| Description |
xi, 232 pages : 169 fig. ; 24 cm |
| Contents |
Machine derived contents note: 1 Differential Mathematical Models ..1 -- 1.1 Introduction 1 -- 1.2 Laws in the Differential Form 1 -- 1.3 Models of Growth 8 -- 1.4 Conservation Laws 14 -- 1.5 Conservation Law for Traffic Problem 23 -- 1.6 One-Dimensional Stationary Models: Fuel Element 29 -- 1.7 Conclusion 33 -- 2 Integrable Differential Equations 37 -- 2.1 Introduction 37 -- 2.2 First-Order Linear Equations 37 -- 2.3 Linear Homogeneous Equations with Constant Coefficients 41 -- 2.4 Linear Inhomogeneous Equations 43 -- 2,5 Equations with Separable Variables 43 -- 2.6 Homogeneous Differential Equations 44 -- 2.7 Depression of Equation 44 -- 2.8 Conclusion 46 -- 3 Dynamic Model of Systems with Heat Generation 47 -- 3.1 Introduction 47 -- 3.2 Mathematical Model 47 -- 3.3 Phase-Plane Portrait. Stable and Unstable Equilibrium 50 -- 3.4 State Set Representation 51 -- 3.5 Plotting the Bifurcation Set 53 -- 3.6 Fold Catastrophe -- 3.7 Catastrophic Jumps at Smooth Variation of Parameters 55 -- 3.8 Time Evolution of System with Heat Generation 57 -- 3.9 Conclusion 60 -- 4 Stiff Differential Equations 61 -- 4.1 Introduction 61 -- 4.2 Model Differential Equation 62 -- 4.3 Method rkfixed. Numerical Instability 63 -- 4.4 Method rkadapt. Integration Step Problem 68 -- 4.5 Method stiffr. Solution of Stiff Model Equation 70 -- 4.6 Method stiffr. Solution of Chemical Kinetics Equations 72 -- 4.7 Explicit and Implicit Methods 74 -- 4.8 Jacobian Matrix 76 -- 4.9 Conclusion 77 -- -- -- -- 5 Heat Transfer near the Stagnation Point at Cross Tube Flow 79 -- 5.1 Introduction 79 -- 5.2 The Integral Equation of a Thermal Boundary Layer 80 -- 5.3 Mathematical Formulation of the Problem 82 -- 5.4 External Flow Velocity Distribution 84 -- 5.5 Analysis for the Stagnation Point 86 -- 5.6 Dimensionless Formulation 89 -- 5.7 Optimization Algorithm for the Right-Hand Side 90 -- 5.8 Numerical Integration with the Built-in Function Odesolve 91 -- 5.9 Conclusion 93 -- 6 The Falkner-Skan Equation of Boundary Layer 95 -- 6.1 Introduction 95 -- 6.2 Model Construction 99 -- 6.3 Boundary-Value Problem. Method sbval 101 -- 6.4 The Solution of the Initial Problem. Method rkfixed 103 -- 6.5 Flow Field Imaging 105 -- 6.6 Boundary Layer on Permeable Walls 107 -- 6.7 Thermal Boundary Layer. Heat Transfer Law 112 -- 6.8 Troubles with Odesolve 119 -- 6.9 C onclusion 120 -- 7 Rayleigh's Equation: Hydrodynamical Instability 123 -- 7.1 Introduction 123 -- 7.2 Hydrodynamic Equations for Free Shear Flow 124 -- 7.3 Perturbation Method. Linearization 125 -- 7.4 Transition to Complex Domain 127 -- 7.5 Numerical Integration in the Complex Domain: Program Euler 130 -- 7.6 Integration and Search of Eigenvalues 131 -- 7.7 Returning to the Real Domain 134 -- 7.8 Conclusion 136 -- 8 Kinematic Waves of Concentration in Ion-Exchange Filter 137 -- 8.1 Introduction 137 -- 8.2 Conservation Equation for Concentration in Filter 138 -- 8.3 Wave Equation for Concentration 140 -- 8.4 Dimensionless Formulation 141 -- 8.5 Isotherm of Adsorption 142 -- 8.6 Solving a Wave Equation Using Method of Characteristics 146 -- 8.7 Conclusion 150 -- 9 Kinematic Shock Waves 151 -- 9.1 Introduction 151 -- 9.2 Conservation Equation in Finite-Difference Form 152 -- 9.3 Discontinuous Solutions. Shock Waves 154 -- 9.4 MacCormack Method. Computing Program McCrm 156 -- -- -- -- 9.5 Shock Waves of Concentration in a Filter 158 -- 9.6 Shock Waves on a Motorway 165 -- 9.7 Gravitational Bubble Flow. Steam-Content Shock Waves 170 -- 9.8 Conclusion 180 -- 10 Numerical Modeling of the CPU-Board Temperature Field 181 -- 10.1 Introduction 181 -- 10.2 Built-in Functions for Partial Differential Equations 183 -- 10.3 Finite-Difference Approximation 184 -- 10.4 Iteration Method of Solution. Program Plate 186 -- 10.5 Thermal Model of the CPU-Board 187 -- 10.6 Problem of Orbital Platform. Function bvalfit 191 -- 10.7 Conclusion 196 -- 11 Temperature Waves 203 -- 11.1 Introduction 203 -- 11.2 Formulation of the Boundary-Value Problem 204 -- 11.3 Discretization 204 -- 11.4 TDMA: Computing Programs Coef and SYSTRD 208 -- 11.5 Computational Modeling of Cyclical Thermal Action 211 -- 11.6 Built-in Function Pdesolve 214 -- 11.7 Conclusion 217 -- LitLiterature 219 -- AppAppendix: Built-in Solvers for ODE 223 -- IndIndex 231 |
| Summary |
Differential equations are often used in mathematical models for technological processes or devices. However, the design of a differential mathematical model is crucial and difficult in engineering. As a hands-on approach to learn how to pose a differential mathematical model the authors have selected 9 examples with important practical application and treat them as following: - Problem-setting and physical model formulation - Designing the differential mathematical model - Integration of the differential equations - Visualization of results Each step of the development of a differential model is enriched by respective Mathcad 11 commands, todays necessary linkage of engineering significance and high computing complexity. To support readers of the book with respect to changes that might occur in future versions of Mathcad (Mathcad 12 for example), updates of examples, codes etc. can be downloaded from the following web page www.thermal.ru. Readers can work with Mathcad-sheets of the book without any Mathcad by help Mathcad Application Server Technology |
| Bibliography |
Includes bibliographical references (pages [219]-221) and index |
| Notes |
Print version record |
| SUBJECT |
MathCAD. http://id.loc.gov/authorities/names/n91108721
|
| Subject |
Differential equations.
|
|
Engineering -- Mathematical models.
|
| Author |
Ochkov, V. F. (Valeriĭ Fedorovich)
|
| LC no. |
2004111785 |
| ISBN |
3540208526 |
|
