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## Description

** Transmutations; Singular and Fractional Differential Equations with Applications to Mathematical Physics**;

*(PDF)*hyperlinks troublesome issues with related extra easy ones. The ebook’s technique works for differential and integral equations and techniques and for a lot of theoretical and utilized issues in mathematical physics; arithmetic; chance and statistics; utilized pc science and numerical strategies. Along with being uncovered to current advances; readers study to use transmutation strategies not simply as sensible instruments; but in addition as autos that present theoretical insights.

- Enables lecturers; researchers; and college students to discover materials beneath the only “roof”
- Integrates mathematical rigor with an illuminating exposition filled with historic notes and fascinating particulars
- Provides the common transmutation methodology as probably the most highly effective for fixing many issues in arithmetic; chance and statistics; mathematical physics; utilized pc science and numerical strategies

**NOTE: The product solely consists of the ebook; ***Transmutations; Singular and Fractional Differential Equations with Applications to Mathematical Physics* in PDF. No access codes are included.

## Additional information

book-author | Elina Shishkina, Sergei Sitnik |
---|---|

publisher | Academic Press; 1st edition |

file-type | |

pages | Pages |

language | English |

asin | B08DQTNK67 |

isbn10 | 0128197811; 028204079 |

isbn13 | 9780128197813/ 9780128204078 |

## Table of contents

Table of contents :

Contents

Acknowledgments and thanks

Introduction

1 Basic deﬁnitions and propositions

1.1 Special functions

1.1.1 Gamma function, beta function, Pochhammer symbol, and error function

1.1.2 Bessel functions

1.1.3 Hypergeometric type functions

1.1.4 Polynomials

1.2 Functional spaces

1.2.1 Orthant Rn+, Cevm, Sev, and Lpγ spaces

1.2.2 Weighted measure, space L∞γ, and deﬁnition of weak (p,q)γ type operators

1.2.3 Space of weighted generalized functions Sev’, absolutely continuous functions, and unitary operators

1.2.4 Mixed case

1.3 Integral transforms and Lizorkin-Samko space

1.3.1 One-dimensional integral transforms with Bessel functions in the kernels and Mellin transform

1.3.2 Properties of composition of integral transforms with Bessel functions in the kernel

1.3.3 Multi-dimensional integral transforms

1.4 Basic facts and formulas

1.4.1 Kipriyanov’s classiﬁcation of second order linear partial differential equations

1.4.2 Divergence theorem and Green’s second identity for B-elliptic and B-hyperbolic operators

1.4.3 Tricomi equation

1.4.4 Abstract Euler-Poisson-Darboux equation

2 Basics of fractional calculus and fractional order differential equations

2.1 Short history of fractional calculus and fractional order differential equations

2.1.1 One-dimensional fractional derivatives and integrals

2.1.2 Fractional derivatives in mechanics

2.1.3 Fractional powers of multi-dimensional operators

2.1.4 Differential equations of fractional order

2.2 Standard fractional order integro-differential operators

2.2.1 Riemann-Liouville fractional integrals and derivatives on a segment

2.2.2 Riemann-Liouville fractional integrals and derivatives on a semiaxis

2.2.3 Gerasimov-Caputo fractional derivatives

2.2.4 Dzrbashian-Nersesyan fractional operators and sequential order fractional operators

2.3 Some more fractional order integro-differential operators

2.3.1 The Erdélyi-Kober operators

2.3.2 Fractional integrals and fractional derivatives of a function with respect to another function

2.3.3 Averaged or distributed order fractional operators

2.3.4 Saigo, Love, and other fractional operators with special function kernels

2.4 Integral transforms and basic differential equations of fractional order

2.4.1 Integral transforms of fractional integrals and derivatives

2.4.1.1 Laplace transform of Riemann-Liouville fractional integrals and derivatives on semiaxes

2.4.1.2 Mellin transform of Riemann-Liouville fractional integrals and derivatives on semiaxes

2.4.1.3 Laplace transform of Gerasimov-Caputo fractional derivatives on semiaxes

2.4.2 Laplace transform method for the homogeneous equations with constant coefﬁcients with the left-sided Riemann-Liouville fractional derivatives of the order α on a semiaxis (0,∞)

2.4.3 Laplace transform method for homogeneous equations with constant coefﬁcients with the left-sided Gerasimov-Caputo fractional derivatives of the order α on a semiaxis [0,∞)

2.4.4 Mellin integral transform and nonhomogeneous linear differential equations of fractional order

3 Essentials of transmutations

3.1 Deﬁnition of the transmutation operator, some examples of classical transmutations

3.1.1 Introduction to transmutation theory

3.1.2 Some examples of classical transmutations

3.2 Transmutations for Sturm-Liouville operator

3.2.1 Description of the problem and terminology

3.2.2 Transmutations in the form of the second kind Fredholm operators

3.2.3 Transmutations in the form of the second kind Volterra operators

3.2.4 Transmutations in the form of the ﬁrst kind Volterra operators

3.3 Transmutations for different potentials

3.3.1 Kernel of transmutation intertwining operators of the Sturm-Liouville type

3.3.2 Cases when potential q(x) is an exponential function

3.3.3 Cases when potential q(x) is constant

3.3.4 Estimates of kernels and point formulas for estimating the error for calculating transmutation operators

3.4 Transmutations for singular Bessel operator

3.4.1 One-dimensional Poisson operator

3.4.2 Multi-dimensional Poisson operator

3.4.3 Generalized translation

3.4.4 Weighted spherical mean

4 Weighted generalized functions generated by quadratic forms

4.1 The weighted generalized function associated with a positive quadratic form and concentrated on a part of a cone

4.1.1 B-ultrahyperbolic operator

4.1.2 Weighted generalized function associated with a positive quadratic form

4.1.3 Weighted generalized function δγ(P)

4.2 Weighted generalized functions realized by the degrees of quadratic forms

4.2.1 Weighted generalized functions Pγ,±λ

4.2.2 The weighted generalized function Pλγ and (P±i 0)γλ associated with a quadratic form with complex coefﬁcients

4.3 Other weighted generalized functions associated with a quadratic form

4.3.1 Functions (w2-|x|2)+,γλ and (c2+P±i0)λγ

4.3.2 General weighted generalized functions connected with quadratic form

4.4 Hankel transform of weighted generalized functions generated by the quadratic form

4.4.1 Hankel transform of rλγ

4.4.2 Hankel transforms of functions Pλγ, (P±i0)λγ, and Pλγ,±

4.4.3 Hankel transforms of functions (w2-|x|2)+,γλ and (c2+P±i0)λγ

5 Buschman-Erdélyi integral and transmutation operators

5.1 Buschman-Erdélyi transmutations of the ﬁrst kind

5.1.1 Sonine-Poisson-Delsarte transmutations

5.1.2 Deﬁnition and main properties of Buschman-Erdélyi transmutations of the ﬁrst kind

5.1.3 Factorizations of the ﬁrst kind Buschman-Erdélyi operators and the Mellin transform

5.2 Buschman-Erdélyi transmutations of the second and third kind

5.2.1 Second kind Buschman-Erdélyi transmutation operators

5.2.2 Sonine-Katrakhov and Poisson-Katrakhov transmutations

5.2.3 Buschman-Erdélyi transmutations of the third kind with arbitrary weight function

5.2.4 Some applications of Buschman-Erdélyi transmutations

5.3 Multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels

5.3.1 Basic deﬁnitions

5.3.2 The n-dimensional Mellin transform and its properties

5.3.3 Lν,2-theory and the inversion formulas for the modiﬁed H-transform

5.3.4 Inversion of H1σ,κ

5.4 Representations in the form of modiﬁed H-transform

5.4.1 Mellin transform of auxiliary functions K1( x) and K2( x)

5.4.2 Mellin transform of Pγδ,1( x) and Pγδ,2( x)

5.4.3 Lν,2-theory of the transforms Pγδ,kf (k=1,2)

5.4.4 Inversion formulas for transforms Pγδ,kf (k=1,2)

6 Integral transforms composition method for transmutations

6.1 Basic ideas and deﬁnitions of the integral transforms composition method for the study of transmutations

6.1.1 Background of ITCM

6.1.2 What is ITCM and how to use it?

6.2 Application of the ITCM to derive transmutations connected with the Bessel operator

6.2.1 Index shift for the Bessel operator

6.2.2 Poisson and “descent” operators, negative fractional power of the Bessel operator

6.2.3 ITCM for generalized translation and the weighted spherical mean

6.2.4 Integral representations of transmutations for perturbed differential Bessel operators

6.3 Connection formulas for solutions to singular differential equations via the ITCM

6.3.1 Application of transmutations for ﬁnding general solutions to Euler-Poisson-Darboux type equations

6.3.2 Application of transmutations for ﬁnding solutions to general Euler-Poisson-Darboux type equations

6.3.3 Application of transmutations for ﬁnding general solutions to singular Cauchy problems

7 Differential equations with Bessel operator

7.1 General Euler-Poisson-Darboux equation

7.1.1 The ﬁrst Cauchy problem for the general Euler-Poisson-Darboux equation

7.1.2 The second Cauchy problem for the general Euler-Poisson-Darboux equation

7.1.3 The singular Cauchy problem for the generalized homogeneous Euler-Poisson-Darboux equation

7.1.4 Examples

7.2 Hyperbolic and ultrahyperbolic equations with Bessel operator in spaces of weighted distributions

7.2.1 The generalized Euler-Poisson-Darboux equation and the singular Klein-Gordon equation

7.2.2 Iterated ultrahyperbolic equation with Bessel operator

7.2.3 Generalization of the Asgeirsson theorem

7.2.4 Descent method for the general Euler-Poisson-Darboux equation

7.3 Elliptic equations with Bessel operator

7.3.1 Weighted homogeneous distributions

7.3.2 Extension of the weighted homogeneous distributions

7.3.3 Weighted fundamental solution of the Laplace-Bessel operator

7.3.4 The Dirichlet problem for an elliptic singular equation

8 Applications of transmutations to different problems

8.1 Inverse problems and applications of Buschman-Erdélyi transmutations

8.1.1 Inverse problems

8.1.2 Copson lemma

8.1.3 Norm estimates and embedding theorems in Kipriyanov spaces

8.1.4 Other applications of Buschman-Erdélyi operators

8.2 Applications of the transmutation method to estimates of the solutions for differential equations with variable coefﬁcients and the problem of E. M. Landis

8.2.1 Applications of the transmutations method to the perturbed Bessel equation with a potential

8.2.2 The solution of the basic integral equation for the kernel of the transmutation operator

8.2.3 Application of the method of transmutation operators to the problem of E. M. Landis

8.2.4 The solution to the E. M. Landis problem belongs to T (λ+ε)

8.3 Applications of transmutations to perturbed Bessel and one-dimensional Schrödinger equations

8.3.1 Formulation of the problem

8.3.2 Solution of the basic integral equation for the kernel of a transmutation operator

8.3.3 Estimates for the case of a power singular at zero potential

8.3.4 Asymptotically exact inequalities for Legendre functions

8.4 Iterated spherical mean in the computed tomography problem

8.4.1 Iterated weighted spherical mean and its properties

8.4.2 Application of identity for an iterated spherical mean to the task of computed tomography

9 Fractional powers of Bessel operators

9.1 Fractional Bessel integrals and derivatives on a segment

9.1.1 Deﬁnitions

9.1.2 Basic properties of fractional Bessel integrals on a segment

9.1.3 Fractional Bessel integrals and derivatives on a segment of elementary and special functions

9.1.4 Fractional Bessel derivatives on a segment as inverse to integrals

9.2 Fractional Bessel integral and derivatives on a semiaxis

9.2.1 Deﬁnitions

9.2.2 Basic properties of fractional Bessel integrals on a semiaxis

9.2.3 Factorization

9.2.4 Fractional Bessel integrals on semiaxes of elementary and special functions

9.3 Integral transforms of fractional powers of Bessel operators

9.3.1 The Mellin transform

9.3.2 The Hankel transform

9.3.3 The Meijer transform

9.3.4 Generalized Whittaker transform

9.4 Further properties of fractional powers of Bessel operators

9.4.1 Resolvent for the right-sided fractional Bessel integral on a semiaxis

9.4.2 The generalized Taylor formula with powers of Bessel operators

10 B-potentials theory

10.1 Deﬁnitions of hyperbolic B-potentials, absolute convergence, and boundedness

10.1.1 Negative fractional powers of the hyperbolic expression with Bessel operators

10.1.2 Absolute convergence and boundedness

10.1.3 Semigroup properties

10.1.4 Examples

10.2 Method of approximative inverse operators applied to inversion of the hyperbolic B-potentials

10.2.1 Method of approximative inverse operators

10.2.2 General Poisson kernel

10.2.3 Representation of the kernel gαε,δ

10.2.4 Inversion of the hyperbolic B-potentials

10.3 Mixed hyperbolic Riesz B-potentials

10.3.1 Deﬁnition and basic properties of the mixed hyperbolic Riesz B-potential

10.3.2 Homogenizing kernel

10.4 Inversion of the mixed hyperbolic Riesz B-potentials

10.4.1 Auxiliary lemma

10.4.2 Property of Lrγ-boundedness of the function gα,γ,ε

10.4.3 Inversion theorems

11 Fractional differential equations with singular coefﬁcients

11.1 Meijer transform method for the solution to homogeneous fractional equations with left-sided fractional Bessel derivatives on semiaxes of Gerasimov-Caputo type

11.1.1 General case

11.1.2 Particular cases and examples

11.2 Mellin transform method

11.2.1 Ordinary linear nonhomogeneous differential equations of fractional order on semiaxes

11.2.2 Example

11.3 Hyperbolic Riesz B-potential and its connection with the solution of an iterated B-hyperbolic equation

11.3.1 General algorithm

11.3.2 Deﬁnition

11.3.3 Variables in Lorentz space

11.3.4 Identity operator

11.4 The Riesz potential method for solving nonhomogeneous equations of Euler-Poisson-Darboux type

11.4.1 General nonhomogeneous iterated Euler-Poisson-Darboux equation

11.4.2 Mixed truncated hyperbolic Riesz B-potential

11.4.3 Nonhomogeneous general Euler-Poisson-Darboux equation with homogeneous conditions

11.4.4 Examples

12 Conclusion

References

Index

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