Solving Initial Value Problems in Nabla Fractional Calculus Kevin - PowerPoint PPT Presentation

Solving Initial Value Problems in Nabla Fractional Calculus Kevin Ahrendt, Lucas Castle, Katy Yochman University of Nebraska-Lincoln, Lamar University, Rose-Hulman Institute of Technology Summer Research Program 2011 Mentors: Dr. Peterson and Dr. Holm July 28, 2011 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 1 / 63

Sample Initial Value Problems Example (Continuous Example)  y ′′ ( t ) = t 2   y (0) = 3  y ′ (0) = 5  y ( t ) = t 4 12 + 5 t + 3 Example (Discrete Fractional Example)  ∇ 1 . 4 0 f ( t ) = t 2 , t ∈ N a +2   f (2) = 1  ∇ f (2) = 2  f ( t ) ≈ − 7 . 44 t . 4 + 1 . 62 t − . 6 + 1 . 61 t 3 . 4 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 2 / 63

Outline Outline Introduction to the Nabla Discrete Calculus 1 Fractional Sums and Differences 2 Taylor Monomials 3 Composition Rules 4 Laplace Transforms 5 Solving Initial Value Problems 6 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 3 / 63

Introduction to the Nabla Discrete Calculus Outline Introduction to the Nabla Discrete Calculus 1 Fractional Sums and Differences 2 Taylor Monomials 3 Composition Rules 4 Laplace Transforms 5 Solving Initial Value Problems 6 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 4 / 63

Introduction to the Nabla Discrete Calculus Domains of Functions in the Discrete Case Definition (Domain of N a ) N a := { a , a + 1 , a + 2 , · · · } . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 5 / 63

Introduction to the Nabla Discrete Calculus Nabla Difference Operator Definition (Nabla Difference Operator) ∇ f ( t ) := f ( t ) − f ( t − 1) , t ∈ N a +1 ∇ f ( a + 4) = f ( a + 4) − f ( a + 3) K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 6 / 63

Introduction to the Nabla Discrete Calculus Nabla Definite Integrals Definition ∇ t = � d t = c +1 f ( t ), for c < d . � a +5 f ( t ) a K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 7 / 63

Fractional Sums and Differences Outline Introduction to the Nabla Discrete Calculus 1 Fractional Sums and Differences 2 Taylor Monomials 3 Composition Rules 4 Laplace Transforms 5 Solving Initial Value Problems 6 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 8 / 63

Fractional Sums and Differences Example of a Nabla Difference Example Consider ∇ t 2 = t 2 − ( t − 1) 2 = t 2 − ( t 2 − 2 t + 1) = 2 t − 1 . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 9 / 63

Fractional Sums and Differences Rising Factorial Function Definition (Rising Factorial Function) For k , n ∈ N , the rising factorial function is k n := k ( k + 1) · · · ( k + n − 1) = ( k + n − 1)! . ( k − 1)! Example 3 2 = 3 · (3 + 1) = 3 · 4 = 12 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 10 / 63

Fractional Sums and Differences Difference of a Rising Factorial Function Example Consider t 2 = t · ( t + 1) . ∇ t 2 = t · ( t + 1) − ( t − 1) · t = t 2 + t − t 2 + t = 2 t . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 11 / 63

Fractional Sums and Differences Integral Sums for Integers Theorem (Repeated Integrals) � t 1 ∇ − n ( t − ( s − 1)) n − 1 f ( s ) ∇ s a f ( t ) := ( n − 1)! a K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 12 / 63

Fractional Sums and Differences Gamma Function The gamma function is an extension of the factorial functions for non-integer values. Definition (Gamma Function) � ∞ t z − 1 e − t dt . Γ( z ) = 0 Properties Γ( n ) = ( n − 1)! , for n ∈ N . x · Γ( x ) = Γ( x + 1). K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 13 / 63

Fractional Sums and Differences Gamma Function Figure: http://en.wikipedia.org/wiki/Gamma function K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 14 / 63

Fractional Sums and Differences Extending the Rising Factorial Function Definition (Rising Function) For k , ν ∈ R , the rising function is k ν := Γ( k + ν ) . Γ( k ) Example 3 2 = 3 · (3 + 1) = Γ(3 + 2) = Γ(5) Γ(3) = 4! 2! = 12 Γ(3) 3 2 . 05 = Γ(3 + 2 . 05) = Γ(5 . 05) ≈ 12 . 94 Γ(3) Γ(3) K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 15 / 63

Fractional Sums and Differences Extending the Repeated Integrals Formula Definition (Nabla Fractional Sum) Let ν > 0, then the v th -order fractional sum is � t 1 ( t − ( s − 1)) ν − 1 f ( s ) ∇ s . ∇ − ν a f ( t ) := Γ( ν ) a K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 16 / 63

Fractional Sums and Differences Defining the Fractional Difference Definition (Nabla Fractional Difference) Let ν > 0, and choose N ∈ N such that N − 1 < ν ≤ N . Then the ν th -order fractional difference is a f ( t ) := ∇ N ∇ − ( N − ν ) ∇ ν f ( t ) , t ∈ N a + N . a K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 17 / 63

Fractional Sums and Differences Fractional Difference Notation Example Consider the 1 . 9 th -order difference. � t � � 1 ∇ 1 . 9 a f ( t ) = ∇ 2 ( t − ( s − 1)) 1 . 9 − 1 f ( s ) ∇ s . Γ(1 . 9) a Example Consider the 2 nd -order difference. ∇ 2 f ( t ) = f ( t ) − 2 f ( t − 1) + f ( t − 2). Non-whole order differences must denote a base, i.e. ∇ 1 . 9 a f ( t ). Whole order differences do not depend on a base, thus the subscript is omitted, i.e. ∇ 2 f ( t ). K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 18 / 63

Fractional Sums and Differences Alternative Definition of the Fractional Difference Theorem (Alternative Definition of a Fractional Difference) The following statements are equivalent for ν > 0 and N ∈ N chosen such that N − 1 < ν < N . a f ( t ) = ∇ N ∇ − ( N − ν ) ∇ ν f ( t ) , a t 1 � ∇ ν ( t − ( s − 1)) − ν − 1 f ( s ) , a f ( t ) = Γ( − ν ) s = a +1 for ν �∈ N 0 . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 19 / 63

Fractional Sums and Differences Unified Definition of the Fractional Sums and Differences 1 The ν th -order fractional sum of f is given by t 1 � ( t − ( s − 1)) ν − 1 f ( s ) , for t ∈ N a . ∇ − ν a f ( t ) := Γ( ν ) s = a +1 2 The ν th -order fractional difference of f is given by � � t 1 s = a +1 ( t − ( s − 1)) − ν − 1 f ( s ) , ν / ∈ N 0 ∇ ν Γ( − ν ) a f ( t ) := ∇ N f ( t ) , ν = N ∈ N 0 for t ∈ N a + N . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 20 / 63

Fractional Sums and Differences Continuity of the Fractional Difference Theorem (Continuity of the Nabla Fractional Difference) Let f : N a → R be given. Then the fractional difference ∇ ν a f is continuous with respect to ν ≥ 0 . Consider the sequence {∇ 1 . 9 a f ( a + 3) , ∇ 1 . 99 f ( a + 3) , ∇ 1 . 999 f ( a + 3) , · · · } . a a This theorem implies that as the difference approaches 2, it depends less and less on its base and behaves more and more like the whole order ∇ 2 f ( a + 3) K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 21 / 63

Taylor Monomials Outline Introduction to the Nabla Discrete Calculus 1 Fractional Sums and Differences 2 Taylor Monomials 3 Composition Rules 4 Laplace Transforms 5 Solving Initial Value Problems 6 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 22 / 63

Taylor Monomials Fractional Taylor Monomials Definition (Fractional Order Taylor Monomials) For ν ∈ R \{− 1 , − 2 , ... } , define the Taylor monomial as ν ( t ) = h ν ( t , a ) := ( t − a ) ν h a Γ( ν + 1) . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 23 / 63

Taylor Monomials Taylor Monomial Example Example Consider t 2 h 2 ( t , 0) = Γ(3) ∇ h 2 ( t , 0) = t 2 − ( t − 1) 2 t 1 = t = Γ(2) = h 1 ( t , 0) 2 ∇ 2 h 2 ( t , 0) = t − ( t − 1) = 1 = h 0 ( t , 0) . K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 24 / 63

Taylor Monomials Generalized Power Rule Theorem (Generalized Power Rule) ∇ ν a h µ ( t , a ) := h µ − ν ( t , a ) or Γ( µ + 1) a ( t − a ) µ := ∇ ν Γ( µ − ν + 1)( t − a ) µ − ν K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 25 / 63

Taylor Monomials Power Rule Example Example Γ(2 + 1) ∇ . 95 t 2 = Γ(2 − . 95 + 1) t (2 − . 95) ≈ 1 . 957 t 1 . 05 K. Ahrendt, L. Castle, K. Yochman Solving IVPs in the Discrete Fractional Calculus July 28, 2011 26 / 63