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1 Contents Preface Notation xi xv 1 The fractional Laplacian in one dimension Random walkers with constant steps Particle number density distribution Numerical simulation Ordinary diffusion Evolution of the variance Population balance Random jumpers Central limit theorem and stable distributions Ordinary diffusion Anomalous diffusion Continuous-time random walks (CTRW) Power-law probability jump lengths A principal-value integral Regularization by singularity subtraction Regularization by second-order differences Evolution equations Wires and springs The fractional Laplacian Definition in terms of a principal-value integral Definition in terms of a regularized integral Evaluation of the coefficient c 1,α Unsteady fractional diffusion equation Fourier transform The fractional Laplacian as a Fourier integral Gaussian distribution Computation of a degenerate hypergeometric function Effect of fractional order Gaussian distribution v
2 vi The Fractional Laplacian Brinkman s approximation Numerical computation of the fractional Laplacian Green s function of the fractional Laplace equation Solution by the Fourier transform The Riesz potential Fractional Laplacian as a self-induced Riesz potential Inverse of the fractional Laplacian Fractional Poisson equation in a restricted domain Homogeneous extended Dirichlet boundary condition Arbitrary extended Dirichlet boundary condition Periodicity condition Green s function of unsteady fractional diffusion Solution by the Fourier transform Evolution of the variance Solution of the initial-value problem Brinkman s approximation Numerical discretization in one dimension Computation of a principal-value integral Mid-point integration rule Influence coefficients Infinite discretization Fractional Laplacian differentiation matrix Infinite discretization Brinkman s approximation Fractional Poisson equation Homogeneous extended Dirichlet boundary condition Arbitrary extended Dirichlet boundary conditions Evolution under fractional diffusion Implicit time integration Evolution of the Gaussian distribution Arbitrary extended Dirichlet boundary condition Jumping creatures Denoising Fisher s equation Numerical stability Advanced time-integration schemes Differentiation by spectral expansion Infinite discretization Boundary conditions Further concepts in one dimension Fractional first derivative Definite integral of the fractional Laplacian
3 Contents vii Effect of fractional order Gaussian distribution Numerical evaluation Properties of the fractional first derivative Even functions Odd functions Arbitrary functions Fractional diffusive flux The Laplacian potential Numerical evaluation The Laplacian potential in terms of the Riesz potential Fractional derivatives from finite-difference stencils Second-order finite-difference stencils Fourth-order finite-difference stencils Fractional third derivative Fourier transform Numerical evaluation Fractional fourth derivative Fourier transform Numerical evaluation Periodic functions Sine, cosines, and the complete Fourier series Complete Fourier series Square wave Cosine Fourier series Sine Fourier series Numerical evaluation of the Fourier coefficients Fractional Laplacian Poisson equation Green s functions Solution of the Poisson equation Periodic Moore Penrose Green s function Integral representation of the periodic Laplacian Representation as a regularized integral Numerical discretization Periodic differentiation matrix Fractional Poisson equation Evolution under fractional diffusion Differentiation by spectral expansion Fractional differentiation matrix Fractional graph Laplacian Embedding of the fractional Poisson equation
4 viii The Fractional Laplacian 5 The fractional Laplacian in three dimensions Stipulation on the Fourier transform Integral representation Notion of the principal-value integral Local contribution Regularized integral representations Determination of the coefficient c 3,α Fractional gradient Integral representation Reduction to the ordinary gradient Numerical evaluation Fractional diffusive flux Laplacian potential Green s function of the fractional Laplace equation The Riesz potential Inverse of the fractional Laplacian Laplacian potential Triply periodic Green s function Green s function as a Fourier series Ewald summation Fractional Poisson equation Exact solution in a sphere Numerical methods Evolution under fractional diffusion Periodic functions and arbitrary domains Fractional Stokes flow Equations of fractional Stokes flow Flow due to a point force Fractional Stokeslet The fractional Laplacian in two dimensions Stipulation on the Fourier transform Integral representation Notion of the principal-value integral Local contribution Regularized integral representations Determination of the coefficient c 2,α Fractional gradient Integral representation Reduction to the ordinary gradient Numerical evaluation Laplacian potential
5 Contents ix 6.5 Green s function of the fractional Laplace equation The Riesz potential Inverse of the fractional Laplacian Laplacian potential Doubly periodic Green s function Green s function as a Fourier series Ewald summation Fractional Poisson equation Exact solution inside a circular disk Numerical methods Evolution due to fractional diffusion Periodic functions and arbitrary domains A Selected definite integrals 231 B The Gamma function 235 C The Gaussian distribution 239 D The fractional Laplacian in arbitrary dimensions 241 D.1 Fourier transform D.2 Integral representation of the fractional Laplacian D.3 Fractional gradient D.4 Green s function of the fractional Laplace equation D.5 The Riesz potential D.5.1 Inverse of the fractional Laplacian D.5.2 Laplacian potential D.6 Fractional Poisson equation in a finite solution domain D.6.1 Homogeneous extended Dirichlet boundary condition 248 D.7 Green s function of unsteady fractional diffusion D.7.1 One dimension D.7.2 Two dimensions D.7.3 Three dimensions D.7.4 Arbitrary dimensions D.8 Embedding of the fractional Poisson equation D.9 Fractional graph Laplacian E Fractional derivatives 259 E.1 The Riemann Liouville fractional derivative E.2 Grünwald Letnikov derivative E.3 Caputo fractional derivative F Aitken extrapolation of an infinite sum 267
6 x The Fractional Laplacian F.1 Riemann s zeta function F.2 Kernel of the periodic fractional Laplacian References 273 Index 275
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ENGINEERING MATHEMATICS (For ESE & GATE Exam) (CE, ME, PI, CH, EC, EE, IN, CS, IT) Salient Features : 89 topics under 31 chapters in 8 units 67 Solved Examples for comprehensive understanding 1386 questions