Shishkina E., Sitnik S. Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics

New York: Academic Press, 2020. — 586 p.Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics connects difficult problems with similar more simple ones. The book's strategy works for differential and integral equations and systems and for many theoretical and applied problems in mathematics, mathematical physics, probability and statistics, applied computer science and numerical methods. In addition to being exposed to recent advances, readers learn to use transmutation methods not only as practical tools, but also as vehicles that deliver theoretical insights.Acknowledgments and thanksBasic definitions and propositions
Special functions
Gamma function, beta function, Pochhammer symbol, and error function
Bessel functions
Hypergeometric type functions
Polynomials
Functional spaces
Orthant Rn+, Cevm, Sev, and Lpγ spaces
Weighted measure, space L∞γ, and definition of weak (p,q)γ type operators
Space of weighted generalized functions Sev', absolutely continuous functions, and unitary operators
Mixed case
Integral transforms and Lizorkin-Samko space
One-dimensional integral transforms with Bessel functions in the kernels and Mellin transform
Properties of composition of integral transforms with Bessel functions in the kernel
Multi-dimensional integral transforms
Basic facts and formulas
Kipriyanov's classification of second order linear partial differential equations
Divergence theorem and Green's second identity for B-elliptic and B-hyperbolic operators
Tricomi equation
Abstract Euler-Poisson-Darboux equation
Basics of fractional calculus and fractional order differential equations
Short history of fractional calculus and fractional order differential equations
One-dimensional fractional derivatives and integrals
Fractional derivatives in mechanics
Fractional powers of multi-dimensional operators
Differential equations of fractional order
Standard fractional order integro-differential operators
Riemann-Liouville fractional integrals and derivatives on a segment
Riemann-Liouville fractional integrals and derivatives on a semiaxis
Gerasimov-Caputo fractional derivatives
Dzrbashian-Nersesyan fractional operators and sequential order fractional operators
Some more fractional order integro-differential operators
The Erdélyi-Kober operators
Fractional integrals and fractional derivatives of a function with respect to another function
Averaged or distributed order fractional operators
Saigo, Love, and other fractional operators with special function kernels
Integral transforms and basic differential equations of fractional order
Integral transforms of fractional integrals and derivatives
Laplace transform of Riemann-Liouville fractional integrals and derivatives on semiaxes
Mellin transform of Riemann-Liouville fractional integrals and derivatives on semiaxes
Laplace transform of Gerasimov-Caputo fractional derivatives on semiaxes
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 (,∞)
Laplace transform method for homogeneous equations with constant coefficients with the left-sided Gerasimov-Caputo fractional derivatives of the order α on a semiaxis [,∞)
Mellin integral transform and nonhomogeneous linear differential equations of fractional order
Essentials of transmutations
Definition of the transmutation operator, some examples of classical transmutations
Introduction to transmutation theory
Some examples of classical transmutations
Transmutations for Sturm-Liouville operator
Description of the problem and terminology
Transmutations in the form of the second kind Fredholm operators
Transmutations in the form of the second kind Volterra operators
Transmutations in the form of the first kind Volterra operators
Transmutations for different potentials
Kernel of transmutation intertwining operators of the Sturm-Liouville type
Cases when potential q(x) is an exponential function
Cases when potential q(x) is constant
Estimates of kernels and point formulas for estimating the error for calculating transmutation operators
Transmutations for singular Bessel operator
One-dimensional Poisson operator
Multi-dimensional Poisson operator
Generalized translation
Weighted spherical mean
Weighted generalized functions generated by quadratic forms
The weighted generalized function associated with a positive quadratic form and concentrated on a part of a cone
B-ultrahyperbolic operator
Weighted generalized function associated with a positive quadratic form
Weighted generalized function δγ(P)
Weighted generalized functions realized by the degrees of quadratic forms
Weighted generalized functions Pγ,±λ
The weighted generalized function Pλγ and (P±i )γλ associated with a quadratic form with complex coefficients
Other weighted generalized functions associated with a quadratic form
Functions (w-|x|)+,γλ and (c+P±i)λγ
General weighted generalized functions connected with quadratic form
Hankel transform of weighted generalized functions generated by the quadratic form
Hankel transform of rλγ
Hankel transforms of functions Pλγ, (P±i)λγ, and Pλγ,±
Hankel transforms of functions (w-|x|)+,γλ and (c+P±i)λγ
Buschman-Erdélyi integral and transmutation operators
Buschman-Erdélyi transmutations of the first kind
Sonine-Poisson-Delsarte transmutations
Definition and main properties of Buschman-Erdélyi transmutations of the first kind
Factorizations of the first kind Buschman-Erdélyi operators and the Mellin transform
Buschman-Erdélyi transmutations of the second and third kind
Second kind Buschman-Erdélyi transmutation operators
Sonine-Katrakhov and Poisson-Katrakhov transmutations
Buschman-Erdélyi transmutations of the third kind with arbitrary weight function
Some applications of Buschman-Erdélyi transmutations
Multi-dimensional integral transforms of Buschman-Erdélyi type with Legendre functions in kernels
Basic definitions
The n-dimensional Mellin transform and its properties
Lν,-theory and the inversion formulas for the modified H-transform
Inversion of Hσ,κ
Representations in the form of modified H-transform
Mellin transform of auxiliary functions K( x) and K( x)
Mellin transform of Pγδ,( x) and Pγδ,( x)
Lν,-theory of the transforms Pγδ,kf (k=,)
Inversion formulas for transforms Pγδ,kf (k=,)
Integral transforms composition method for transmutations
Basic ideas and definitions of the integral transforms composition method for the study of transmutations
Background of ITCM
What is ITCM and how to use it?
Application of the ITCM to derive transmutations connected with the Bessel operator
Index shift for the Bessel operator
Poisson and "descent" operators, negative fractional power of the Bessel operator
ITCM for generalized translation and the weighted spherical mean
Integral representations of transmutations for perturbed differential Bessel operators
Connection formulas for solutions to singular differential equations via the ITCM
Application of transmutations for finding general solutions to Euler-Poisson-Darboux type equations
Application of transmutations for finding solutions to general Euler-Poisson-Darboux type equations
Application of transmutations for finding general solutions to singular Cauchy problems
Differential equations with Bessel operator
General Euler-Poisson-Darboux equation
The first Cauchy problem for the general Euler-Poisson-Darboux equation
The second Cauchy problem for the general Euler-Poisson-Darboux equation
The singular Cauchy problem for the generalized homogeneous Euler-Poisson-Darboux equation
Examples
Hyperbolic and ultrahyperbolic equations with Bessel operator in spaces of weighted distributions
The generalized Euler-Poisson-Darboux equation and the singular Klein-Gordon equation
Iterated ultrahyperbolic equation with Bessel operator
Generalization of the Asgeirsson theorem
Descent method for the general Euler-Poisson-Darboux equation
Elliptic equations with Bessel operator
Weighted homogeneous distributions
Extension of the weighted homogeneous distributions
Weighted fundamental solution of the Laplace-Bessel operator
The Dirichlet problem for an elliptic singular equation
Applications of transmutations to different problems
Inverse problems and applications of Buschman-Erdélyi transmutations
Inverse problems
Copson lemma
Norm estimates and embedding theorems in Kipriyanov spaces
Other applications of Buschman-Erdélyi operators
Applications of the transmutation method to estimates of the solutions for differential equations with variable coefficients and the problem of E M Landis
Applications of the transmutations method to the perturbed Bessel equation with a potential
The solution of the basic integral equation for the kernel of the transmutation operator
Application of the method of transmutation operators to the problem of E M Landis
The solution to the E M Landis problem belongs to T (λ+ε)
Applications of transmutations to perturbed Bessel and one-dimensional Schrödinger equations
Formulation of the problem
Solution of the basic integral equation for the kernel of a transmutation operator
Estimates for the case of a power singular at zero potential
Asymptotically exact inequalities for Legendre functions
Iterated spherical mean in the computed tomography problem
Iterated weighted spherical mean and its properties
Application of identity for an iterated spherical mean to the task of computed tomography
Fractional powers of Bessel operators
Fractional Bessel integrals and derivatives on a segment
Definitions
Basic properties of fractional Bessel integrals on a segment
Fractional Bessel integrals and derivatives on a segment of elementary and special functions
Fractional Bessel derivatives on a segment as inverse to integrals
Fractional Bessel integral and derivatives on a semiaxis
Definitions
Basic properties of fractional Bessel integrals on a semiaxis
Factorization
Fractional Bessel integrals on semiaxes of elementary and special functions
Integral transforms of fractional powers of Bessel operators
The Mellin transform
The Hankel transform
The Meijer transform
Generalized Whittaker transform
Further properties of fractional powers of Bessel operators
Resolvent for the right-sided fractional Bessel integral on a semiaxis
The generalized Taylor formula with powers of Bessel operators
B-potentials theory
Definitions of hyperbolic B-potentials, absolute convergence, and boundedness
Negative fractional powers of the hyperbolic expression with Bessel operators
Absolute convergence and boundedness
Semigroup properties
Examples
Method of approximative inverse operators applied to inversion of the hyperbolic B-potentials
Method of approximative inverse operators
General Poisson kernel
Representation of the kernel gαε,δ
Inversion of the hyperbolic B-potentials
Mixed hyperbolic Riesz B-potentials
Definition and basic properties of the mixed hyperbolic Riesz B-potential
Homogenizing kernel
Inversion of the mixed hyperbolic Riesz B-potentials
Auxiliary lemma
Property of Lrγ-boundedness of the function gα,γ,ε
Inversion theorems
Fractional differential equations with singular coefficients
Meijer transform method for the solution to homogeneous fractional equations with left-sided fractional Bessel derivatives on semiaxes of Gerasimov-Caputo type
General case
Particular cases and examples
Mellin transform method
Ordinary linear nonhomogeneous differential equations of fractional order on semiaxes
Example
Hyperbolic Riesz B-potential and its connection with the solution of an iterated B-hyperbolic equation
General algorithm
Definition
Variables in Lorentz space
Identity operator
The Riesz potential method for solving nonhomogeneous equations of Euler-Poisson-Darboux type
General nonhomogeneous iterated Euler-Poisson-Darboux equation
Mixed truncated hyperbolic Riesz B-potential
Nonhomogeneous general Euler-Poisson-Darboux equation with homogeneous conditions
Examples