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Title Kindergarten of fractional calculus
Creators Das Shantanu.
Collection Электронные книги зарубежных издательств ; Общая коллекция
Subjects Fractional calculus. ; EBSCO eBooks
Document type Other
File type PDF
Language English
Rights Доступ по паролю из сети Интернет (чтение, печать, копирование)
Record key on1143468793
Record create date 3/9/2020

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  • Table of Contents
  • List of Photographs
  • List of Figures
  • List of Tables
  • Foreword
  • Preface
  • Acknowledgements
  • Chapter One
    • 1.1 Introduction
    • 1.2 The thought problem for constructing a ‘half-coin’
      • 1.2.1 The idea of Negative Probability through the generalisation of existing theories
      • 1.2.2 The tossing of coins and Probability Generation Functions (PGFs)
      • 1.2.3 Generalizing the PGF to get a half-coin construct: a paradox?
    • 1.3 The question posed by L’Hospital to Leibniz
    • 1.4 A recall for analytic functions
      • 1.4.1 Real analytic functions
      • 1.4.2 Complex analytic functions
    • 1.5 Recalling the Cauchy integral formula
      • 1.5.1 The derivative is an integration process
      • 1.5.2 The Taylor series from Cauchy’s integral formula
    • 1.6 Singularity pole branch point and branch cut
      • 1.6.1 Isolated singularity
      • 1.6.2 Pole singularity (removable singularity)
      • 1.6.3 Essential singularity
      • 1.6.4 Multivalued functions, branch points and branch-cuts
    • 1.7 Recalling residue calculus
      • 1.7.1 Finding the residue by power series expansion
      • 1.7.2 The formula for calculating residue
    • 1.8 The basics of analytic continuation
    • 1.9 The factorial and its several representations
      • 1.9.1 Representation of a falling factorial
      • 1.9.2 Double factorial
      • 1.9.3 Truncated factorial
      • 1.9.4 Some other historical representations of factorials
      • 1.9.5 Increasing truncated factorials (Pochhammer numbers)
      • 1.9.6 Euler’s integral representation for factorials
    • 1.10 The gamma function
      • 1.10.1 The recurring relationship of the gamma function
      • 1.10.2 Factorials related to the gamma function
      • 1.10.3 Calculation of the factorial of non-integer numbers by gamma function
      • 1.10.4 Poles of the gamma function
      • 1.10.5 Extending the gamma function for negative arguments by analytic continuation
      • 1.10.6 Representations of the gamma function via the contour integral (Hankel’s formula)
      • 1.10.7 Ratio of gamma functions at negative integer points
      • 1.10.8 The duplication formula for the gamma function
      • 1.10.9 Reflection formula of the gamma function
      • 1.10.10 Asymptotic representation of the gamma function
      • 1.10.11 Reciprocal gamma function
      • 1.10.12 Representing the gamma function as a limit form
      • 1.10.13 Some interesting values of gamma functions that we frequently encountered
    • 1.11 Ratio of gamma functions
    • 1.12 Generalising binomial coefficients by gamma function
    • 1.13 Incomplete gamma functions
      • 1.13.1 Tricomi’s incomplete gamma function and the complementary incomplete gamma function
      • 1.13.2- Incomplete gamma function
      • 1.13.3 The elementary relationships of Tricomi’s incomplete gamma function
    • 1.14 Power numbers
    • 1.15 The beta function and the incomplete beta function
      • 1.15.1 Integral representations of the beta function
      • 1.15.2 The derivation for the reflection formula of the gamma function
      • 1.15.3 Application of analytic continuation to the reflection formula
      • 1.15.4 Incomplete beta functions
    • 1.16 Psi function
      • 1.16.1 The recurring series formula for psi function
      • 1.16.2 Analytical continuation of a finite harmonic series by using psi function
    • 1.17 A unified notation to represent multiple differentiation and integration
    • 1.18 Differentiation and integration of series
    • 1.19 Revising the concepts of mixed differentiation and integration
    • 1.20 The lower limit as an important factor in multiple integrals
    • 1.21 Generalising product rule for multiple integration and differentiations
    • 1.22 Chain rule for multiple derivatives
    • 1.23 Differentiation and integration of power functions: unification & generalisation
    • 1.24 Non-differentiable functions: the motivation for fractional calculus
      • 1.24.1 Hurst exponent
      • 1.24.2 Self-similarity
      • 1.24.3 Nowhere differentiable functions
    • 1.25 Short summary
    • 1.26 References
  • Chapter Two
    • 2.1 Introduction
    • 2.2 Iterative integration: a review
      • 2.2.1 Double integration
      • 2.2.2 Double integration with a changed order of integration giving the idea of convolution
      • 2.2.3 Generalising the double integration to an n - fold integration
    • 2.3 Repeated integration formula by induction
    • 2.4 Explaining the convolution process in repeated integration numerically
    • 2.5 Generalising the expression for a repeated integration to get the Riemann-Liouville fractional integration formula
    • 2.5.1 A generalisation of a repeated integration formula by the gamma function
    • 2.5.2 Applying the Riemann-Liouville formula to get double-integration andsemi-integration for the function f (x) 􀀠 x
    • 2.6 Fractional integration of a power function
      • 2.6.1 Fractional integration of power functions with the lower limit of integration as zero
      • 2.6.2 Fractional integration of power functions with the lower limit of integration as non-zero
      • 2.6.3 Region of validity of the integer order integration of the power function
    • 2.7 The extended region of fractional integration of a power function: analytic continuation
    • 2.8 An analytic continuation of a finite harmonic series-derived via a fractional integration concept
    • 2.9 Fractional integral of exponential function by series
      • 2.9.1 Fractional integration of the exponential function with a start point of integration as zero
      • 2.9.2 Introducing a higher transcendental function in the result of the fractional integration of the exponential function
      • 2.9.3 Fractional integration of an exponential function with the start point of integration as non-zero
      • 2.9.4 Fractional integration of a one-parameter Mittag-Leffler function as conjugation to classical integration of the exponential function
    • 2.10 Tricks required for solving fractional integrals
      • 2.10.1 Obtaining a useful identity for fractional integration with a lower limit of integration as zero
      • 2.10.2 Obtaining a useful identity for a fractional integration with a lower limit of integration as non-zero
      • 2.10.3 Application of the obtained identity for acquiring a fractional integration of a power function
      • 2.10.4 Application of the obtained identity for acquiring fractional integration of the exponential function in terms of Tricomi’s incomplete gamma function
    • 2.11 Fractional integration of the analytical function
      • 2.11.1 Fractional integration related to a series of ordinary whole derivatives of an analytical function
      • 2.11.2 Application of the Fractional integration of an analytical function to an exponential function for integer order integration
      • 2.11.3 An alternate representation and generalisation of the Riemann-Liouville fractional integration formula of analytical function
    • 2.12 A simple hydrostatic problem of water flowing through the weir of a dam with the use of a fractional integration formula
    • 2.13 Introducing an initialising process of fractional integration
    • 2.14 Can we change the sign of fractional order in a fractional integration formula to write a formula for the fractional derivative?
      • 2.14.1 Divergence of the Riemann-Liouville integral for a change of sign of order of integration
      • 2.14.2 Using Reisz’s logic of analytic continuation to geta meaningful representation of the Riemann-Liouville fractional integration formula with a changed sign of the order of integration
      • 2.14.3 Attaching meaning to the Riemann-Liouville fractional integration formula for a positive order elementary approach
    • 2.15 Fractional integration of the Weyl type
      • 2.15.1 Weyl transform of the fractional integration of a function on future points to infinity
      • 2.15.2 Weyl fractional integration of the power function
      • 2.15.3 Weyl fractional integration of the exponential function
    • 2.16 Applications using the Weyl fractional integration
    • 2.17 Use of the kernel function in a convolution integral in the fractional integration formula
      • 2.17.1 Defining kernel functions for forward and backward fractional integration
      • 2.17.2 Additive rule for fractional integrals
      • 2.17.3 The choice of a different convolution kernel defines different types of fractional integration operators
      • 2.17.4 Generalisation of the fractional integration formula with respect to function
    • 2.18 Scaling law in a fractional integral
    • 2.19 The formula for fractional integration is derived from the generalisation of Cauchy’s integral theorem of complex variables
    • 2.20 Repeated integrals as a limit of a sum and its generalisation
    • 2.20.1 Limit of a sum formula to have classical repeated n - fold integration
    • 2.20.2 Limit of a sum formula to have fractional integration: the generalisation of classical formula
    • 2.21 Fractional integration is the area under the shape changing curve
      • 2.21.1 Analytical explanation of the area under the shape changing curve for the Riemann-Liouville fractional integration
      • 2.21.2 Numerical demonstration of fractional integration as the area under a shape changing curve
      • 2.21.3 A fractional integration is the case of fading memory while a classical one-whole integration is a case with a constant past memory
    • 2.22 The functions which can be fractionally integrated and differentiated
    • 2.23 Discrete numerical evaluation of the Riemann-Liouville fractional integration
      • 2.23.1 Using the average of function values at the end of intervals to have a formula for numerical evaluation of the Riemann-Liouville fractional integration
      • 2.23.2 Using the weighted average of the function values at the end of intervals to have a formula for the numerical evaluation of the Riemann-Liouville fractional integration
    • 2.24 Short summary
    • 2.25 References
  • Chapter Three
    • 3.1 Introduction
    • 3.2 Iterative differentiation and its generalisation
      • 3.2.1 Euler’s scheme of generalisation for repeated differentiation
      • 3.2.2 A seeming paradox in Euler’s scheme of generalisation for repeated differentiation
      • 3.2.3 Euler’s formula applied to find the fractional derivative and fractional integration of power functions
      • 3.2.4 That Euler’s formula gives us non-zero as a fractional derivative for a constant function seems to be a paradox
      • 3.2.5 The observation of Euler’s formula: the fractional derivative of a constant becomes zero when the fractional order of the derivative tends to one-whole number in the classical case
      • 3.2.6 Some interesting formulas for the semi-derivative and semi-integration with Euler’s formula
      • 3.2.7 Applying Euler’s formula for a function represented as power series expansion
      • 3.2.8 Generalising the iterated differentiation for x- p to get a fractional derivative formula, a different result is obtained by Euler’s scheme
    • 3.3 A natural extension of normal integer order classical derivative to obtain a fractional derivative
      • 3.3.1 Obtaining the fractional derivative by exponential approach for x-􀁅 by using the Laplace integral
      • 3.3.2 Representing x-􀁅 by using the Laplace integral in terms of the exponential functionby using the Laplace integral derivation
      • 3.3.3 A fractional derivative of the trigonometric and exponential function susing an exponential approach
      • 3.3.4 An expomemtial approach postulate of Leibniz and extended by Liouville
    • 3.4 Liouville's way of looking at a fractional derivative contradicts Euler's generalisation
      • 3.4.1 Liouville and Euler's fractional derivatives approach is applied to an exponential function
      • 3.4.2 Discussing the paradoxes of Euler’s and Leibniz’s by considering formulas to get classical one-whole integration–and a resolution
      • 3.4.3 The difference between Euler’s and Leibniz’s postulate is due to the limits of integration
      • 3.4.4 The fractional derivative requires the limits of a start-point and end-point for integration
    • 3.5 Applying Liouville’s logic to get a fractional derivative of x-a and a cosine function
    • 3.6 Liouville’s approach of a fractional derivative to arrive at formulas as a limit of difference quotients
      • 3.6.1 Liouville’s formula for fractional integration
      • 3.6.2 Liouville’s formula for a fractional derivative in integral representation
      • 3.6.3 Derivation of Liouville’s fractional derivative formulas
      • 3.6.4 Extension of Liouville’s approach to arrive at formulas of the fractional derivative as a limit of difference quotients
    • 3.7 A repeated integration approach to get the fractional derivative in Riemann-Liouville and Caputo formulations
      • 3.7.1 Fractional integration is a must to get a fractional derivative
      • 3.7.2 Different ways to apply a fractional integration formula to obtain a fractional derivative
    • 3.8 Fractional derivatives for the Riemann-Liouville (RL): ‘left hand definition’ (LHD)
    • 3.9 Fractional derivatives of the Caputo: ‘right hand definition’ (RHD)
    • 3.10 Relation between Riemann-Liouville and Caputo derivatives
    • 3.10.1 Riemann-Liouvelli and Caputo derivatives related by initial value of the function at start point of fractional derivative process
    • 3.10.2 A generalisation of a fundamental theorem of integral calculus is the relation between Riemann-Liouville and Caputo fractional derivatives
    • 3.10.3 Further generalisation of the relationship between the Caputo and Riemann-Liouville fractional derivatives
    • 3.10.4 The Caputo and Riemann-Liouville fractional derivatives need an area under the shape changing curve
    • 3.11 Can we change the sign of order of the Riemann-Liouville fractional derivative to write the RL fractional integration formula?
    • 3.12 The Weyl fractional derivative
    • 3.13 The most fundamental approach for repeated differentiation
    • 3.14 The unified context differentiation/integration Grunwald-Letnikov formula
      • 3.14.1 Generalisation by use of the backward shift operator defining the repeated differentiation and integration
      • 3.14.2 Using the generalised formula obtained by a backward shift operator of fractional integration/differentiation on an exponential function
      • 3.14.3 Generalisation by use of the forward shift operator defining repeated differentiation and integration
      • 3.14.4 Using the generalised formula obtained by a forward shift operator of fractional integration/differentiation on an exponential function
      • 3.14.5 The final unified formula for fractional derivatives and integration
      • 3.14.6 Reduction of a unified fractional derivative/integration unified formula to a classical derivative and integration
    • 3.15 The whole derivative is local property whereas the fractional derivative is non-local property
    • 3.16 Applying the limit of a finite difference formula result to get the answer to L’Hospital’s question
    • 3.17 Fractional finite difference (from local finite difference to non-local finite difference)
    • 3.18 Dependence on the lower limit
    • 3.19 Translation property
    • 3.20 Scaling property
    • 3.21 Fractional differentiation-integration behavior near the lower limit
    • 3.22 The fractional differentiation-integration behavior far from the lower limit
    • 3.23 A numerical evaluation of fractional differ-integrals using the Grunwald-Letnikov formula
    • 3.24 The discrete numerical evaluation of the Riemann-Liouville fractional derivative formula
    • 3.25 Generalisation of the Riemann-Liouville and Caputo derivatives with the choice of a different kernel, weights and base-function
      • 3.25.1 Generalisation of the left and right fractional derivatives with respect to the base function z(x) and weights w(x)
      • 3.25.2 The relationship between the generalised Riemann-Liouville and Caputo fractional derivatives with weight and base function
      • 3.25.3 The composition properties of generalised fractional derivatives
      • 3.25.4 The various forms of generalised fractional derivatives
    • 3.26 The Caputo fractional derivative with a non-singular exponential type kernel
    • 3.27 Short summary
    • 3.28 References
  • Chapter Four
    • 4.1 Introduction
    • 4.2 Applying the Grunwald-Letnikov (GL) formula of fractional derivatives to simple functions
      • 4.2.1 Finding the fractional derivative of a constant using the Grunwald-Letnikov (GL) formula
      • 4.2.2 Finding a fractional derivative of a linear function using the Grunwald-Letnikov (GL) formula
      • 4.2.3 Verifying the GL fractional derivative of a linear function using the Riemann-Liouville (RL) formula of the fractional integration
      • 4.2.4 Verifying the GL fractional derivative of a linear function by using the Riemann-Liouville (RL) formula of the fractional derivative
    • 4.3 Applying the Riemann-Liouville formula to get the fractional derivative for some simple functions
      • 4.3.1 Left fractional derivative of the power function through application of the integral formula of the Riemann-Liouville fractional derivative
      • 4.3.2 Right fractional derivative of the power function through application of the integral formula of the Riemann-Liouville fractional derivative
    • 4.4 Applying the Caputo formula to get a fractional derivative for simple functions
      • 4.4.1 Left fractional derivative of the power function xb with respect to x through application of the integral formula of the Caputo fractional derivative
      • 4.4.2 Left fractional derivative of the power function 􀀋 (x)􀀌􀁘 􀁜 with respect to x through application of the integral formula of the generalized Caputo fractional derivative
    • 4.5 The fractional differ-integration of the binomial function
    • 4.6 Fractional differ-integration exponential function using the Riemann-Liouville fractional derivative formula
    • 4.7 Fractional differ-integration for logarithmic functions
    • 4.8 Fractional differ-integration for some complicated functions described by the power series expansion
    • 4.9 Fractional differ-integration of the hyperbolic and trigonometric function using the series expansion method
    • 4.10 Fractional differ-integration of the Bessel function using the series expansion method
    • 4.11 Fractional differ-integration of the distribution functions using the Riemann-Liouville formula
      • 4.11.1 Uniform distribution function and its fractional differ-integration
      • 4.11.2 The delta distribution function and its fractional differ-integration
      • 4.11.3 The relationship between uniform distribution and the delta distribution function
    • 4.12 Fractional differ-integration of the saw-tooth function
    • 4.13 Fractional differ-integration of the generalized periodic function
    • 4.14 The Eigen-functions for the Riemann-Liouville and Caputo fractional derivative operators
      • 4.14.1 Caputo derivative of the order 􀁄 for the ‘one-parameter Mittag-Leffler function’
      • 4.14.2 Riemann-Liouville derivative of the order 􀁄 for the ‘one-parameter Mittag-Leffler function’
      • 4.14.3 Riemann-Liouville derivative of the order 􀁄 for a function of the ‘two-parameter Mittag-Leffler function’
    • 4.15 The fractional derivative of the zero-corrected function and relationship to Caputo derivative
      • 4.15.1 The Riemann-Liouville fractional derivative of the zero-corrected functionf (x) = a - x with a derivative start point at x = 0
      • 4.15.2 The Caputo fractional derivative function f (x) = a - x with a derivative start point at x = 0
      • 4.15.3 The Riemann-Liouville fractional derivative of the zero-correctedfunction f (t) = tv from the start point as t 􀁺 0
      • 4.15.4 The Riemann-Liouville fractional derivative of the order a for the zero-corrected ‘one-parameter Mittag-Leffler function'
      • 4.15.5 The Caputo fractional derivative of the order a for the ‘one-parameter Mittag-Leffler function’
      • 4.15.6 The Riemann-Liouville fractional derivative of the order 􀁄 of the zero-correctedfunction f (t) cos (t􀁄 )􀁄 􀀠 and f (t) sin (t􀁄 )􀁄 􀀠 with a derivative start point at t = 0
      • 4.15.7 The Riemann-Liouville fractional derivative of the order 􀁅 of the zero-corrected ‘one parameter Mittag-Leffler function’ f (x) E (x􀁄 )􀁄 􀀠 with a derivative start point at x 􀀠 0 and 􀁄 􀁺 􀁅
      • 4.15.8 The Riemann-Liouville and Caputo fractional derivatives of the order 􀁄 for the function 1f (x) x cos , (x ) 􀁅 􀁄􀁄 􀁅􀀠 􀀐 and 1f (x) x sin , (x ) 􀁅 􀁄􀁄 􀁅􀀠 􀀐 with a derivative start point at x 􀀠 0
      • 4.15.9 The Riemann-Liouville fractional derivative of the zero-corrected exponential function
      • 4.15.10 The Riemann-Liouville fractional derivative of the zero-corrected f (x) 􀀠 cos(ax) and f (x) 􀀠 sin(ax) functions with a derivative start point at x 􀀠 0 and its relationship to the Caputo derivative
    • 4.16 The fractional derivative at a non-differentiable point of the continuous function
    • 4.17 Computation of the fractional derivative and integration-a review and comparison of the various schemes
    • 4.18 Fractional derivatives of the same-order but of different RL-Caputo types
    • 4.19 Generalizing Leibniz’s rule
    • 4.20 Chain rule for fractional differ-integration
    • 4.21 Analytical continuation of the differ-integral operator from integer to real order
    • 4.22 Analytic continuation of the differ-integral operator from a real to complex order
    • 4.23 Short summary
    • 4.24 References
  • Chapter Five
    • 5.1 Introduction
    • 5.2 Initialization fractional integration (the Riemann-Liouville approach)
    • 5.2.1 The origin of the initialization function (or history function) in the fractional integration process
    • 5.2.2 The initialization function is a constant function in the classical integration process
    • 5.2.3 Types of initialization for fractional integration process
    • 5.3 Terminal and side initialization for fractional integration
    • 5.3.1 Terminal charging for fractional integration
    • 5.3.2 Evaluation of initialization function for f (t) = t for semi-integration process starting at t = 1
    • 5.3.3 The initialization concept of classical integration in the context of a developed concept of the initialization function for fractional integration
    • 5.3.4 Side charging for fractional integration
    • 5.4 Initializing the fractional derivative Riemann-Liouville approach
      • 5.4.1 Fractional derivative for a non-local operation requires initialization
      • 5.4.2 The generalization of the initialization process
    • 5.5 Terminal initialization for the fractional derivative
      • 5.5.1 Derivation of terminal charging for the fractional derivative
      • 5.5.2 Evaluation of the initialized semi-derivative for f (t) 􀀠 (t 􀀎 2)2 starting at t 􀀠 0
    • 5.6 Side-initialization of the fractional derivative
    • 5.7 Initializing fractional differ-integrals using the Grunwald-Letnikov approach
    • 5.8 Criteria for the generalized differ-integration composition
      • 5.8.1 Composition rules
      • 5.8.2 Demonstration of the composition of a fractional derivative, and the fractional integration and mixed operation that is applied to functions 1f (t) 􀀠 t 2 & 1f (t) t 2 􀀠 􀀐 , and discussion
      • 5.8.3 Composition rules with initialized differ-integrations
    • 5.9 The Relationship between uninitialized Caputo and Riemann-Liouville (RL) fractional derivatives
      • 5.9.1 Expression for the RL fractional integration of f (t) from 0 􀀐 t as integration of function ( ) q f t 􀀐 x from 0 􀀐 tq
      • 5.9.2 Finding an un-initialized Caputo derivative and demonstrating that no singularity term appears at the start point of fractional differentiation
      • 5.9.3 Finding an un-initialized Riemann-Liouville (RL) derivative and demonstrating the appearance of a singularity term at the start point of a fractional derivative
      • 5.9.4 Adding singularity function at the start point of a fractional derivative to equate uninitialized RL and Caputo derivatives
      • 5.9.5 Finding out the relationship of the RL-Caputo fractional derivative via Leibniz’s formula
      • 5.9.6 Evaluation of the RL and Caputo derivatives from a non-zero start point where the value of the function is zero
    • 5.10 Initialization of Caputo derivatives
      • 5.10.1 Initialization of Caputo derivatives and the associated difficulties
      • 5.10.2 The initialized fractional derivative theory
      • 5.10.3 Demonstration of the initialized function for a fractional derivative (RL and Caputo) as zero when the function value at the start point is zero and the function is zero before the start point
      • 5.10.4 Making the initialized Caputo derivative equal to the initialized RL derivative in order to provide an initialization function for the Caputo derivative
      • 5.10.5 The Caputo derivative initialization function
    • 5.11 Generalization of the RL and Caputo formulations and the initialization function
    • 5.12 Observations regarding difficulties in Caputo initialization and demanding physical conditions vis-à-vis RL initialization conditions
    • 5.13 The Fractional Derivative of a sinusoidal function with a lower terminal not at minus infinity, and an initialization function
    • 5.14 The Laplace transform of fractional differ-integrals
      • 5.14.1 The generalization of classical Laplace transform formulas for differentiation and integration
      • 5.14.2 Generalization is not possible for a few classical Laplace transform identities
      • 5.14.3 Evaluating the Laplace transform for function ( ) kx 􀀋0 kx ( ) 􀀌f x e Dx e f x 􀀠 􀀐 􀁄 􀂪 􀂺 􀂬 􀂼 , 􀁄 􀁤 0 by using the Riemann-Liouville integral formula
    • 5.15 Approximate representation of a fractional Laplace variable by rational polynomials in the integer power of the Laplace variable
    • 5.16 Generalized Laplace transform
      • 5.16.1 Laplace transform for the Riemann-Liouville fractional derivative
      • 5.16.2 Laplace transform for the Caputo fractional derivative
      • 5.16.3 Requirement of the fractional order of the initial states of the RL derivative and the integer order (classical) initial states for Caputo derivative initializations
      • 5.16.4 Generalized Laplace transforms formula for a fractional derivative with a type parameter
      • 5.16.5 Generalized Laplace transform demonstrated for use in a fractional differential equation
    • 5.17 Generalized stationary conditions
    • 5.18 Demonstration of the generalized Laplace transform for solving the initialized fractional differential equation
      • 5.18.1 Composition of a fractional differential equation for a Riemann-Liouville fractional derivative
      • 5.18.2 Composition of the fractional differential equation for the Caputo fractional derivative
      • 5.18.3 Applying the concept of the initialization function of fractional derivatives in a fractional differential equation
    • 5.19 Fourier transform of a fractional derivative operator
      • 5.19.1 Fourier transform from a Laplace transform
      • 5.19.2 Similarity and dissimilarity in Laplace and Fourier transforms
    • 5.20 Complex order differ-integrations described via the Laplace transform
    • 5.21 Short summary
    • 5.22 References
  • Chapter Six
    • 6.1 Introduction
    • 6.2 Tricks in solving some fractional differential equations
      • 6.2.1 A method using the Laplace transform for a fractional differential equation
      • 6.2.2 Alternative method to solve a fractional differential equation
      • 6.2.3 A method using the Laplace transform for a fractional integral equation
      • 6.2.4 An alternative method to solve fractional integral equations
    • 6.3 Abel’s fractional integral equation of ‘tautochrone’-a classical problem
    • 6.4 Using the power series expansion method to obtain inverse Laplace transforms
    • 6.5 Using Laplace transform techniques and power series expansion to solve simple fractional differential equations
    • 6.6 The contour integration method for obtaining inverse Laplace transforms
    • 6.7 Operational calculus of applying Heaviside units to a partial differential equation
    • 6.8 The response of a fractional differential equation with detailed analysis of power law functions
      • 6.8.1 Demonstration of power law functions as a solution to a simple fractional differential equation excited by a constant step function
      • 6.8.2 Demonstration of a power series function as a solution to the fractional differential equation excited by a constant step function
      • 6.8.3 A demonstration of the power series function as a solution to a fractional differential equation excited by the constant step function through use of the Laplace transform
      • 6.8.4 Obtaining approximate short-time and long-time responses from a power series solution for a fractional differential equation excited by a constant step function
    • 6.9 An analytical method to obtain inverse Laplace transforms ‘without contour integration’ - the Berberan-Santos method
      • 6.9.1 Development of the Berberan-Santos technique to obtain a distribution function for the decay rates for the relaxation function of time
      • 6.9.2 Derivation of the Berberan-Santos method
    • 6.10 A few examples of the inverse Laplace transform of functions available in standard Laplace transform tables obtained using the Berberan-Santos method
      • 6.10.1 The inverse Laplace transform of function
      • 6.10.2 The inverse Laplace transform of the function
      • 6.10.3 The inverse Laplace transform of function
      • 6.10.4 The inverse Laplace transform of function
      • 6.10.5 The inverse Laplace transform of function
      • 6.10.6 The inverse Laplace transform of the function
    • 6.11 Examples of using the Berberan-Santos method to obtain the Laplace inversion of a few functions that are not provided in standard Laplace Transform tables
      • 6.11.1 The inverse Laplace transform of function
      • 6.11.2 The inverse Laplace transform of function
      • 6.11.3 The inverse Laplace transform of the function
      • 6.11.4 An integral representation of the Mittag-Leffler function using the Berberan-Santos method
      • 6.11.5 The inverse Laplace transform of the function
      • 6.11.6 The inverse Laplace transform of function
      • 6.11.7 The inverse Laplace transform of function
    • 6.12 The relaxation-response with the Mittag-Leffler function vis-à-vis the power law function as obtained for fractional differential equation analysis
    • 6.13 Use of several fractional order derivatives in order to obtain a generalized fractional differential equation
    • 6.14 Short summary
    • 6.15 References
  • Chapter Seven
    • 7.1 Introduction
    • 7.2 A first order linear differential equation and its fractional generalization using the Caputo derivative
      • 7.2.1 Demonstrating the similarity of a classical differential operator with the Caputo fractional differential operator in a differential equation solution pattern
      • 7.2.2 The first order differential equation of a relaxation process with an initial condition
      • 7.2.3 A first order differential equation’s homogeneous and complementary parts
      • 7.2.4 A fractional order differential equation composed using the Caputo derivative’s homogeneous and complementary parts
      • 7.2.5 Can we just replace the exp(􀀐􀁏t) in the classical case with E ( t􀁄 )􀁄 􀀐􀁏 in order to get the total solution for the fractional differential equation composed using the Caputo derivative?
      • 7.2.6 Getting a particular solution to a fractional differential equation composed by the Caputo derivative through a fractional integration operation using a classical convolution process
      • 7.2.7 A homogeneous fractional differential equation composed using a Caputo derivative is driven by a fractional RL derivative of a Heaviside unit step function
    • 7.3 The first order linear differential equation and its fractional generalization using the Riemann-Liouville (RL) derivative
      • 7.3.1 Demonstrating the similarity of a classical differential operator with the Riemann-Liouville fractional differential operator in a differential equation solution
      • 7.3.2 A fractional order differential equation composed with the Riemann-Liouville derivative is fundamental and a particular solution is offered by a given fractional order initial state
      • 7.3.3 Verification of a fractional initial state from the obtained solution for a fractional order differential equation composed using the Riemann-Liouville derivative
      • 7.3.4 Modification in a solution for a fractional differential equation composed using the Riemann-Liouville derivative solution with classical integer order initial states instead of fractional order initial states
      • 7.3.5 The requirement of one Green’s function for a fractional order differential equation composed using the Riemann-Liouville derivative and two Green’s functions for the Caputo derivative
    • 7.4 Formal description of a fractional differential and integral equation
      • 7.4.1 Concepts of an ordinary differential equation as extended to fractional differential and integral equations
      • 7.4.2 The indicial polynomial corresponding to the fractional differential equation
      • 7.4.3 The order of the fractional differential equation and the number of linearly independent solutions
    • 7.5 Finding a solution to a homogeneous fractional differential equation
      • 7.5.1 An ordinary homogeneous classical differential equation and solution in terms of an exponential function
      • 7.5.2 A fractional differential equation and solution candidate in terms of a higher transcendental function
      • 7.5.3 A direct approach using the Miller-Ross function as a solution for a fractional differential equation (FDE)
      • 7.5.4 Getting a solution from the roots of an indicial polynomial corresponding to the FDE using the Miller-Ross function
    • 7.6 Motivation for the Laplace transform technique
      • 7.6.1 The indicial polynomial is the same as we get from the Laplace transform of a fractional differential equation
      • 7.6.2 A solution with distinct roots in an indicial polynomial
      • 7.6.3 The solution with equal roots for indicial polynomials
    • 7.7 A linearly independent solution of the fractional differential equation (FDE)
    • 7.8 The explicit solution for a homogeneous fractional differential equation (FDE)
    • 7.9 The non homogeneous fractional differential equation and its solution
    • 7.10 Fractional integral equations and their solution
      • 7.10.1 Describing a fractional integral equation
      • 7.10.2 Extension of the final value theorem of integral calculus
      • 7.10.3 The solution to fractional integral equations
    • 7.11 Examples of fractional integral equations with explicit solutions
    • 7.12 Sequential fractional derivative of the Miller-Ross type ka x􀁄 D and sequential fractional differential equations (SFDE)
      • 7.12.1 The sequential fractional differential equation (SFDE)
      • 7.12.2 The matrix form representation of SFDE
    • 7.13 Solution of the ordinary differential equation using state transition matrices -a review
      • 7.13.1 Origin of the state transition matrix in a solution to systems of linear differential equations
      • 7.13.2 The state transition matrix as Green’s function is a solution to the homogeneous system of differential equations
      • 7.13.3 A demonstration of the ways to represent the state transition matrix and its usage in multivariate dynamic systems
    • 7.14 ‘Alpha-exponential functions’ as eigen-functions for the Riemann-Liouville (RL) and Caputo fractional derivative operators
      • 7.14.1 The alpha-exponential functions (1 & 2) for fractional derivative operators are similar to the exponential function as an eigen-function for a classical derivative
      • 7.14.2 The alpha-exponential functions 􀁄 (t) e􀁄At 􀀩 􀀠 and 􀁄 (t) e􀁄At 􀀩􀀄 􀀠 􀀄 as related to the Mittag-Leffler function
      • 7.14.3 Defining the alpha-exponential functions (1 & 2) via a kernel of convolution as the power law functions and their Laplace transforms
      • 7.14.4 The alpha-exponential functions (1 & 2) are related via the convolution relationship
    • 7.15 Fractional derivatives of the ‘alpha-exponential functions (1 & 2)’
      • 7.15.1 The Caputo derivative of the alpha-exponential function-2
      • 7.15.2 The Riemann-Liouville derivative of the alpha-exponential function-1
      • 7.15.3 The backward Riemann-Liouville derivative of the alpha-exponential function-1
      • 7.15.4 The relationship between the state transition matrices
    • 7.16 The general solution to a sequential fractional differential equation using ‘alpha-exponential functions’
      • 7.16.1 The homogeneous SFDE and its characteristic equation (or indicial polynomial)
      • 7.16.2 A generalized Wronskian
      • 7.16.3 Linearly independent solutions of SFDEs
      • 7.16.4 A demonstration of linearly independent solutions of the SFDE in an equation of motion with fractional order damping
    • 7.17 The solution of a multivariate system of a fractional order differential equation with the RL Caputo derivative using a state transition matrix of the ‘alpha-exponential functions -1 & 2’
      • 7.17.1 The multivariate system with an RL derivative and its solution with the state transition matrix
      • 7.17.2 A Multivariate system with the Caputo derivative and its solution with state transition matrices
      • 7.17.3 Formalizing the multivariate problem to get a state trajectory as a solution with the defined state transition matrices
      • 7.17.4 An application to obtain the state trajectory solution for a multivariate fractional differential equation system with a given initial condition and a forcing function
      • 7.17.5 An application to obtain the state trajectory solution for a multivariate fractional differential equation with matrix A as a skew-symmetric system with a given initial condition and forcing function
    • 7.18 The solution to a fractional differential equation of type 􀀋* 􀀌 􀀋* 􀀌b D E0 E1 D 􀁖 􀀎 􀁄 􀁖 􀀠 􀁈 􀀎 􀁄 􀁈 with RL or Caputo formulations
      • 7.18.1 Using an RL-derivative formulation for a solution of 􀁈 with a known 􀁖 input
      • 7.18.2 Using the Caputo derivative formulation for a solution of 􀁈 with a known 􀁖 input
      • 7.18.3 Using the RL-derivative formulation for a solution of 􀁖 with a known 􀁈 input
      • 7.18.4 Using the Caputo derivative formulation for a solution of 􀁖 with a known 􀁈 input
    • 7.19 A generalization of a fractional differential equation with a sequential fractional derivative
    • 7.20 Short summary
    • 7.21 References
  • Chapter Eight
    • 8.1 Introduction
    • 8.2 Time fractional diffusion wave equation (TFDWE)
    • 8.3 Green’s function
      • 8.3.1 Green’s function for the diffusion equation with auxiliary functions as defined by a similarity variable
      • 8.3.2 Green’s function for the wave equation with auxiliary functions as defined by the similarity variable
    • 8.4 Solution of TFDWE via the Laplace transform for the Green’s function for Cauchy and signalling problems
      • 8.4.1 The Cauchy problem for a TFDWE solution in the Laplace domain
      • 8.4.2 Signalling problem for TFDWE in Laplace domain
      • 8.4.3 The solution of Cauchy and the signalling problems of TFDWE in the time domain by inverting the Laplace domains solution by using auxiliary functions
    • 8.5 Reciprocal relation between Green’s function of Cauchy and signalling problem
    • 8.6 The origin of the auxiliary function from the Green’s function of the Cauchy problem
      • 8.6.1 An inverse Laplace transform of the Green’s function
      • 8.6.2 The description of the Hankel contour from the Bromwich path integral used for an integral representation of the auxiliary function
    • 8.7 The integral and series representation of auxiliary functions
      • 8.7.1 Getting a series representation of the auxiliary function M􀁘 (z) from its integral representation
      • 8.7.2 The relationship of Wright’s function to M-Wright’s function or the auxiliary function
      • 8.7.3 Proof of the property of the auxiliary function
      • 8.7.4 Moments of the M-Wright function or the auxiliary function
    • 8.8 The auxiliary functions M (z) 􀁘 and F (z) 􀁘 as a fractional generalisation of the Gaussian function
      • 8.8.1 The series representation of the Wright function derived from its integral representation formula
      • 8.8.2 A further series representation of auxiliary functions
      • 8.8.3 The relationship between auxiliary functions
      • 8.8.4 An auxiliary function M􀁘 (z) for 􀁘 at 12 , 13 and 0
    • 8.9 Laplace and Fourier transforms of the auxiliary functions
      • 8.9.1 The inverse Laplace transform of function X (s) e s􀀠 􀀐 􀁘 giving F (t 􀁘 )􀁘􀀐 and M (t 􀁘 )
      • 8.9.2 The Mittag-Leffler function as an integral representation on Hankel’s path
      • 8.9.3 The Laplace transform of an auxiliary function M􀁘 (t) as the Mittag-Leffler function
      • 8.9.4 The Fourier Transform of the auxiliary function
    • 8.10 A graphical representation of the M-Wright or auxiliary function
    • 8.11 Auxiliary function in two variables M (x,t) 􀁘 and its Laplace & Fourier transforms
      • 8.11.1 Defining M􀁘 (x, t) in relation to M (xt 􀁘 )
      • 8.11.2 Laplace Transform of M􀁘 (x, t)
      • 8.11.3 Fourier transform of M􀁘 (x, t)
      • 8.11.4 Special cases for M􀁘 (x, t) for 􀁘 as 12 and one
    • 8.12 The use of the two variable auxiliary functions in TFDWE
      • 8.12.1 Time fractional diffusion equation and its solution with M􀁘 (x, t)
      • 8.12.2 A time fractional drift equation and its solution with M (x,t)
    • 8.13 Reviewing time fractional diffusion equations with several manifestations
      • 8.13.1 Classical diffusion equation
      • 8.13.2 The Green’s function for a classical diffusion equation and its moment
      • 8.13.3 A classical diffusion equation with a stretched time variable in its Green’s function and moment
      • 8.13.4 Time fractional diffusion equation
      • 8.13.5 The Green’s function for a time fractional diffusion equation and its moments
      • 8.13.6 A time fractional diffusion equation with a stretched time variable
      • 8.13.7 The Green’s function for a time fractional diffusion equation with a stretched time variable
    • 8.14 Short summary
    • 8.15 References
  • Chapter Nine
    • 9.1 Introduction
    • 9.2 Condition of term-by-term differ-integration of a series
      • 9.2.1 Linearity and homogeneity of fractional differ-integral operators
      • 9.2.2 Convergence of a differ-integration of series
      • 9.2.3 Discussion on convergence by application of the Riemann-Liouville fractional integration formula from term-by-term to a differ-integrable series
      • 9.2.4 Discussion on convergence by application of the Euler fractional derivative formula for power functions term-by-term into differ-integrable series
    • 9.3 Composition rule in fractional calculus
      • 9.3.1 Condition for the inverse operation f 􀀠 D􀀐QDQ f to be satisfied
      • 9.3.2 Condition DqDQ f 􀀠 Dq􀀎Q f to be satisfied
      • 9.3.3 Condition for DNDq f 􀀠 DqDN f , N as positive integer
      • 9.3.4 Generalising the composition DqDQ f to Dq􀀎Q f by initial values of f
      • 9.4 The reversibility of a differ-integral operator DQ f 􀀠 g to f 􀀠 D􀀐Qg
    • 9.5 An alternate representation for fractional differ-integration for real analytic functions
      • 9.5.1 A representation using the fractional differ-integral formula of the Riemann-Liouville
      • 9.5.2 Representation using the Grunwald-Letnikov formula for differ-integration
    • 9.6 The fractional derivative for non-differentiable functions is defined
      • 9.6.1 Defining a quotient i.e. x / 􀀋 t 􀀌􀁄 􀀧 􀀧 for a fractional derivative such that the fractional derivative of the constant is zero
      • 9.6.2 Corollary to the definition of a fractional derivative
    • 9.7 Utilising the Mittag-Leffler function to get suitable fractional derivatives as a conjugation to the classical derivative
    • 9.8 The fractional derivative via fractional difference and its Laplace Transform
      • 9.8.1 Defining the fractional derivative f (􀁄 ) via the Forward Shift Operator
      • 9.8.2 Getting the fractional derivative of the constant as a non-zero by the Laplace transform technique
      • 9.8.3 Getting the Laplace Transform
      • 9.8.4 Getting the Laplace Transform
      • 9.8.5 The definition of fractional differentiability is satisfactory for self-similar functions
    • 9.9 The modified fractional derivative for a function with non-zero initial condition-Jumarie type
      • 9.9.1 Defining the fractional derivative via the offset function construction to have a fractional derivative of the constant as a zero-Jumarie type
      • 9.9.2 The Laplace Transform of the Modified RL fractional Derivative and its conjugation to a classical derivative
    • 9.10 Integral representation of modified RL fractional derivative of Jumarie type
    • 9.11 The application of the modified fractional RL derivative of the Jumarie type to various functions
      • 9.11.1 Modified RL derivative of order 􀁄 for function
      • 9.11.2 A modified RL derivative of order 􀁅 􀀡 0 for function
      • 9.11.3 The RL fractional derivative of order 􀁄 for function
      • 9.11.4 The modified RL derivative of order 􀁄 for function
      • 9.11.5 Applying the modified RL fractional derivative to a non-differentiable point of a function: 1f (x) 􀀠 x 2 􀀎 b
      • 9.11.6 Defining the critical order for a non-differentiable point by use of the modified RL fractional derivative
    • 9.12 Fractional Taylor series
    • 9.13 The use of fractional Taylor series
    • 9.14 Conversion formulas for fractional differentials
    • 9.15 Integration with respect to fractional differentials
      • 9.15.1 Integration w.r.t. (d􀁛 )􀁄 and its relationship to the Riemann-Liouville fractional integration formula
      • 9.15.2 A demonstration of the integration w.r.t. (d􀁛 )􀁄 for f (x) 􀀠 x􀁊 ,f (x) 􀀠 1, f (x) 􀀠􀁇 (x) and f (x) E ( x􀁄 )
      • 9.15.3 Some useful identities for the integration w.r.t.
    • 9.16 Leibniz’s rule for the product of two functions for a modified fractional derivative
    • 9.17 Integration by parts for a fractional differential
    • 9.18 The chain rule for a modified fractional derivative
      • 9.18.1 The derivation of three formulas for the chain rule with a modified fractional derivative
      • 9.18.2 The application of three formulas for the chain rule with a modified fractional derivative for different cases
    • 9.19 Coarse grained system
      • 9.19.1 Need for coarse graining
      • 9.19.2 The fractional velocity u􀁄 with coarse grained time differential
      • 9.19.3 Fractional velocity v􀁄 with a coarse grained space differential
    • 9.20 The solution for a fractional differential equation with a modified fractional derivative
      • 9.20.1 Fractional differential equation defined with modified fractional derivative
      • 9.20.2 Defining the function Ln􀁄 (x) in conjugation to ln x and obtaining the solution to a fractional differential equation with a modified fractional derivative
      • 9.20.3 Proof of the identity 􀀋E (ax􀁄 )􀀌􀀋E (ay􀁄 )􀀌 E 􀀋a(x y)􀁄 􀀌􀁄 􀁄 􀁄 􀁴 􀀎 in respect of a modified fractional derivative definition
      • 9.20.4 The fractional differential equation x(􀁄 ) (t) 􀀠 􀁏 􀀋a(t)􀀌 x(t) 􀀎 b(t) with a modified fractional derivative
      • 9.20.5 The fractional differential equation x(􀁄 ) (t) 􀀠 􀁏t1􀀐􀁄 x(t) with a modified fractional derivative
      • 9.20.6 The fractional differential equation: a 􀀋x(2􀁄 ) (t)􀀌􀀎 b􀀋x(􀁄 ) (t)􀀌􀀎 c 􀀠 0 with a modified fractional derivative
    • 9.21 The application to dynamics close to the equilibrium position are subjected to coarse graining
      • 9.21.1 The dynamic system at equilibrium
      • 9.21.2 Dynamic system subjected to coarse graining in time differential dt
      • 9.21.3 The dynamic system subjected to coarse graining in space differential dx
      • 9.21.4 The system when both differentials dx and dt are subjected to coarse graining
    • 9.22 Using the Mittag-Leffler function in Integral Transform formulas
      • 9.22.1 Defining the fractional Laplace transform by using the Mittag-Leffler function
      • 9.22.2 Defining a fractional convolution integration process by use of the fractional Laplace Transforms
      • 9.22.3 Demonstrating fractional Laplace Transform for the Heaviside unit step function
      • 9.22.4 Defining the fractional delta distribution function (x) 􀁄 􀁇 and its fractional Fourier integral representation by use of the Mittag-Leffler function
      • 9.22.5 The inverse fractional Laplace transformation by the Mittag-Leffler Function
      • 9.22.6 The Mittag-Leffler function to define the fractional order Gamma function
    • 9.23 Derivatives with the order as a continuous distributed function
      • 9.23.1 The fractional differential equation generalised from integer order to fractional order
      • 9.23.2 The concept of order distribution
      • 9.23.3 From the discrete delta distributed function to a continuous distribution function for describing fractional orders
      • 9.23.4 The continuous order distributed differential equation
    • 9.24 Short summary
    • 9.25 References
  • Appendix A
    • A.1 Hyper-geometric functions
    • A.2 Mittag-Leffler function
      • A.2.1 One parameter Mittag-Leffler function
      • A.2.2 The two parameter Mittag-Leffler function
      • A.2.3 Graphical representations of Mittag-Leffler function
      • A.2.4 Generalized Hyperbolic and Trigonometric Functions
    • A.3 The error function and its fractional generalization
    • A.4 Variants of the Mittag-Leffler function
    • A.5 The Laplace integral and its connection to the Mittag-Leffler function
    • A.6 The Laplace transforms of the Mittag-Leffler function and several other variants
    • A.7 Agrawal Function
    • A.8 Erdelyi’s Function
    • A.9 The Robotnov-Hartley function
    • A.10 Miller-Ross function
    • A.11 The generalized cosine and sine function in the Miller-Ross formulation
    • A.12 Generalized R function and G function
    • A.13-Bessel Function
    • A.14 Wright Function
    • A.15 Prabhakar Function
      • A-15.1 The three parameter Mittag-Leffler function
      • A.15.2 Prabhakar Integral
      • A.15.3 The Prabhakar integral as a series-sum of the Riemann-Liouville fractional integrals
  • Appendix B
  • Appendix C
  • Appendix D
  • Appendix E
    • E.1 Branch-points and branches in multi-valued function
    • E.2 The branch point of order (n 􀀐1) and branch-cut for n 􀀐 valued function
    • E.3 Multiple Riemann sheets connected along the branch-cut for a multi-valued function
    • E.4 An example elaborating branches and branch-cut for a two-valued function
    • E.5 Choosing the contour with branch-cut for a function having branch-point
  • Appendix F
    • F.1 Restriction in Series Representation of the Mittag-Leffler Function
    • F.2 The Integral Representation of the two-parameter Mittag-Leffler function
    • F.3 The formula for a two-parameter Mittag-Leffler function with a negative order
    • F.4 The series representation of the two-parameter Mittag-Leffler function with a negative order
    • F.5 The graphical representation of the Mittag-Leffler function with the negative order
    • F.6 Comparisons of the functional relationship of , E (x) 􀁄 􀁅 and , E (x)
  • Appendix G
    • G.1 Revising basics of Laplace transforms
    • G.2-The inverse Laplace transform via contour integration
    • G.3 Jordan’s Lemma
    • G.4 The application of Residue Calculus to get an Inverse Laplace Transform of X (s) 􀀠 2e􀀐2s / (s2 􀀎 4) by contour integration
    • G.5 The application of residue calculus to get the inverse Laplace transform of X (s) 􀀠 s 􀀐 a by contour integration on the branch cut in a complex plane
    • G.6 The application of residue calculus to get the inverse Laplace transform of ln( ) ( ) b ass s X s 􀀮 􀀐 􀀠 by contour integration on the branch cut in a complex plane
    • G.7 The application of residue calculus to get the inverse Laplace transform of ln ( ) sb sas X s 􀀠 􀀐 by contour integration on branch cut in the complex plane
  • Bibliography

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