On the Application of Genetic Algorithms to Differential Equations

Genetic algorithms can be used in order to solve optimization problems. Such a technique may be used in order to solve differential equations. Key words: differential equations; genetic algorithms; JEL Classification: C61, C63 The main issue There are some fundamental econometric issues which lead to differential equations. For example, taking into account the demand-supply equilibrium, we may consider the variation of the price as being a ratio of the difference between demand and supply. () ( ) ()P a D t S t t where a is a positive constant. Assuming small values for t, we obtain the differential model: ()' ( ) ( ) ()P t a D t S t = − When demand exceeds supply, the right part represents a positive value, which implies that P is an increasing function. Formally, we may consider the general Cauchy problem 0 0 ' ( ,), ()y f x y y x y = =, where x is the independent variable and ()y y x = is the dependent variable. By using the classical assumption: [ ] []0 0 0 0: ,,f x X x X y Y y Y − + × − + →R is continuous and satisfies the Lipschitz condition 1 2 1 2 ( , ) ( ,)f x y f x y L y y − ≤