Chapter8
1 of 7
http:/www.mpri.lsu.edu/textbook/Chapter8.htm#intr
Chapter 8
CALCULUS OF VARIATIONS
Introduction
Euler Equation
Functions, Functionals and Neighborhoods
More Complex Problems
Functional with Higher Derivatives in the Integrand
Functional wi
Solutions Chapters 15
Section 1.1
1. Under multiplication, the positive integers form a monoid but not a group, and the
positive even integers form a semigroup but not a monoid.
2. With |a| denoting the order of a, we have |0| = 1, |1| = 6, |2| = 3, |3| =
Phase Plane Analysis
Phase-plane: The -plane is called the phase-plane.
Equilibrium Points: They correspond to solutions of a coupled system of differential equations where
the solutions are constant, i.e. where and , simultaneously.
E.g.: Consider a syst
Predator and Prey Model
There are several types of predator-prey interactions: that of herbivores (which eat plant species),
carnivores (which eat animal species), parasites (which lives on or in another species called host), and
cannibals (which eat thei
Chapter8
1 of 9
file:/E:/Study Material/College/Optimization/chapter 8 Ralph 4_files/Ch.
Chapter 8
CALCULUS OF VARIATIONS
Introduction
Euler Equation
Functions, Functionals and Neighborhoods
More Complex Problems
Functional with Higher Derivatives in the
Chapter8
1 of 8
http:/www.mpri.lsu.edu/textbook/Chapter8-b.htm#closure
Chapter 8
CALCULUS OF VARIATIONS
Introduction
Euler Equation
Functions, Functionals and Neighborhoods
More Complex Problems
Functional with Higher Derivatives in the Integrand
Function
Chapter8
1 of 4
file:/E:/Study Material/College/Optimization/problems_files/Chapter8-c.htm
Chapter 8
CALCULUS OF VARIATIONS
Introduction
Euler Equation
Functions, Functionals and Neighborhoods
More Complex Problems
Functional with Higher Derivatives in th
Solutions to Chapter8
1 of 6
file:/E:/Study Material/College/Optimization/solution 2_files/Chpt8-a.htm
Chapter 8
CALCULUS OF VARIATION
Problems 8.1 to 8.5 | Problems 8.6 to 8.8
problem8.6 | problem8.7 | problem8.8
8-6(9). For steady flow of an incompressi
Compartmental Model
In this model, there is a compartment along with certain related input and/or output. The processes
having inputs to and outputs from a compartment over time may be considered as compartmental
models. The following diagram depicts the
Lake Pollution Model
Pollution in our lakes and rivers has become a major problem particularly over the last 50 years. In order
to improve this situation in the future, it is necessary to gain a
This problem can be considered as a compartmental model with
Exponential Growth Model
One important factor in modeling populations is whether the population grows continuously with time or
in discrete jumps. Another important factor in deciding how to model population growth is the size of
the population. Small pop