Advances in Applied Mathematical Analysis and Applications by Mangey Ram and Tadashi Dohi

Advances in Applied Mathematical Analysis and Applications Contents

Preface xiii

Acknowledgements xv

List of Contributors xvii

List of Figures xxi

List of Tables xxv

List of Abbreviations xxvii

1 Similarity Solutions of Spherical Shock Waves in a

Self-Gravitating Ideal Gas 1

Astha Chauhan and Rajan Arora

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Formulation of Problem . . . . . . . . . . . . . . . . . . . 3

1.3 Similarity Analysis . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Similarity Solutions . . . . . . . . . . . . . . . . . . . . . 8

1.5 Imploding Shocks . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . 15

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 19

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Dual Solutions for Finite Element Analysis of Unsteady

Hydromagnetic Stagnation Point Flow of Cu – Water Nanofluid

Generated by Stretching Sheet 23

Santosh Chaudhary and KM Kanika

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Formulation of the Problem . . . . . . . . . . . . . . . . . 26

2.3 Similarity Transformation . . . . . . . . . . . . . . . . . . 28

2.4 Local Skin Friction and Local Nusselt Number . . . . . . . 29

2.5 Method of Solution . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Method Validation . . . . . . . . . . . . . . . . . . . . . . 32

2.7 Numerical Results and Discussion . . . . . . . . . . . . . . 32

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Multiparametric Modeling of Carbon Cycle in Temperate

Wetlands for Regional Climate Change Analysis Using

Satellite Data 51

Anna Kozlova, Lesia Elistratova, Yuriy V. Kostyuchenko,

Alexandr Apostolov and Igor Artemenko

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 On the Methodology of Emission Analysis . . . . . . . . . 54

3.2.1 Generalized Approach . . . . . . . . . . . . . . . . 54

3.2.2 On the Uncertainty Control . . . . . . . . . . . . . 56

3.3 Modeling of the Carbon Cycle . . . . . . . . . . . . . . . . 57

3.3.1 Key Model Variables and Parameters . . . . . . . . 57

3.3.2 From Local to Global Methane Cycle Modeling: Variables and Parameters . . . . . . . . . . . . . . 61

3.3.3 Uncertainty Analysis in the Carbon Models . . . . . 65

3.3.4 On the Local Models Integration into the Global

Models: A Methodology . . . . . . . . . . . . . . . 66

3.4 Satellite Tools and Data for the Carbon Cycle Control . . . 72

3.4.1 Satellite Tools for GHG Monitoring . . . . . . . . . 72

3.4.2 Satellite Tools for Plant Productivity and Carbon

Stock Assessment . . . . . . . . . . . . . . . . . . 73

3.4.3 Satellite Tools for Uncertainty Assessment in Crops Productivity . . . . . . . . . . . . . . . . . . . . . 75

3.5 Approach to Emission Assessment and Control . . . . . . . 79

3.5.1 Stochastic Tools for Decision Making in Carbon

Emissions Control . . . . . . . . . . . . . . . . . . 80

3.6 Results of Multiparametric Modeling of Methane Cycle in

Temperate Wetlands in View of Regional Climate Change

Using Satellite Data . . . . . . . . . . . . . . . . . . . . . 82

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 An Intelligent Neuro Fuzzy System for Pattern Classification 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Artificial Neural Networks Approach . . . . . . . . . . . . 97

4.3 Statistical Approach . . . . . . . . . . . . . . . . . . . . . 97

4.4 Design of an Intelligent Neuro Fuzzy System . . . . . . . . 97

4.5 Hybrid Learning Algorithm of Proposed Method . . . . . . 98

4.6 Forward Pass . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.7 Backward Pass . . . . . . . . . . . . . . . . . . . . . . . . 100

4.7.1 Rules’ Index Vector . . . . . . . . . . . . . . . . . 101

4.8 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 104

4.9 Graphs of Performance Error Vs No. of Iterations . . . . . . 106

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5 Fuzzy Inventory Model with Demand, Deterioration

and Inflation: A Comparative Study Through

NGTFN and CNTFN 113

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Preliminary Concept . . . . . . . . . . . . . . . . . . . . . 115

5.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4 Mathematical Model . . . . . . . . . . . . . . . . . . . . . 118

5.5 Formulation of Fuzzy Mathematical Model . . . . . . . . . 121

5.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . 129

5.7 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . 130

5.8 Concluding Remarks and Future Studies . . . . . . . . . . . 130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Summability and Its Application for the Stability of the System 139

Smita Sonker and Alka Munjal

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.2 Summability Process . . . . . . . . . . . . . . . . . . . . . 141

6.3 Types of Summability . . . . . . . . . . . . . . . . . . . . 142

6.3.1 Ordinary Summability . . . . . . . . . . . . . . . . 142

6.3.2 Strong Summability . . . . . . . . . . . . . . . . . 142

6.3.3 Absolute Summability . . . . . . . . . . . . . . . . 143

6.4 Regularity of a Summability Process . . . . . . . . . . . . . 143

6.5 Silverman-Toeplitz Theorem . . . . . . . . . . . . . . . . . 143

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6.6 Application of Absolute Summable Factor for Stability . . . 144

6.6.1 Stability of the Frequency Response of the Moving

Average System . . . . . . . . . . . . . . . . . . . 144

6.6.2 Stability of the Frequency Response of the

Oscillating Impulse System . . . . . . . . . . . . . 147

6.6.3 Stability of the Frequency Response of the

Exponential System . . . . . . . . . . . . . . . . . 150

6.6.3.1 Stability of frequency response of the exponential system up to

n = 0, 1, 2,. . . , 88 . . . . . . . . . . . . . 151

6.6.3.2 Stability of frequency response of the exponential system up to

n = 0, 1, 2, . . . , 176 . . . . . . . . . . . . 153

6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 156

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7 Design of Manufacturing, Control, and Automation Systems 159

Mohit Pant, Mohit Sood, Aeshwarya Dixit and Sahil Garg

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2 Steps in Design of Manufacturing Systems . . . . . . . . . 161

7.3 Types of Manufacturing . . . . . . . . . . . . . . . . . . . 162

7.4 Additive Manufacturing (AM) . . . . . . . . . . . . . . . . 162

7.4.1 Design of Additive Manufacturing System . . . . . 163

7.4.2 Subtractive Type Manufacturing . . . . . . . . . . . 163

7.4.3 Machining Process in Subtractive Manufacturing. . 164

7.4.4 Cutting Tools for Subtractive Manufacturing . . . . 165

7.5 Rapid Prototyping . . . . . . . . . . . . . . . . . . . . . . 166

7.5.1 Advantages and Disadvantages of Rapid

Prototyping . . . . . . . . . . . . . . . . . . . . . 166

7.5.2 Types of Rapid Prototypes . . . . . . . . . . . . . . 167

7.6 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.6.1 Feedback in Control System . . . . . . . . . . . . . 172

7.7 Automation . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.7.1 Application of Automation in Industries . . . . . . 174

7.7.2 Type of Automation System . . . . . . . . . . . . . 174

7.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8 SEIR – Application for Crop through Water and Soil Texture 179

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.1.1 Dynamical System . . . . . . . . . . . . . . . . . . 179

8.1.2 Dynamics of Crop using Fertile Soil . . . . . . . . 180

8.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . 181

8.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 184

8.3.1 Local Stability . . . . . . . . . . . . . . . . . . . . 184

8.3.2 Global Stability . . . . . . . . . . . . . . . . . . . 185

8.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . 186

8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 189

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9 Advances in Radial Basis Functions 193

Geeta Arora and Gurpreet Singh Bhatia

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.2 Requirement of Mesh-free Methods . . . . . . . . . . . . . 194

9.3 Radial Basis Function . . . . . . . . . . . . . . . . . . . . 195

9.4 Properties of Radial Basis Functions . . . . . . . . . . . . . 195

9.5 Developments in Radial Basis Functions . . . . . . . . . . . 196

9.6 Shape Parameters . . . . . . . . . . . . . . . . . . . . . . . 196

9.7 RBF Interpolation . . . . . . . . . . . . . . . . . . . . . . 198

9.8 Solution of Differential Equation . . . . . . . . . . . . . . . 200

9.8.1 Kansa Method . . . . . . . . . . . . . . . . . . . . 200

9.8.2 Symmetric Collocation Method . . . . . . . . . . . 201

9.8.3 RBF-pseudospectral Approach . . . . . . . . . . . 201

9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

10 Modeling for Time Period of Natural Frequency for Non-Homogeneous Square Plate with Variable

Thickness and Temperature Effect 209

Reeta Bhardwaj, Naveen Mani and Amit Sharma

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . 210

10.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

10.4 Construction of Problem . . . . . . . . . . . . . . . . . . . 214

10.5 Solution of Problem . . . . . . . . . . . . . . . . . . . . . 216

10.6 Numerical Illustration and Discussions . . . . . . . . . . . 218

10.7 Results Comparison . . . . . . . . . . . . . . . . . . . . . 223

10.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 224

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

11 A Study on Metric Fixed Point Theorems Satisfying Integral

Type Contractions 229

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 230

11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 230

11.3 Fixed Point Theorem Satisfying Weak Integral Type

Rational Contractions . . . . . . . . . . . . . . . . . . . . 234

11.4 (π; φ)− Integral Type Weak Contractions and Fixed-Point

Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 246

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

12 Objective Function – in Radiometric Studies – Application to AGRS Surveys Associated with Radon 249

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 249

12.2 Ordinary and Weighted Linear Regression Models . . . . . 250

12.3 Computations of Compton Factors . . . . . . . . . . . . . . 252

12.3.1 Study of Variations in Stripping Factors when the

Correlation among Radioelements Exists . . . . . . 253

12.3.2 Estimation of Stripping Factors and their Variations with Flying Heights . . . . . . . . . . . . . . . . . 255

12.4 Objective Function for Study of Random Errors in Stripping

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 256

12.5 Effects of Airborne Bi24 and Error Bias due to It . . . . . . 258

12.6 Ground and Airborne Experiments . . . . . . . . . . . . . . 259

12.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . 260

12.7.1 Test Area A . . . . . . . . . . . . . . . . . . . . . 260

12.7.2 Test Area B . . . . . . . . . . . . . . . . . . . . . 260

12.7.3 Area of Very High Thorium Concentration . . . . . 261

12.7.4 Area of High Uranium Concentration . . . . . . . . 261

12.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 261

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

13 Modeling Kernel Function in Blackbody Radiation Inversion 265

13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 265

13.2 Nature and Statement of the Problem . . . . . . . . . . . . 266

13.3 Towards a Solution to the Problem . . . . . . . . . . . . . . 269

13.3.1 Reducing Φ(x) Function to Gaussians . . . . . . . . 272

13.3.2 Convolution of Equation (13.4) . . . . . . . . . . . 274

13.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 276

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Index 277

About the Editors 281