Differential Equations in Engineering: Research and Applications

Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
Acknowledgments
Editors
Contributors
Chapter 1 Element-Free Galerkin Method for Computational Fracture Mechanics
	1.1 Introduction
	1.2 Historical Developments in Meshfree Methods
	1.3 Element-Free Galerkin Method
	1.4 Moving Least Square (MLS) Approximations
	1.5 Efficient Calculation of the Shape Function
	1.6 Weight Function
	1.7 Numerical Integration
	1.8 Domain of Influence
	1.9 Imposition of Boundary Conditions
	1.10 Governing Equation
	1.11 Crack Modeling in the Element-Free Galerkin Method
		1.11.1 Extrinsic MLS Enrichment
		1.11.2 Intrinsic MLS Enrichment
	1.12 Integration Integral
	1.13 Applications of Element-Free Galerkin Methods to Computational Fracture Mechanics
		1.13.1 Crack Modeling under Mechanical Loads
		1.13.2 Modeling of Vertical Bi-Material Interface
		1.13.3 Modelling of Bi-Metallic Interfacial Edge Crack
		1.13.4 Modeling of Thermoelastic Fracture
			1.13.4.1 Centre Crack in Square Domain
		1.13.5 Thermal Fracture in Coatings
			1.13.5.1 Edge Crack with a Thermal Load
	1.14 Conclusion
	References
Chapter 2 Evaporative Capillary Instability of Swirling Fluid Layer with Mass Transfer
	2.1 Introduction
	2.2 Mathematical Description
		2.2.1 Basic State
		2.2.2 Perturbed State
	2.3 Dimensionless Form of the Dispersion Relationship
	2.4 Numerical Results and Discussions
	2.5 Conclusions
	Acknowledgment
	References
Chapter 3 Control Instruments of Regularized Problems Based on Mathematical Modeling of Structural Perturbations with Applications at the Nodes of 25-Bar Truss Systems
	3.1 Introduction
	3.2 Family of Linear Elastic Partial Differential Equations with Explicit Consideration of Structural Perturbations
		3.2.1 Family of Constrained Ill-Posed Optimal Control Problems Due to Structural Perturbations
	3.3 Family of Regularized Ill-Posed Optimal Control Problems with State and Structural Perturbation Constraints
	3.4 Applications to Real-world Measurements: Structural Perturbation Models Imposed at the Nodes of 25-Bar Truss Systems and Regularization of the Control Instruments
		3.4.1 Interpretations of Results: Control Instruments of the Optimal Mass Design of 25-Bar Truss Systems with Loading Conditions Imposed at the Node Elements
	3.5 Discussion
	3.6 Conclusion
	Conflict of Interest
	Acknowledgments
	References
Chapter 4 Numerical Simulation of Singularly Perturbed Differential Equation with Large Delay Using Exponential B-Spline Collocation Method
	4.1 Introduction
	4.2 Analysis of Recent Numerical Work Carried out on SPDDE
	4.3 Considered Boundary Value Problem
	4.4 The Exponential Cubic B-spline Collocation Method
	4.5 Convergence Analysis
	4.6 Numerical Examples
	4.7 Discussion and Conclusions
	References
Chapter 5 Application of Differential Equations to Instability of Nanofluids
	5.1 Introduction
	5.2 Formulation of the Problem and Conservation Equations
	5.3 Solution for Model 1: Initially, Volume Fraction Varies in the Vertical Direction
	5.4 Solution for Model 2: Initially, Volume Fraction Remains Constant
	5.5 Discussions and Comparative Studies of the Results
	5.6 Numerical Results and Discussions
	5.7 Conclusions
	References
Chapter 6 Analysis of Prey–Predator Model
	6.1 Introduction
	6.2 Description of Method
		6.2.1 Case 1
		6.2.2 Theorem 1
		6.2.3 Case 2
		6.2.4 Theorem 2
		6.2.5 Case 3
	6.3 Stability Analysis
		6.3.1 Theorem 3
	6.4 Stability Analysis for Prey–Predator Model
	6.5 Applications
		6.5.1 Disease Model
			6.5.1.1 Case 1
			6.5.1.2 Case 2
			6.5.1.3 Case 3
		6.5.2 Numerical Illustration
			6.5.2.1 Case 1
			6.5.2.2 Case 2
			6.5.2.3 Case 3
	6.6 Results and Discussion
	6.7 Conclusion
	References
Chapter 7 Incremental Harmonic Balance Method for Multi-Degree-of-Freedom System with Time-Delays
	7.1 Introduction
	7.2 Formulation of IHB Method for Delay Differential Equations
	7.3 Path-Following and Parametric Continuation
	7.4 Stability Analyses of Periodic Solutions
		7.4.1 Floquet’s Theory for an Uncontrolled System, Using Hsu’s Scheme
		7.4.2 Floquet’s Theory for a Time-Delay System by the Semi-Discretization Method
	References
Chapter 8 Solution to the Dirac Equation
	8.1 Introduction
	8.2 Preliminaries
	8.3 Solution to the Massless Field
	8.4 Solution to the Anti-Massless Field
	8.5 Results
	8.6 Discussion
	8.7 Conclusions
	8.8 Acknowledgments
	References
Chapter 9 Periodic Solution of a Nonlinear Economic Cycle Model with a Generic Investment Function
	9.1 Introduction
	9.2 Economic Cycle Model
	9.3 Implicit Harmonic Balance Procedure
	9.4 Numerical Analysis
		9.4.1 The Comparison of the Periodic Solution with the Simulation Result
		9.4.2 The Periodic Solution of the Nonlinear Economic Cycle Model
		9.4.3 The Effects of the Quadratic Term on the Periodic Solution
	9.5 Conclusions
	Appendix
		Trigonometric Identities
	References
Chapter 10 Response Evolution of a Marine Riser in Random Sea Waves
	10.1 Introduction
	10.2 Marine Riser System
	10.3 Path Integration Procedure
	10.4 Numerical Analysis
		10.4.1 The Case of Slight Geometric Nonlinearity
		10.4.2 The Case of Strong Geometric Nonlinearity
		10.4.3 The Case of Strong Correlation between Excitations
	10.5 Conclusion
	References
Chapter 11 Solution of System of PDE Governed in Natural Convective Flow in a Rectangular Porous Cavity
	11.1 Introduction
	11.2 Model Formulation
	11.3 Governing Equations
	11.4 Non-Dimensional Equations
	11.5 Solution Procedure
	11.6 Stream Function and Nusselt Number
	11.7 Interpretation of Results
	11.8 Conclusions
	References
Index