Solodov A., Ochkov V. Differential Models: An Introduction with Mathcad

Springer, 2004. — 246 p. — ISBN: 3540208526Differential 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 resultsEach 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.Differential Mathematical Models.Laws in the Differential Form.
Models of Growth.
Conservation Laws.
Conservation Law for Traffic Problem.
One-Dimensional Stationary Models: Fuel Element.Integrable Differential Equations.First-Order Linear Equations.
Linear Homogeneous Equations with Constant Coefficients.
Linear Inhomogeneous Equations.
Equations with Separable Variables.
Homogeneous Differential Equations.
Depression of Equation.Dynamic Model of Systems with Heat Generation.Mathematical Model.
Phase-Plane Portrait. Stable and Unstable Equilibrium.
State Set Representation.
Plotting the Bifurcation Set.
Fold Catastrophe.
Catastrophic Jumps at Smooth Variation of Parameters.
Time Evolution of System with Heat Generation.Stiff Differential Equations.Model Differential Equation.
Method rkfixed. Numerical Instability.
Method rkadapt. Integration Step Problem.
Method stiffr. Solution of Stiff Model Equation.
Method stiffr. Solution of Chemical Kinetics Equations.
Explicit and Implicit Methods.
Jacobian Matrix.Heat Transfer near the Stagnation Point at ross Tube Flow.The Integral Equation of a Thermal Boundary Layer.
Mathematical Formulation of the Problem.
External Flow Velocity Distribution.
Analysis for the Stagnation Point.
Dimensionless Formulation.
Optimization Algorithm for the Right-Hand Side.
Numerical Integration with the Built-in Function Odesolve.The Falkner–Skan Equation of Boundary Layer.Model Construction.
Boundary-Value Problem. Method sbval.
The Solution of the Initial Problem. Method rkfixed.
Flow Field Imaging.
Boundary Layer on Permeable Walls.
Thermal Boundary Layer. Heat Transfer Law.
Troubles with Odesolve.Rayleigh’s Equation: Hydrodynamical Instability.Hydrodynamic Equations for Free Shear Flow.
Perturbation Method. Linearization.
Transition to Complex Domain.
Numerical Integration in the Complex Domain: Program Euler.
Integration and Search of Eigenvalues.
Returning to the Real Domain.Kinematic Waves of Concentration in Ion-Exchange Filter.Conservation Equation for Concentration in Filter.
Wave Equation for Concentration.
Dimensionless Formulation.
Isotherm of Adsorption.
Solving a Wave Equation Using Method of Characteristics.Kinematic Shock Waves.Conservation Equation in Finite-Difference Form.
Discontinuous Solutions. Shock Waves.
MacCormack Method. Computing Program McCrm.
Shock Waves of Concentration in a Filter.
Shock Waves on a Motorway.
Gravitational Bubble Flow. Steam-Content Shock Waves.Numerical Modeling of the CPU-Board Temperature Field.Built-in Functions for Partial Differential Equations.
Finite-Difference Approximation.
Iteration Method of Solution. Program Plate.
Thermal Model of the CPU-Board.
Problem of Orbital Platform. Function bvalfit.Temperature Waves.Formulation of the Boundary-Value Problem.
Discretization.
TDMA: Computing Programs Coef and SYSTRD.
Computational Modeling of Cyclical Thermal Action.
Built-in Function Pdesolve.Literature.
Appendix: Built-in Solvers for ODE.