Transmutations, Singular and Fractional Differential Equations with Applications to Mathematical Physics (Mathematics in Science and Engineering)

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