Book contents
- Frontmatter
- Contents
- Preface
- 1 Overviews
- 2 Setting Up Dynamic Models
- 3 The Master Equation
- 4 Introductory Simple and Simplified Models
- 5 Aggregate Dynamics and Fluctuations of Simple Models
- 6 Evaluating Alternatives
- 7 Solving Nonstationary Master Equations
- 8 Growth and Fluctuations
- 9 A New Look at the Diamond Search Model
- 10 Interaction Patterns and Cluster Size Distributions
- 11 Share Market with Two Dominant Groups of Traders
- Appendix
- References
- Index
Appendix
Published online by Cambridge University Press: 15 October 2009
- Frontmatter
- Contents
- Preface
- 1 Overviews
- 2 Setting Up Dynamic Models
- 3 The Master Equation
- 4 Introductory Simple and Simplified Models
- 5 Aggregate Dynamics and Fluctuations of Simple Models
- 6 Evaluating Alternatives
- 7 Solving Nonstationary Master Equations
- 8 Growth and Fluctuations
- 9 A New Look at the Diamond Search Model
- 10 Interaction Patterns and Cluster Size Distributions
- 11 Share Market with Two Dominant Groups of Traders
- Appendix
- References
- Index
Summary
Deriving Generating Functions via Characteristic Curves
We follow Hildebrand (1976, Chap. 8) in summarizing the method for deriving generating functions defined by quasilinear partial differential equations.
We only consider equations with two independent variables, x and y, and a dependent variable z, of the form
An important special case is
We put this into a more symmetrical form. Suppose that G(x, y, z) = c defines a solution implicitly, i.e., this equation determines z as a function of x and y that satisfies the partial differential equation. Assume that ∂G/∂z ≠ 0.
Then,
and
Substituting these into the original equation, we arrive at
We can interpret this equation geometrically as saying that the vector (P, Q, R) is orthogonal to the gradient ∇ G, i.e., the vector lies in the tangent plane to G(x, y, z) = const. At any point on the solution (integral) surface, the vector (P, Q, R) is tangent to any curve on the surface passing through at the point. Such curves are called characteristic curves of the differential equation.
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- Information
- Modeling Aggregate Behavior and Fluctuations in EconomicsStochastic Views of Interacting Agents, pp. 195 - 244Publisher: Cambridge University PressPrint publication year: 2001