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.
Author(s): Alexander Solodov, Valery Ochkov
Edition: 1st ed. Softcover of orig. ed. 2005
Publisher: Springer
Year: 2004
Language: English
Pages: 238
Contents......Page 8
1.2 Laws in the Differential Form......Page 11
1.3 Models of Growth......Page 18
1.4 Conservation Laws......Page 24
1.5 Conservation Law for Traffic Problem......Page 33
1.6 One-Dimensional Stationary Models: Fuel Element......Page 39
1.7 Conclusion......Page 43
2.2 First-Order Linear Equations......Page 46
2.3 Linear Homogeneous Equations with Constant Coefficients......Page 50
2.5 Equations with Separable Variables......Page 52
2.7 Depression of Equation......Page 53
2.8 Conclusion......Page 55
3.2 Mathematical Model......Page 56
3.3 Phase-Plane Portrait. Stable and Unstable Equilibrium......Page 59
3.4 State Set Representation......Page 60
3.5 Plotting the Bifurcation Set......Page 62
3.6 Fold Catastrophe......Page 63
3.7 Catastrophic Jumps at Smooth Variation of Parameters......Page 64
3.8 Time Evolution of System with Heat Generation......Page 66
3.9 Conclusion......Page 69
4.1 Introduction......Page 70
4.2 Model Differential Equation......Page 71
4.3 Method rkfixed. Numerical Instability......Page 72
4.4 Method rkadapt. Integration Step Problem......Page 77
4.5 Method stiffr. Solution of Stiff Model Equation......Page 79
4.6 Method stiffr. Solution of Chemical Kinetics Equations......Page 81
4.7 Explicit and Implicit Methods......Page 83
4.8 Jacobian Matrix......Page 85
4.9 Conclusion......Page 86
5.1 Introduction......Page 88
5.2 The Integral Equation of a Thermal Boundary Layer......Page 89
5.3 Mathematical Formulation of the Problem......Page 91
5.4 External Flow Velocity Distribution......Page 93
5.5 Analysis for the Stagnation Point......Page 95
5.6 Dimensionless Formulation......Page 98
5.7 Optimization Algorithm for the Right-Hand Side......Page 99
5.8 Numerical Integration with the Built-in Function Odesolve......Page 100
5.9 Conclusion......Page 102
6.1 Introduction......Page 104
6.2 Model Construction......Page 108
6.3 Boundary-Value Problem. Method sbval......Page 110
6.4 The Solution of the Initial Problem. Method rkfixed......Page 112
6.5 Flow Field Imaging......Page 114
6.6 Boundary Layer on Permeable Walls......Page 116
6.7 Thermal Boundary Layer. Heat Transfer Law......Page 121
6.8 Troubles with Odesolve......Page 128
6.9 Conclusion......Page 129
7.1 Introduction......Page 131
7.2 Hydrodynamic Equations for Free Shear Flow......Page 132
7.3 Perturbation Method. Linearization......Page 133
7.4 Transition to Complex Domain......Page 135
7.5 Numerical Integration in the Complex Domain: Program Euler......Page 138
7.6 Integration and Search of Eigenvalues......Page 139
7.7 Returning to the Real Domain......Page 142
7.8 Conclusion......Page 144
8.1 Introduction......Page 145
8.2 Conservation Equation for Concentration in Filter......Page 146
8.3 Wave Equation for Concentration......Page 148
8.4 Dimensionless Formulation......Page 149
8.5 Isotherm of Adsorption......Page 150
8.6 Solving a Wave Equation Using Method of Characteristics......Page 154
8.7 Conclusion......Page 158
9.1 Introduction......Page 159
9.2 Conservation Equation in Finite-Difference Form......Page 160
9.3 Discontinuous Solutions. Shock Waves......Page 162
9.4 MacCormack Method. Computing Program McCrm......Page 164
9.5 Shock Waves of Concentration in a Filter......Page 166
9.6 Shock Waves on a Motorway......Page 173
9.7 Gravitational Bubble Flow. Steam-Content Shock Waves......Page 178
9.8 Conclusion......Page 188
10.1 Introduction......Page 189
10.2 Built-in Functions for Partial Differential Equations......Page 191
10.3 Finite-Difference Approximation......Page 192
10.4 Iteration Method of Solution. Program Plate......Page 194
10.5 Thermal Model of the CPU-Board......Page 195
10.6 Problem of Orbital Platform. Function bvalfit......Page 199
10.7 Conclusion......Page 204
11.1 Introduction......Page 210
11.3 Discretization......Page 211
11.4 TDMA: Computing Programs Coef and SYSTRD......Page 215
11.5 Computational Modeling of Cyclical Thermal Action......Page 218
11.6 Built-in Function Pdesolve......Page 221
11.7 Conclusion......Page 224
Literature......Page 226
Appendix: Built-in Solvers for ODE......Page 229
P......Page 237
W......Page 238
