On solutions of linear fractional differential equations of a variational type - Małgorzata Klimek

[SPIS TREŚCI]

CONTENTS

Notation index

1. Introduction

2. Fractional operators and Mellin transform

2.1.Introduction
2.2. Riemann-Liouville fractional operators in finite interval
2.3. Liouville fractional operators on the halfaxis
2.4. Caputo derivative
2.5. Composition rules
2.6. Riesz potentials
2.7. Mellin transform and its properties

3. Mellin transform method applied to fractional equations with Riemann-Liouville or Caputo derivatives

3.1. Introduction
3.2. Fractional linear equation with left-sided Riemann-Liouville derivative and tβ- potential
3.2.1. Example: Solution for case α + β = α
3.2.2. Example: Solution for case α + β = α/2
3.2.3. Example: Solution for case α + β = α/3
3.3. Fractional linear equation with Caputo derivative and tβ- potential
3.3.1. Example: Solution for case α + β = α
3.3.2. Example: Solution for case α + β = α/2
3.4. Nonhomogeneous fractional equations with tβ-potential
3.5. Fractional linear equation with right-sided Riemann-Liouville derivative and tβ potential
3.5.1. Example: Solution for case α + β = α
3.5.2. Example: Solution for case α + β = α/2
3.5.3. Example: Solution for case α + β = α/J
3.6. Generalized linear sequential fractional equation with variable coefficients
3.6.1. Example: Solution for case M = 2 and β = 0
3.6.2. Example: Solution for arbitrary M and β = 0

4. Euler-Lagrange equations in fractional mechanics

4.1. Fractional mechanics - non-sequential formulation
4.1.1. Example: Simple model with left-sided Riemann-Liouville derivative
4.1.2. Example: Fractional oscillator equation
4.2. Euler-Lagrange equations in sequential fractional mechanics
4.2.1. Example: Simple model with left-sided derivative
4.2.2. Example: Free motion in model with order α ∈ (1, 2)
4.2.3. Example: Fractional oscillator
4.3. Remarks on sequential mechanics with alternative integration by parts formula
4.4. Fractional models with constraints
4.4.1. Example: Simple fractional optimal control problem
4.5. Fractional embedding in derivation of Euler-Lagrange equations
4.5.1. Fractional operators of order (α, β)
4.5.2. Euler-Lagrange equations - two approaches
4.5.3. Generalized fractional Euler-Lagrange equations
4.5.4. Example: Equation for linear friction

5. Stationary functions for fractional derivatives

5.1. Introduction
5.1.1. Example: Application of polynomial function in transformation and solving procedure of certain ordinary differential equations
5.1.2. Example: Application of stationary functions in transformation and solving procedure of certain fractional differential equations
5.2. Stationary functions for left- and right-sided fractional derivatives
5.3. Stationary functions for symmetric fractional derivative in finite time interval
5.3.1. Properties of stationary functions for symmetric fractional derivative
5.3.2. Classical limits α −→ (n − 1)+
5.3.3. Example: Stationary functions of symmetric fractional derivative for α′ ∈ (0, 1) and α ∈ (1, 2)
5.4. Stationary functions for antisymmetric fractional derivative in finite time interval
5.4.1. Properties of stationary functions for antisymmetric fractional derivative
5.4.2. Example: Stationary functions for antisymmetric fractional derivative of order α ∈ (1, 2)
5.5. Stationary functions for composition cDαb−Dα0+
5.6. Stationary functions for composition Dαb−Dα0+

6. Equations with symmetric and antisymmetric fractional derivatives

6.1. Introduction
6.2. Equations with fractional symmetric derivative
6.2.1. Solution in case α′ ∈ (n − 1, n) with n - an odd number
6.2.1.1. Condition α′ + β = ǫ′/J and J ∈ N
6.2.1.2. Example α′ + β = ǫ′
6.2.2. Solution in case α ∈ (n − 1, n) with n- being an even number
6.2.3. Boundary conditions and particular solutions of equations with symmetric derivative
6.3. Equations with fractional antisymmetric derivative
6.3.1. Solution in case α ∈ (n − 1, n) with n - being an even number
6.3.2. Example: case α ∈ (1, 2) and classical limit α −→ 1+
6.3.3. Solution in case α′ ∈ (n − 1, n) with n-odd
6.3.4. Boundary conditions and particular solution of eigenfunction equation with antisymmetric derivative

7. Linear fractional differential equations with symmetric and antisymmetric fractional derivatives

7.1. Introduction
7.2. Linear fractional differential equations with variational derivatives
7.3. Linear equation with symmetric fractional derivative
7.3.1. Example: case α ∈ (1, 2) and N = 2
7.4. Linear equation with antisymmetric fractional derivative
7.4.1 Example: case α ∈ (1, 2) and N = 2

8. Fixed point theorem in solving simple and generalized eigenfunction equations for fractional operators of a variational type

8.1. Introduction
8.2. Eigenfunction equation for composition cDαb−Dαa+
8.2.1. Continuous solutions of eigenfunction equation
8.2.2. Singular solutions of eigenfunction equation
8.2.3. Boundary conditions and particular solutions of eigenfunction equation
8.2.3.1. Example: Solutions for order α = 1
8.2.3.2. Example: Continuous solutions for order α ∈ (0, 1)
8.2.4. Approximate solutions of eigenfunction equation: continuous case
8.2.4.1. Conditions for error of approximation || fap –Faλ ||< ǫ
8.3. Eigenfunction equation for composition Dαb−Dαa+
8.3.1. Continuous solutions of eigenfunction equation
8.3.2. Singular solutions of eigenfunction equation
8.3.3. Boundary conditions and particular solutions of eigenfunction equation
8.3.4. Approximate solutions of eigenfunction equation: continuous case
8.4. Generalized eigenfunction equation for composition cDαb−Dαa+
8.4.1. Continuous solutions of generalized eigenfunction equations
8.4.2. Singular solutions of generalized eigenfunction equation
8.4.3. Boundary conditions and particular solutions for generalized eigenfunction equation
8.4.3.1. Example: Solutions of a generalized eigenfunction equation for α ∈ (0, 1) and M = 2
8.4.4. Approximate solutions of generalized eigenfunction equation: continuous case
8.5. Generalized eigenfunction equation for composition Dαb−Dαa+
8.5.1. Continuous and singular solutions of generalized eigenfunction equation
8.5.2. Boundary conditions and particular solutions of generalized eigenfunction equation
8.5.3. Example: Solutions of generalized eigenfunction equation for α ∈ ( 12 , 1) and M = 2
8.5.4. Approximate solutions for generalized eigenfunction equations: continuous case

9. Linear equations with compositions of left- and right-sided fractional derivatives

9.1. Linear equation with constant coefficients and fractional operator cDαb−Dαa+
9.1.1. Example: Case α ∈ (0, 1) and N = 2
9.2. Linear equation with constant coefficients and fractional operator Dαb−Dαa+
9.2.1. Example: Case α ∈ ( 12 , 1) and N = 2

Bibliography

A. Function spaces
B. Fox and Meijer functions
C. Proofs of convergence

C.1. Proofs of convergence for complex series from Chapter 3
C.1.1. Equation with left-sided Riemann-Liouville derivative and tβ potential
C.1.2. Equation with left-sided Caputo derivative and tβ potential
C.1.3. Equation with right-sided Riemann-Liouville derivative and tβ potential
C.1.4. Generalized linear sequential fractional equation with variable coefficients
C.2. Convergence of series representing solutions
C.2.1. Convergence of series representing solution for equations with left-sided Riemann-Liouville derivative
C.2.2. Convergence of series representing solution for equations with left-sided Caputo derivative
C.2.3. Convergence of series representing solution for equations with right-sided Riemann-Liouville derivative
C.3. Proofs of convergence for complex series from Chapter 6
C.3.1. Equation with symmetric fractional derivative
C.3.2. Equation with antisymmetric fractional derivative
C.4. Convergence of series representing solutions
C.4.1. Solution of equations with the symmetric derivative of order α′ ∈ (n − 1, n) and n being an odd number
C.4.2. Remarks on convergence of series representing solutions in Theorems 6.2, 6.4 and 6.5

D. Banach theorem applied to eigenfunction equations with fractional operators

D.1. Proof of Proposition 8.1
D.2. Proof of Proposition 8.14