Numerical Solutions of Realistic Nonlinear Phenomena (Nonlinear Systems and Complexity, 31)

Preface
Contents
1 Monotone Iterative Technique for Non-autonomous Semilinear Differential Equations with Non-instantaneous Impulses
	1.1 Introduction
	1.2 Preliminaries
	1.3 Main Results
	1.4 Nonlocal Problem
	1.5 Integro-Differential Equations
	1.6 Example
	References
2 An Extrapolated Crank Nicholson VMS-POD Method for Darcy Brinkman Equations
	2.1 Introduction
	2.2 Full Order Model for Darcy Brinkman System
	2.3 Reduced Order Modelling with POD
	2.4 Error Estimates
	2.5 Numerical Experiments
		2.5.1 Problem Description
		2.5.2 Test 1: Convergence Rates with Respect to Δt and R
		2.5.3 Test 2: Comparison of POD Solution and VMS-POD Solution
	2.6 Conclusion
	References
3 Comparison of Exact and Numerical Solutions for the Sharma–Tasso–Olver Equation
	3.1 Introduction
	3.2 Analysis of the Exact Solution Method
	3.3 An Application of the Exact Solution Method
		3.3.1 Case 1
		3.3.2 Case 2
	3.4 Analysis of Finite Difference Method
		3.4.1 Truncation Error and Stability Analysis
		3.4.2 L2  and L∞ Error Norms
	3.5 Implementation of the FDEs
	3.6 Conclusions
	References
4 A Linear B-Spline Approximation for a Class of Nonlinear Time and Space Fractional Partial Differential Equations
	4.1 Introduction
	4.2 Discretization of the Fractional Operators
	4.3 Numerical Approach for the FPDE
		4.3.1 Space-FPDE
		4.3.2 Time-FPDE
	4.4 Applications
	4.5 Conclusion
	References
5 Escaping from Current Minimizer by Using an Auxiliary Function Smoothed by Bezier Curves
	5.1 Introduction
	5.2 Basic Concepts and Assumptions
		5.2.1 Bezier Curves
	5.3 The Auxiliary Function Method
		5.3.1 Literature Review of the Filled Function
		5.3.2 A New Auxiliary Function and Its Properties
	5.4 Numerical Experiments
	5.5 Conclusion
	References
6 A Modified Laguerre Matrix Approach for Burgers–Fisher Type Nonlinear Equations
	6.1 Introduction
	6.2 The Model
		6.2.1 Singular Perturbed Generalized Burgers–Fisher Equation
		6.2.2 Burgers–Fisher Equation
		6.2.3 Burgers Equation
		6.2.4 Fisher Equation
	6.3 The Method
	6.4 Error Analysis
		6.4.1 Residual Error Estimation
	6.5 Numerical Results
	6.6 Concluding Remarks
	References
7 Increasing the Effects of Auxiliary Function by Multiple Extrema in Global Optimization
	7.1 Introduction
	7.2 Preliminaries
		7.2.1 Overview of the Filled Functions
		7.2.2 Filled Functions with Two-Parameter
		7.2.3 Filled Functions with One Parameter
	7.3 A New Filled Function Method and Its Properties
	7.4 The Algorithm
	7.5 Numerical Results
	7.6 Conclusions
	References
8 A New Approach for the Solution of the Generalized Abel Integral Equation
	8.1 Introduction
	8.2 Derivations and Main Results
	8.3 Illustrative Examples
	8.4 Conclusions
	References
9 NPSOG: A New Hybrid Method for Unconstrained Differentiable Optimization
	9.1 Introduction
	9.2 Nonmonotone Spectral Gradient Method
		9.2.1 Iterative Search Method and Step Size
		9.2.2 Nonmonotone Spectral Gradient Methods
			9.2.2.1 The Barzilai–Borwein Method
			9.2.2.2 The Nonmonotone Line Search Method
			9.2.2.3 Nonmonotone Spectral Gradient (NSG) Algorithm
	9.3 Particle Swarm Optimization (PSO)
		9.3.1 Metaheuristic Methods
		9.3.2 Swarm Intelligence
		9.3.3 PSO as a Member of Swarm Intelligence
		9.3.4 PSO Basic Algorithm
		9.3.5 Modification to the Basic Algorithm
			9.3.5.1 Velocity Bounds
			9.3.5.2 Inertia Weight
			9.3.5.3 Constriction Factor
		9.3.6 Neighborhood Topologies
	9.4 The Proposed Hybrid Approach
	9.5 Simulation Results
	9.6 Conclusion
	References
10 Detection of HIV-1 Protease Cleavage Sites via Hidden Markov Model and Physicochemical Properties of Amino Acids
	10.1 Introduction
	10.2 Hidden Markov Model
		10.2.1 Toy Example
		10.2.2 Calculation of Likelihood
			10.2.2.1 Naive Approach
			10.2.2.2 Forward Algorithm
			10.2.2.3 Backward Algorithm
		10.2.3 Viterbi Algorithm: Inference of the Most Probable Path
		10.2.4 Baum–Welch Algorithm: Estimating the Model Parameters
	10.3 Application
		10.3.1 Data Description
		10.3.2 Creation of States
		10.3.3 Initialization of the EM Algorithm
		10.3.4 Modeling the Data via HMM
		10.3.5 Results
	10.4 Conclusion
	References
11 A Numerical Approach for Variable Order Fractional Equations
	11.1 Introduction
	11.2 Problem Definition and Integration Method for Variable Order Fractional Differential Equations
		11.2.1 Numerical Method
		11.2.2 Stability Criteria of the Method
		11.2.3 Numerical Example
	11.3 Multi-term Variable Order Fractional Equations
	11.4 Addition of y(t) Term to Variable Order Fractional Differential Equations
	11.5 Conclusion
	References
12 Evolution of Plane Curves via Lie Symmetry Analysis in the Galilean Plane
	12.1 Introduction
	12.2 Evolution of Plane Curves by Flow in G2
	12.3 Application via Lie symmetry Analysis
	12.4 Conclusion
	References
Index