Numerical Treatment and Analysis of Time-Fractional Evolution Equations

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	Preface
	Contents
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	1 Existence, Uniqueness, and Regularity of Solutions
		1.1 Basics of Fractional Calculus
		1.2 Mittag–Leffler Function
		1.3 Existence, Uniqueness, and Sobolev Regularity
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	2 Spatially Semidiscrete Discretization
		2.1 Galerkin Finite Element Method
		2.2 Error Analysis via Mittag–Leffler Functions
		2.3 Error Analysis via Laplace Transform
		2.4 Lumped Mass FEM
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	3 Convolution Quadrature
		3.1 Convolution Quadrature Generated by BDF
		3.2 BDFk CQ with Initial Correction
		3.3 Fractional Crank–Nicolson Scheme
		3.4 Parallel in Time Algorithm
		3.5 Fast Convolution
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	4 Finite Difference Methods: Construction and Implementation
		4.1 Construction of Time-Stepping Schemes
		4.2 Sum of Exponential Approximation
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	5 Finite Difference Methods on Uniform Meshes
		5.1 Error Analysis of L1 Scheme
		5.2 Corrected L1 Scheme
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	6 Finite Difference Methods on Graded Meshes
		6.1 Error Analysis via Nonuniform Gronwall's Inequality
		6.2 Error Analysis of the L1 Scheme via Barrier Functions
		6.3 Error Analysis of Alikhanov's Scheme via Barrier Functions
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	7 Nonnegativity Preservation
		7.1 Nonnegativity Preservation
		7.2 Spatially Semidiscrete Methods
		7.3 Fully Discrete Scheme
		7.4 Maximum-Norm Contractivity
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	8 Discrete Maximal Regularity
		8.1 R-Boundedness, UMD Spaces, and Fourier Multiplier Theorems
		8.2 Convolution Quadrature Generated by BDF
		8.3 L1 Scheme
		8.4 Explicit Euler Method
		8.5 Fractional Crank–Nicolson Method
		8.6 Inhomogeneous Initial Condition
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	9 Subdiffusion with Time-Dependent Coefficients
		9.1 Regularity Theory
		9.2 Semidiscrete Galerkin FEM
		9.3 Time Discretization by Backward Euler CQ
		9.4 Time Discretization by Corrected BDF2 CQ
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	10 Semilinear Subdiffusion
		10.1 Discrete Gronwall's Inequality
		10.2 Error Estimates for the Linearized Scheme
		10.3 High-Order Time-Stepping Schemes
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	11 Time-Space Finite Element Approximation
		11.1 Time-Space Petrov–Galerkin Formulation
		11.2 Petrov–Galerkin FEM on Tensor-Product Meshes
		11.3 Error Estimates
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	12 Spectral Galerkin Approximation
		12.1 Time-Space Galerkin Formulation
		12.2 Log Orthogonal Functions
		12.3 Spectral Galerkin Method
		12.4 Fully Discrete Scheme
		12.5 Fast Linear Solver
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	13 Incomplete Iterative Solution at Time Levels
		13.1 Incomplete Iterative Scheme
		13.2 Error Analysis for Smooth Initial Data
		13.3 Error Analysis for Nonsmooth Initial Data
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	14 Optimal Control with Subdiffusion Constraint
		14.1 Regularity Theory
		14.2 Numerical Approximation of the Forward Problem
		14.3 Numerical Approximation of the Optimal Control Problem
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	15 Backward Subdiffusion
		15.1 Stability and Regularization
		15.2 Spatially Semidiscrete Scheme
		15.3 Fully Discrete Scheme
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	Appendix A Mathematical Preliminaries
	A.1  Gamma Function
	A.2 Polylogarithmic Function
	A.3 Integral Transforms
	Appendix  References
	Index