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

Description

This book features original research articles on the topic of mathematical modelling and fractional differential equations. The contributions, written by leading researchers in the field, consist of chapters on classical and modern dynamical systems modelled by fractional differential equations in physics, engineering, signal processing, fluid mechanics, and bioengineering, manufacturing, systems engineering, and project management. The book offers theory and practical applications for the solutions of real-life problems and will be of interest to graduate level students, educators, researchers, and scientists interested in mathematical modelling and its diverse applications. Features Presents several recent developments in the theory and applications of fractional calculus Includes chapters on different analytical and numerical methods dedicated to several mathematical equations Develops methods for the mathematical models which are governed by fractional differential equations Provides methods for models in physics, engineering, signal processing, fluid mechanics, and bioengineering Discusses real-world problems, theory, and applications Cover Half Title Series Page Title Page Copyright Page Contents Preface Editors Contributors 1 Mathematical Analysis and Simulation of Chaotic Tritrophic Ecosystem Using Fractional Derivatives with Mittag-Leffler Kernel 1.1 Introduction 1.2 Method of Approximation of Fractional Derivative 1.3 Model Equations and Stability Analysis 1.3.1 Fractional Food Chain Dynamics with Holling Type II Functional Response 1.3.2 Multi-Species Ecosystem with a Beddington-DeAngelis Functional Response 1.4 Numerical Experiment for Fractional Reaction-Diffusion Ecosystem 1.5 Conclusion References 2 Solutions for Fractional Diffusion Equations with Reactive Boundary Conditions 2.1 Introduction 2.2 The Problem: Diffusion and Kinetics 2.3 Discussion and Conclusions Acknowledgement References 3 An Efficient Computational Method for Non-Linear Fractional Lienard Equation Arising in Oscillating Circuits 3.1 Introduction 3.2 Preliminaries 3.3 Method of Solution 3.4 Numerical Experiments and Discussion 3.5 Conclusions 3.6 Application Appendix References 4 A New Approximation Scheme for Solving Ordinary Differential Equation with Gomez-Atangana-Caputo Fractional Derivative 4.1 Introduction 4.2 A New Numerical Approximation 4.2.1 Error Estimate 4.3 Application 4.3.1 Example 1 4.3.2 Example 2 4.3.3 Example 3 4.4 Conclusion References 5 Fractional Optimal Control of Diffusive Transport Acting on a Spherical Region 5.1 Introduction 5.2 Preliminaries 5.3 Formulation of Axis-Symmetric FOCP 5.3.1 Half Axis-Symmetric Case 5.3.2 Complete Axis-Symmetric Case 5.4 Numerical Results 5.5 Conclusions References 6 Integral-Balance Methods for the Fractional Diffusion Equation Described by the Caputo-Generalized Fractional Derivative 6.1 Introduction 6.2 Fractional Calculus News 6.3 Basics Calculus for the Integral-Balance Methods 6.4 Integral-Balance Methods 6.4.1 Approximation with the HBIM 6.4.2 Approximation with DIM 6.5 Approximate Solutions of the Generalized Fractional Diffusion Equations 6.5.1 Quadratic Profile 6.5.2 Cubic Profile 6.6 Myers and Mitchell Approach for Exponent n 6.6.1 Residual Function 6.6.2 At Boundary Conditions 6.6.3 Outsides of Boundary Conditions 6.7 Conclusion References 7 A Hybrid Formulation for Fractional Model of Toda Lattice Equations 7.1 Introduction 7.2 Basic Idea of HATM with Adomian's Polynomials 7.3 Application to the Toda Lattice Equations 7.4 Numerical Result and Discussion 7.5 Concluding Remarks Acknowledgements References 8 Fractional Model of a Hybrid Nanofluid 8.1 Introduction 8.2 Problem's Description 8.3 Generalization of Local Model 8.4 Solution of the Problem 8.4.1 Solutions of the Energy Equation 8.4.2 Solution of Momentum Equation 8.5 Results and Discussion 8.6 Concluding Remarks Acknowledgment References 9 Collation Analysis of Fractional Moisture Content Based Model in Unsaturated Zone Using q-homotopy Analysis Method 9.1 Introduction 9.2 Mathematical Preliminaries 9.3 Fractional Moisture Content Based Model 9.4 Applications 9.5 Numerical Simulation 9.6 Conclusion References 10 Numerical Analysis of a Chaotic Model with Fractional Differential Operators: From Caputo to Atangana-Baleanu 10.1 Introduction 10.2 Basic Definitions of Fractional Calculus 10.3 New Numerical Scheme with Atangana-Baleanu Fractional Derivative 10.4 Numerical Scheme with Caputo Fractional Derivative 10.5 Numerical Scheme for Caputo-Fabrizio Fractional Derivative 10.6 Existence and Uniqueness Condition for Atangana-Baleanu Fractional Derivative 10.7 Existence and Uniqueness Condition for Caputo Fractional Derivative 10.8 Existence and Uniqueness Condition for Caputo-Fabrizio Fractional Derivative 10.9 Conclusion References 11 A New Numerical Method for a Fractional Model of Non-Linear Zakharov-Kuznetsov Equations via Sumudu Transform 11.1 Introduction 11.2 Preliminaries 11.3 Adomian Decomposition Sumudu Transform Method 11.4 Error Analysis of the Proposed Technique 11.5 Test Examples 11.6 Conclusion References 12 Chirped Solitons with Fractional Temporal Evolution in Optical Metamaterials 12.1 Introduction 12.2 Model Description 12.2.1 The Modified Riemann-Liouville Derivative and Bessel's Equation 12.2.2 Solutions of Schrödinger Equation 12.2.3 Soliton Solution 12.3 Conclusion References 13 Controllability on Non-dense Delay Fractional Differential System with Non-Local Conditions 13.1 Introduction 13.2 Preparatory Results 13.3 Results on Controllability 13.4 Conclusion References Index