Preface xi

**Part I Invariant derivatives of functionals and numerical methods for functional differential equations 1**

**1 The invariant derivative of functionals 3**

1 Functional derivatives 3

1.1 The Frechet derivative 4

1.2 The Gateaux derivative 4

**2 Classification of functionals on C[a, b] 5**

2.1 Regular functionals 5

2.2 Singular functionals 6

**3 Calculation of a functional along a line 6**

3.1 Shift operators 6

3.2 Superposition of a functional and a function 7

3.3 Dini derivatives 8

**4 Discussion of two examples 8**

4.1 Derivative of a function along a curve 8

4.2 Derivative of a functional along a curve 9

**5 The invariant derivative 11**

5.1 The invariant derivative 11

5.2 The invariant derivative in the class B[a, b] 12

5.3 Examples 13

**6 Properties of the invariant derivative 16**

6.1 Principles of calculating invariant derivatives 16

6.2 The invariant differentiability and invariant continuity 19

6.3 High order invariant derivatives 20

6.4 Series expansion 21

**7 Several variables 21**

7.1 Notation 21

7.2 Shift operator 21

7.3 Partial invariant derivative 22

**8 Generalized derivatives of nonlinear functionals 22**

8.1 Introduction 22

8.2 Distributions (generalized functions) 24

8.3 Generalized derivatives of nonlinear distributions 25

8.4 Properties of generalized derivatives 27

8.5 Generalized derivative (multidimensional case) 28

8.6 The space SD of nonlinear distributions 29

8.7 Basis on shift 30

8.8 Primitive 31

8.9 Generalized solutions of nonlinear differential equations 34

8.10 Linear differential equations with variables coeffecients 36

**9 Functionals on Q[−t ; 0] 37**

9.1 Regular functionals 39

9.2 Singular functionals 40

9.3 Specific functionals 40

9.4 Support of a functional 41

**10 Functionals on R × Rn × Q[−t; 0] 42**

10.1 Regular functionals 42

10.2 Singular functionals 44

10.3 Volterra functionals 44

10.4 Support of a functional 45

**11 The invariant derivative 45**

11.1 Invariant derivative of a functional 46

11.2 Examples 48

11.3 Invariant continuity and invariant differentiability 58

11.4 Invariant derivative in the class B[−t; 0] 59

**12 Coinvariant derivative 65**

12.1 Coinvariant derivative of functionals 65

12.2 Coinvariant derivative in a class B[−t; 0] 68

12.3 Properties of the coinvariant derivative 71

12.4 Partial derivatives of high order 73

12.5 Formulas of i–smooth calculus for mappings 75

**13 Brief overview of Functional Differential Equation theory 76**

13.1 Functional Differential Equations 76

13.2 FDE types 78

13.3 Modeling by FDE 80

13.4 Phase space and FDE conditional representation 81

**14 Existence and uniqueness of FDE solutions 84**

14.1 The classic solutions 84

14.2 Caratheodory solutions 92

14.3 The step method for systems with discrete delays 94

**15 Smoothness of solutions and expansion into the Taylor series 95**

15.1 Density of special initial functions 98

15.2 Expansion of FDE solutions into Taylor series 100

**16 The sewing procedure 103**

16.1 General case 104

16.2 Sewing (modification) by polynomials 105

16.3 The sewing procedure of the second order 107

16.4 Sewing procedure of the second order for linear delay differential equation 109

**2 Numerical methods for functional differential equations 113**

**17 Numerical Euler method 115**

**18 Numerical Runge-Kutta-like methods 118**

18.1 Methods of interpolation and extrapolation 119

18.2 Explicit Runge-Kutta-like methods 127

18.3 Order of the residual of ERK-methods 132

18.4 Implicit Runge-Kutta-like methods 136

**19 Multistep numerical methods 142**

19.1 Numerical models 143

19.2 Order of convergence 143

19.3 Approximation order. Starting procedure 145

**20 Startingless multistep methods 146**

20.1 Explicit methods 147

20.2 Implicit methods 148

20.3 Startingless multistep methods 150

**21 Nordsik methods 152**

21.1 Methods based on calculation of high order derivatives 155

21.2 Various methods based on the separation of finite-dimensional and infinite-dimensional components of the phase state 158

**22 General linear methods of numerical solving functional differential equations 162**

22.1 Introduction 162

22.2 Methodology of classification numerical FDE models 173

22.3 Necessary and sufficient conditions of convergence with order p 181

22.4 Asymptotic expansion of the global error 186

**23 Algorithms with variable step-size and some aspects of computer realization of numerical models 196**

23.1 ERK-like methods with variable step 197

23.2 Methods of interpolation and extrapolation of discrete model prehistory 202

23.3 Choice of the step size 207

23.4 Influence of the approximate calculating functionals of the right-hand side of FDEs 212

23.5 Test problems 217

**24 Soft ware package Time-delay System Toolbox 230**

24.1 Introduction 230

24.2 Algorithms 230

24.3 The structure of the Time-delay System Toolbox 231

24.4 Descriptions of some programs 232

**Part II Invariant and generalized derivatives of functions and functionals 251**

**25 The invariant derivative of functions 253**

25.1 The invariant derivative of functions 253

25.2 Examples 256

25.3 Relationship between the invariant derivative and the Sobolev generalized derivative 258

**26 Relation of the Sobolev generalized derivative and the generalized derivative of the distribution theory 261**

26.1 Affinitivity of the generalized derivative of the distribution theory and the Sobolev generalized derivative 261

26.2 Multiplication of generalized functions at the Hamel basis 262

Bibliography 267

Index 271