Methods of Mathematical Modelling: Fractional Differential Equations (Mathematics and its Applications)

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
Editors......Page 12
Contributors......Page 14
 1.1 Introduction......Page 18
 1.2 Method of Approximation of Fractional Derivative......Page 20
  1.3.1 Fractional Food Chain Dynamics with Holling Type II Functional Response......Page 25
  1.3.2 Multi-Species Ecosystem with a Beddington-DeAngelis Functional Response......Page 28
 1.4 Numerical Experiment for Fractional Reaction-Diffusion Ecosystem......Page 32
 References......Page 35
 2.1 Introduction......Page 38
 2.2 The Problem: Diffusion and Kinetics......Page 40
 2.3 Discussion and Conclusions......Page 50
 References......Page 51
 3.1 Introduction......Page 56
 3.2 Preliminaries......Page 58
 3.4 Numerical Experiments and Discussion......Page 59
 3.6 Application......Page 63
 Appendix......Page 64
 References......Page 65
 4.1 Introduction......Page 68
 4.2 A New Numerical Approximation......Page 69
  4.2.1 Error Estimate......Page 71
  4.3.2 Example 2......Page 72
  4.3.3 Example 3......Page 74
 4.4 Conclusion......Page 76
 References......Page 77
 5.1 Introduction......Page 80
 5.2 Preliminaries......Page 84
 5.3 Formulation of Axis-Symmetric FOCP......Page 86
  5.3.1 Half Axis-Symmetric Case......Page 87
  5.3.2 Complete Axis-Symmetric Case......Page 90
 5.4 Numerical Results......Page 92
 References......Page 95
 6.1 Introduction......Page 100
 6.2 Fractional Calculus News......Page 102
 6.3 Basics Calculus for the Integral-Balance Methods......Page 103
  6.4.1 Approximation with the HBIM......Page 105
  6.4.2 Approximation with DIM......Page 106
  6.5.1 Quadratic Profile......Page 108
  6.5.2 Cubic Profile......Page 111
 6.6 Myers and Mitchell Approach for Exponent n......Page 115
  6.6.1 Residual Function......Page 116
  6.6.2 At Boundary Conditions......Page 117
  6.6.3 Outsides of Boundary Conditions......Page 118
 References......Page 119
 7.1 Introduction......Page 122
 7.2 Basic Idea of HATM with Adomian\'s Polynomials......Page 124
 7.3 Application to the Toda Lattice Equations......Page 127
 7.4 Numerical Result and Discussion......Page 134
 7.5 Concluding Remarks......Page 135
 References......Page 143
 8.1 Introduction......Page 148
 8.2 Problem\'s Description......Page 150
 8.3 Generalization of Local Model......Page 151
  8.4.1 Solutions of the Energy Equation......Page 154
  8.4.2 Solution of Momentum Equation......Page 157
 8.5 Results and Discussion......Page 161
 8.6 Concluding Remarks......Page 162
 References......Page 163
 9.1 Introduction......Page 168
 9.2 Mathematical Preliminaries......Page 170
 9.3 Fractional Moisture Content Based Model......Page 172
 9.4 Applications......Page 175
 9.5 Numerical Simulation......Page 177
 9.6 Conclusion......Page 179
 References......Page 180
 10.1 Introduction......Page 184
 10.2 Basic Definitions of Fractional Calculus......Page 185
 10.3 New Numerical Scheme with Atangana-Baleanu Fractional Derivative......Page 186
 10.4 Numerical Scheme with Caputo Fractional Derivative......Page 190
 10.5 Numerical Scheme for Caputo-Fabrizio Fractional Derivative......Page 193
 10.6 Existence and Uniqueness Condition for Atangana-Baleanu Fractional Derivative......Page 195
 10.7 Existence and Uniqueness Condition for Caputo Fractional Derivative......Page 198
 10.8 Existence and Uniqueness Condition for Caputo-Fabrizio Fractional Derivative......Page 200
 References......Page 203
 11.1 Introduction......Page 206
 11.2 Preliminaries......Page 207
 11.3 Adomian Decomposition Sumudu Transform Method......Page 208
 11.5 Test Examples......Page 210
 References......Page 217
 12.1 Introduction......Page 222
 12.2 Model Description......Page 223
  12.2.1 The Modified Riemann-Liouville Derivative and Bessel\'s Equation......Page 224
  12.2.2 Solutions of Schrödinger Equation......Page 226
  12.2.3 Soliton Solution......Page 227
 References......Page 231
 13.1 Introduction......Page 236
 13.2 Preparatory Results......Page 238
 13.3 Results on Controllability......Page 240
 13.4 Conclusion......Page 249
 References......Page 250
Index......Page 0