CS 57: Numerical Methods Final Exam Study Guide Version 1.00 Created by Charles Feng http://www.fenguin.net CS 57: Numerical Methods Final Exam Study Guide 1 Contents 1 Introductory Matter 3 1.1 Calculus
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign
More informationarxiv: v1 [math.ca] 28 Feb 2014
Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.11. (213) 2945-2948. arxiv:142.7161v1 [math.ca] 28 Feb 214 No Violation of the Leibniz Rule. No Fractional Derivative. Vasily E.
More informationCalculus of Variations Summer Term 2014
Calculus of Variations Summer Term 2014 Lecture 12 26. Juni 2014 c Daria Apushkinskaya 2014 () Calculus of variations lecture 12 26. Juni 2014 1 / 25 Purpose of Lesson Purpose of Lesson: To discuss numerical
More informationAnomalous diffusion equations
Anomalous diffusion equations Ercília Sousa Universidade de Coimbra. p.1/34 Outline 1. What is diffusion? 2. Standard diffusion model 3. Finite difference schemes 4. Convergence of the finite difference
More informationA Doctor Thesis. High Accuracy Numerical Computational Methods for Fractional Differential Equations. Yuki Takeuchi
A Doctor Thesis High Accuracy Numerical Computational Methods for Fractional Differential Equations ( ) by Yuki Takeuchi Submitted to the Graduate School of the University of Tokyo on December, 4 in Partial
More informationComputational Non-Polynomial Spline Function for Solving Fractional Bagely-Torvik Equation
Math. Sci. Lett. 6, No. 1, 83-87 (2017) 83 Mathematical Sciences Letters An International Journal http://dx.doi.org/10.18576/msl/060113 Computational Non-Polynomial Spline Function for Solving Fractional
More informationINVESTIGATION OF THE BEHAVIOR OF THE FRACTIONAL BAGLEY-TORVIK AND BASSET EQUATIONS VIA NUMERICAL INVERSE LAPLACE TRANSFORM
(c) 2016 Rom. Rep. Phys. (for accepted papers only) INVESTIGATION OF THE BEHAVIOR OF THE FRACTIONAL BAGLEY-TORVIK AND BASSET EQUATIONS VIA NUMERICAL INVERSE LAPLACE TRANSFORM K. NOURI 1,a, S. ELAHI-MEHR
More informationFRACTIONAL-ORDER UNIAXIAL VISCO-ELASTO-PLASTIC MODELS FOR STRUCTURAL ANALYSIS
A fractional-order uniaxial visco-elasto-plastic model for structural analysis XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII E. Oñate, D.R.J. Owen,
More informationOutline. 1 Numerical Integration. 2 Numerical Differentiation. 3 Richardson Extrapolation
Outline Numerical Integration Numerical Differentiation Numerical Integration Numerical Differentiation 3 Michael T. Heath Scientific Computing / 6 Main Ideas Quadrature based on polynomial interpolation:
More informationNumerical Differential Equations: IVP
Chapter 11 Numerical Differential Equations: IVP **** 4/16/13 EC (Incomplete) 11.1 Initial Value Problem for Ordinary Differential Equations We consider the problem of numerically solving a differential
More information5. Ordinary Differential Equations. Indispensable for many technical applications!
Indispensable for many technical applications! Numerisches Programmieren, Hans-Joachim Bungartz page 1 of 30 5.1. Introduction Differential Equations One of the most important fields of application of
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MATHEMATICS ACADEMIC YEAR / EVEN SEMESTER QUESTION BANK
KINGS COLLEGE OF ENGINEERING MA5-NUMERICAL METHODS DEPARTMENT OF MATHEMATICS ACADEMIC YEAR 00-0 / EVEN SEMESTER QUESTION BANK SUBJECT NAME: NUMERICAL METHODS YEAR/SEM: II / IV UNIT - I SOLUTION OF EQUATIONS
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 16. f(x) dx,
Panel integration Week 12: Monday, Apr 16 Suppose we want to compute the integral b a f(x) dx In estimating a derivative, it makes sense to use a locally accurate approximation to the function around the
More informationName of the Student: Unit I (Solution of Equations and Eigenvalue Problems)
Engineering Mathematics 8 SUBJECT NAME : Numerical Methods SUBJECT CODE : MA6459 MATERIAL NAME : University Questions REGULATION : R3 UPDATED ON : November 7 (Upto N/D 7 Q.P) (Scan the above Q.R code for
More informationA THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS
A THEORETICAL INTRODUCTION TO NUMERICAL ANALYSIS Victor S. Ryaben'kii Semyon V. Tsynkov Chapman &. Hall/CRC Taylor & Francis Group Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor
More informationMath 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.
Math 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.8: 1b, 3b, 4 Burden & Faires 5.6. Multistep Methods 1. Use all the Adams-Bashforth
More informationDOCTORAL THESIS Extended Abstract
Lodz University of Technology Faculty of Electrical, Electronic, Computer and Control Engineering DOCTORAL THESIS Extended Abstract Dariusz Brzeziński MSc Eng. Problems of Numerical Calculation of Derivatives
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationAPPLICATION OF HAAR WAVELETS IN SOLVING NONLINEAR FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS
APPLICATION OF HAAR WAVELETS IN SOLVING NONLINEAR FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS H. SAEEDI DEPARTMENT OF MATHEMATICS, SAHID BAHONAR UNIVERSITY OF KERMAN, IRAN, 76169-14111. E-MAILS:
More informationFixed point iteration and root finding
Fixed point iteration and root finding The sign function is defined as x > 0 sign(x) = 0 x = 0 x < 0. It can be evaluated via an iteration which is useful for some problems. One such iteration is given
More informationOn interval predictor-corrector methods
DOI 10.1007/s11075-016-0220-x ORIGINAL PAPER On interval predictor-corrector methods Andrzej Marcinia 1,2 Malgorzata A. Janowsa 3 Tomasz Hoffmann 4 Received: 26 March 2016 / Accepted: 3 October 2016 The
More informationWORKSHOP ON FRACTIONAL CALCULUS AND ITS APPLICATIONS Roma Tre, March 11, 2015 PROGRAM
On March 11, 2015, a workshop on fractional calculus was held, at the Department of Mathematics and Physics, Roma Tre University. The purpose of this one-day event was to present few contributions by several
More informationModeling of Structural Relaxation By A Variable- Order Fractional Differential Equation
University of South Carolina Scholar Commons Theses and Dissertations 2016 Modeling of Structural Relaxation By A Variable- Order Fractional Differential Equation Su Yang University of South Carolina Follow
More informationResearch Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations
Research Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations Ramses van Zon SciNet HPC Consortium November 25, 2014 Ramses van Zon (SciNet HPC Consortium)Research
More informationSEVERAL RESULTS OF FRACTIONAL DERIVATIVES IN D (R + ) Chenkuan Li. Author's Copy
RESEARCH PAPER SEVERAL RESULTS OF FRACTIONAL DERIVATIVES IN D (R ) Chenkuan Li Abstract In this paper, we define fractional derivative of arbitrary complex order of the distributions concentrated on R,
More informationFractional-order attractors synthesis via parameter switchings
Fractional-order attractors synthesis via parameter switchings Marius F Danca 1,2 and Kai Diethelm 3,4 1 Department of Mathematics and Computer Science Avram Iancu University Str. Ilie Macelaru nr. 1A,
More information