While the Bridegroom Is With Them': Marriage, Family, Gender and Violence in the Gospel of Matthew

Preface......Page 5
Acknowledgements......Page 7
Contents......Page 8
About the Author......Page 15
List of Figures......Page 16
List of Tables......Page 25
 1.1 Overview......Page 28
 1.2 Introduction to Fractional Calculus......Page 29
  1.2.1 Fractional Derivative and Integration......Page 30
  1.2.2 Preliminaries of Local Fractional Calculus......Page 38
 1.3 Wavelets......Page 41
  1.3.1 Wavelet Transform......Page 42
  1.3.2 Orthonormal Wavelets......Page 45
  1.3.3 Multiresolution Analysis......Page 47
  1.4.2 Modified Decomposition Method......Page 50
  1.4.3 New Two-Step Adomian’s Decomposition Method......Page 53
  1.4.4 New Approach for Adomian’s Decomposition Method......Page 57
  1.4.5 Modified Homotopy Analysis Method with Fourier Transform......Page 58
  1.4.6 Modified Fractional Reduced Differential Transform Method......Page 59
  1.4.7 Coupled Fractional Reduced Differential Transform Method......Page 61
  1.4.8 Optimal Homotopy Asymptotic Method......Page 65
  1.4.9 First Integral Method......Page 68
  1.4.10 Haar Wavelets and the Operational Matrices......Page 69
 1.5 Numerical Methods for Solving Stochastic Point Kinetics Equation......Page 73
  1.5.1 Wiener Process......Page 74
  1.5.2 Euler–Maruyama Method......Page 75
  1.5.3 Order 1.5 Strong Taylor Method......Page 76
 References......Page 77
 2.1 Introduction......Page 81
  2.2.1 Coupled Nonlinear Klein–Gordon–Schrödinger Equations......Page 82
  2.2.2 Space Fractional Diffusion Equations on Finite Domain......Page 83
  2.2.3 Space Fractional Diffusion Equation with Insulated Ends......Page 84
  2.3.1 A Modified Decomposition Method for Coupled K-G-S Equations......Page 85
  2.3.2 The Two-Step Adomian Decomposition Method......Page 88
  2.3.3 ADM with a Simple Variation for Space Fractional Diffusion Model......Page 91
  2.4.1 Implementation of MDM for Analytical Approximate Solutions of Coupled K-G-S Equations......Page 93
  2.4.2 Numerical Results and Discussion for Coupled K-G-S Equations......Page 94
  2.5.1 Solution of One-Dimensional Space Fractional Diffusion Equation......Page 96
  2.5.2 Solution of Two-Dimensional Space Fractional Diffusion Equation......Page 99
  2.5.3 Numerical Results and Discussion for Space Fractional Diffusion Equations......Page 101
  2.6.1 Implementation of the Present Method......Page 105
  2.6.2 Numerical Results and Discussion......Page 106
 2.7 Conclusion......Page 108
 References......Page 109
 3.1 Introduction......Page 112
 3.2 Outline of the Present Study......Page 113
  3.2.1 Fractional Dynamic Model of Bagley–Torvik Equation......Page 114
  3.3.1 Haar Wavelets......Page 115
  3.3.2 Operational Matrix of the General Order Integration......Page 120
  3.3.3 Function Approximation by Haar Wavelets......Page 121
  3.3.4 Convergence of Haar Wavelet Approximation......Page 123
 3.4 Basic Idea of Optimal Homotopy Asymptotic Method......Page 125
 3.5 Application of Haar Wavelet Method for the Numerical Solution of Bagley–Torvik Equation......Page 128
  3.5.1 Numerical Results and Discussions......Page 130
  3.5.2 Error Estimate......Page 132
  3.6.1 Application of Haar Wavelet to Fractional Fisher-Type Equation......Page 133
  3.6.2 Application of OHAM to Fractional Fisher-Type Equation......Page 136
  3.6.3 Numerical Results and Discussion......Page 138
 3.7 Conclusion......Page 142
 References......Page 143
 4.1 Introduction......Page 144
 4.2 Outline of the Present Study......Page 146
 4.3 Numerical Approximation Techniques for Riesz Space Fractional Derivative......Page 147
  4.3.2 Fractional Centered Difference Approximation Technique for the Riesz Space Fractional Derivative......Page 148
  4.3.3 Inhomogeneous Fractional Diffusion Equation with Riesz Space Fractional Derivative......Page 150
  4.3.4 Time and Space Fractional Fokker–Planck Equation with Riesz Fractional Operator......Page 151
  4.3.5 Numerical Results for Riesz Fractional Diffusion Equation and Riesz Fractional Fokker–Planck Equation......Page 154
  4.3.6 Stability and Convergence of the Proposed Finite Difference Schemes......Page 158
  4.4.1 Basic Idea of Modified Homotopy Analysis Method with Fourier Transform......Page 163
  4.4.2 Implementation of the MHAM-FT Method for Approximate Solution of Nonlocal Fractional SGE......Page 165
 4.5 Conclusion......Page 176
 References......Page 177
 5.1 Introduction......Page 180
 5.2 Outline of the Present Study......Page 183
  5.2.1 Time Fractional Nonlinear Acoustic Wave Equations......Page 184
  5.2.3 Time Fractional (2 + 1)-Dimensional Davey–Stewartson Equations......Page 185
 5.3 Algorithm of the First Integral Method with Fractional Complex Transform......Page 186
 5.4 Algorithm of the Kudryashov Methods Applied with Fractional Complex Transform......Page 188
 5.5 Algorithm of the Mixed Dn-Sn Method with Fractional Complex Transform......Page 190
  5.6.1 The Burgers–Hopf Equation......Page 193
  5.6.2 The Khokhlov–Zabolotskaya–Kuznetsov Equation......Page 196
  5.6.3 Numerical Results and Discussions for Nonlinear Fractional Acoustic Wave Equations......Page 199
  5.7.1 Kudryashov Method for Time Fractional KdV-KZK Equation......Page 201
  5.7.2 Modified Kudryashov Method for Time Fractional KdV-KZK Equation......Page 203
  5.7.3 Numerical Results and Discussions......Page 205
  5.8.1 Exact Solutions of Fractional (2 + 1)-Dimensional Davey–Stewartson Equation......Page 208
  5.8.2 Exact Solutions of the Fractional (2 + 1)-Dimensional New Integrable Davey–Stewartson-Type Equation......Page 212
 5.9 Conclusion......Page 218
 References......Page 219
 6.1 Introduction......Page 223
 6.2 Outline of the Present Study......Page 224
  6.2.2 Tzitzéica-Type Nonlinear Evolution Equations......Page 225
  6.3.1 Algorithm for the Improved Generalized Jacobi Elliptic Function Method......Page 226
  6.3.2 Algorithm for the New Extended Auxiliary Equation Method......Page 229
 6.4 New Explicit Exact Solutions of Coupled Schrödinger–Boussinesq Equations......Page 231
  6.4.1 Numerical Simulations for the Solutions of Coupled Schrödinger–Boussinesq Equations......Page 238
  6.5.1 New Exact Solutions of Dodd–Bullough–Mikhailov (DBM) Equation......Page 240
  6.5.2 New Exact Solutions of Tzitzeica–Dodd–Bullough (TDB) Equation......Page 244
  6.5.3 Physical Interpretations of the Solutions......Page 247
 6.6 Conclusion......Page 250
 References......Page 251
 7.1 Introduction......Page 254
 7.2 Outline of the Present Study......Page 257
  7.2.2 Time Fractional Coupled KdV Equations......Page 259
  7.2.4 Time Fractional Predator–Prey Dynamical System......Page 260
  7.2.5 Fractional Coupled Schrödinger–KdV Equation......Page 261
  7.2.6 Fractional Whitham–Broer–Kaup, Modified Boussinesq, and Approximate Long Wave Equations in Shallow Water......Page 262
  7.3.1 Modified Fractional Reduced Differential Transform Method......Page 263
  7.3.2 Coupled Fractional Reduced Differential Transform Method......Page 266
  7.4.1 Numerical Solutions of Variant Types of Time Fractional KdV Equations......Page 268
  7.4.2 Convergence Analysis and Error Estimate......Page 277
 7.5 Application of CFRDTM for the Solutions of Time Fractional Coupled KdV Equations......Page 278
  7.5.1 Numerical Solutions of Time Fractional Coupled KdV Equations......Page 279
  7.5.2 Soliton Solutions for Time Fractional Coupled Modified KdV Equations......Page 288
  7.5.3 Approximate Solution for Fractional Predator–Prey Equation......Page 303
  7.5.4 Solutions for Time Fractional Coupled Schrödinger–KdV Equation......Page 312
  7.5.5 Traveling Wave Solutions for the Variant of Time Fractional Coupled WBK Equations......Page 330
  7.5.6 Convergence and Error Analysis of CFRDTM......Page 337
 7.6 Conclusion......Page 352
 References......Page 354
 8.1 Introduction......Page 357
  8.2.1 Riesz Fractional Coupled Schrödinger–KdV Equations......Page 358
  8.2.2 Riesz Fractional Chen–Lee–Liu Equation......Page 360
 8.3 The Proposed Numerical Technique for Riesz Fractional Coupled Schrödinger–KdV Equations......Page 362
  8.3.2 Crank–Nicolson Spectral Method for the KdV-like Equation......Page 364
 8.4 Properties of the Numerical Scheme and Stability Analysis for the Coupled Schrödinger–KdV Equations......Page 365
 8.5 Implicit Finite Difference Method for the Riesz Fractional Coupled Schrödinger–KdV Equations......Page 367
  8.5.1 Approximation of Riesz Fractional Derivative by the Fractional Centered Difference......Page 368
  8.5.3 Numerical Experiments and Discussion......Page 369
 8.6 New Proposed Technique for Riesz Fractional Chen–Lee–Liu Equation......Page 370
 8.8 Stability Analysis of Proposed Time-Splitting Spectral Scheme for Riesz Fractional Chen–Lee–Liu Equation......Page 377
  8.9.1 Shifted Grünwald–Letnikov Formula for Riesz Space Fractional Derivative......Page 379
  8.9.2 Weighted Shifted Grünwald–Letnikov Formula for Riesz Space Fractional Derivative......Page 380
  8.9.3 CN-WSGD Numerical Scheme......Page 381
  8.10.1 Stability Analysis......Page 384
  8.10.2 Convergence Analysis......Page 385
  8.10.3 Numerical Experiments and Discussion......Page 387
 References......Page 393
 9.1 Introduction......Page 396
  9.3.1 Strong Convergence......Page 397
 9.4 Evolution of Stochastic Neutron Point Kinetics Model......Page 398
 9.5 Application of Euler–Maruyama Method and Strong Order 1.5 Taylor Method for the Solution of Stochastic Point Kinetics Model......Page 404
  9.5.2 Strong Order 1.5 Taylor Method for the Solution of Stochastic Point Kinetics Model......Page 405
  9.5.3 Numerical Results and Discussion......Page 406
 9.6 Conclusion......Page 408
 References......Page 409