MAP 6336 - Ordinary Differential Equations Day 1: Introduction J.D. Mireles James
What is a differential equation?
What is a differential equation? A question involving an unknown function and its derivatives.
What is a differential equation? A question involving an unknown function and its derivatives. Example: can you find a function which is equal to its own derivative?
What is a differential equation? A question involving an unknown function and its derivatives. Example: can you find a function which is equal to its own derivative? d f(t) =f(t). dt
What is a differential equation? A question involving an unknown function and its derivatives. Example: can you find a function which is equal to its own derivative? d dt f(t) =f(t). Answer: f(t) =ce t (for any c 2 R)
= 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has d dx 2 f(x) =e x. = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx Liouville s theorem: this function has no elementary anti-derivative. = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx Liouville s theorem: this function has no elementary anti-derivative. 1X e x 1 = n=0 n! xn = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx Liouville s theorem: this function has no elementary anti-derivative. 1X e x 1 = n! xn e x 2 n=0 = 1X n=0 ( 1) n n! x 2n = 1X n=0 ( 1) n n! 1 2n +1 x2n+1 + c
Problem: we can t always solve Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx Liouville s theorem: this function has no elementary anti-derivative. 1X 1X e x 1 = n! xn e x ( 1) n 2 = x 2n n! Z n=0 Z 1 n=0! X ( 1) n e x2 dx = x 2n dx n! n=0 1X ( 1) n 1 = n! 2n +1 x2n+1 + c n=0
Problem: we can t always solve = 1X n=0 Example: find a function which has Z d 2 f(x) =e x. f(x) = e x2 dx + b dx Liouville s theorem: this function has no elementary anti-derivative. 1X 1X e x 1 = n! xn e x ( 1) n 2 = x 2n n! Z n=0 Z 1 n=0! X ( 1) n e x2 dx = x 2n dx n! n=0 ( 1) n Z 1X x 2n ( 1) n 1 dx = n! n! 2n +1 x2n+1 + c n=0
Numerical approximation of 1X ( 1) n 1 f(x) = n! 2n +1 x2n+1 n=0 Fifty Taylor coefficients, c = 0.
Some themes:
Some themes: Most differential equations don t have ``closed form solutions. You will ``solve very few differential equations in this class (and in life).
Some themes: Most differential equations don t have ``closed form solutions. You will ``solve very few differential equations in this class (and in life). Will need to develop methods to approximate solutions. Using those approximations often involves the computer.
Some themes: Most differential equations don t have ``closed form solutions. You will ``solve very few differential equations in this class (and in life). Will need to develop methods to approximate solutions. Using those approximations often involves the computer. Will also want to evaluate the quality of approximations. Was the picture correct? (We threw away infinitely many terms!)
Why study differential equations? Theory: Differential Equations:
Why study differential equations? Mathematical modeling. Theory: Differential Equations:
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory
Why study differential equations? Mathematical modeling. Theory: Differential Equations: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Differential Equations: (Newton s Second Law) Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Differential Equations: (Newton s Second Law) (Second Law of Thermo) Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) (Navier-Stokes Equation) Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) (Navier-Stokes Equation) (Hodgkin-Huxley) Weather/Climate Science General Relativity Theory Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) (Navier-Stokes Equation) (Hodgkin-Huxley) (Various!) Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) (Navier-Stokes Equation) (Hodgkin-Huxley) (Various!) (Einstein s Equations) Option/derivative pricing
Why study differential equations? Mathematical modeling. Theory: Newtonian mechanics Thermodynamics Quantum mechanics Electromagnetic theory Fluid Dynamics Neuroscience Weather/Climate Science General Relativity Theory Option/derivative pricing Differential Equations: (Newton s Second Law) (Second Law of Thermo) (Schrodinger s Equation) (Maxwell s Equations) (Navier-Stokes Equation) (Hodgkin-Huxley) (Various!) (Einstein s Equations) (Black-Scholes Equation)
Why study differential equations?
Why study differential equations? Inside mathematics itself
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics Minimal curves/surfaces: calculus of variations
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics Minimal curves/surfaces: calculus of variations Probability/Ergodic theory
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics Minimal curves/surfaces: calculus of variations Probability/Ergodic theory Operator theory/functional analysis
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics Minimal curves/surfaces: calculus of variations Probability/Ergodic theory Operator theory/functional analysis Nonlinear analysis/dynamical systems theory
Why study differential equations? Inside mathematics itself Riemannian geometry: existence of geodesics Minimal curves/surfaces: calculus of variations Probability/Ergodic theory Operator theory/functional analysis Nonlinear analysis/dynamical systems theory Perelman s proof of the Poincare conjecture: Ricci Flow/Curve shortening. Millenium problem - million dollars from Clay Foundation
Why study differential equations?
Why study differential equations? Differential equations are ubiquitous in the physical/biological/social sciences, and inside mathematics itself. So, progress in differential equations advances out understanding of a thousand different areas.
Why study differential equations? Differential equations are ubiquitous in the physical/biological/social sciences, and inside mathematics itself. So, progress in differential equations advances out understanding of a thousand different areas. Example: Kuramoto-Sivashinsky Equation @ @t u + 2 u + u + 1 2 kruk2 =0
Why study differential equations? Differential equations are ubiquitous in the physical/biological/social sciences, and inside mathematics itself. So, progress in differential equations advances out understanding of a thousand different areas. Example: Kuramoto-Sivashinsky Equation @ @t u + 2 u + u + 1 2 kruk2 =0 The equation arises as a model of phase change dynamics in diffusive chemical reactions, flame front propagation, fluid flow in a thin film layer, Benard convection in a long box, wave interaction at the boundary of two vicious fluids, drift waves in plasma. Exhibits very complex patterns in both space and time.
Examples: Newtonian mechanics
Examples: Newtonian mechanics
Examples: Newtonian mechanics A mass of m=10 sits on a frictionless table and is attached to a spring. (The spring itself is fastened to a wall.). At equilibrium the mass rests at x =0. Now the spring is displaced (stretched) 4 units to the right and released. What happens?
Examples: Newtonian mechanics A mass of m=10 sits on a frictionless table and is attached to a spring. (The spring itself is fastened to a wall.). At equilibrium the mass rests at x =0. Now the spring is displaced (stretched) 4 units to the right and released. What happens? Hook s Law: a spring exerts a force on a body which is proportional to the amount by which the spring is displaced.
Examples: Newtonian mechanics A mass of m=10 sits on a frictionless table and is attached to a spring. (The spring itself is fastened to a wall.). At equilibrium the mass rests at x =0. Now the spring is displaced (stretched) 4 units to the right and released. What happens? Hook s Law: a spring exerts a force on a body which is proportional to the amount by which the spring is displaced. F (x) = Kx. (restorative)
Examples: Newtonian mechanics A mass of m=10 sits on a frictionless table and is attached to a spring. (The spring itself is fastened to a wall.). At equilibrium the mass rests at x =0. Now the spring is displaced (stretched) 4 units to the right and released. What happens? Hook s Law: a spring exerts a force on a body which is proportional to the amount by which the spring is displaced. F (x) = Kx. (restorative) It is determined that our spring has K = 20. The position of the mass-spring system at all future times is governed by the equation
Examples: Newtonian mechanics A mass of m=10 sits on a frictionless table and is attached to a spring. (The spring itself is fastened to a wall.). At equilibrium the mass rests at x =0. Now the spring is displaced (stretched) 4 units to the right and released. What happens? Hook s Law: a spring exerts a force on a body which is proportional to the amount by which the spring is displaced. F (x) = Kx. (restorative) It is determined that our spring has K = 20. The position of the mass-spring system at all future times is governed by the equation 10 d2 dt 2 x(t) = 20x(t). x(0) = 4 d dt x(0) = 0
Examples: Newtonian mechanics
Examples: Newtonian mechanics A small body moves in the Earth s gravitational field. The body is initially observed to be located at x=100, y =50, z =0, and moving with a velocity of x =5 y = -3 and z = 0. (Here x and y are measured relative to the center of the earth). Predict the future motion of the body.
Examples: Newtonian mechanics A small body moves in the Earth s gravitational field. The body is initially observed to be located at x=100, y =50, z =0, and moving with a velocity of x =5 y = -3 and z = 0. (Here x and y are measured relative to the center of the earth). Predict the future motion of the body. Newton s Law of universal gravitation says that the Earth exerts a (vector) force on the body whose magnitude is proportional to the product of the masses of the Earth and the small body, and inverse proportional to square of the distance between the two bodies. The force vector points from the body to the center of the Earth.
Examples: Newtonian mechanics A small body moves in the Earth s gravitational field. The body is initially observed to be located at x=100, y =50, z =0, and moving with a velocity of x =5 y = -3 and z = 0. (Here x and y are measured relative to the center of the earth). Predict the future motion of the body. Newton s Law of universal gravitation says that the Earth exerts a (vector) force on the body whose magnitude is proportional to the product of the masses of the Earth and the small body, and inverse proportional to square of the distance between the two bodies. The force vector points from the body to the center of the Earth. F(x, y) = mmc p x2 + y 2 2 u u = 1 x p x2 + y 2 y
Examples: Newtonian mechanics A small body moves in the Earth s gravitational field. The body is initially observed to be located at x=100, y =50, z =0, and moving with a velocity of x =5 y = -3 and z = 0. (Here x and y are measured relative to the center of the earth). Predict the future motion of the body. Newton s Law of universal gravitation says that the Earth exerts a (vector) force on the body whose magnitude is proportional to the product of the masses of the Earth and the small body, and inverse proportional to square of the distance between the two bodies. The force vector points from the body to the center of the Earth. F(x, y) = mmc p x2 + y 2 2 u u = 1 x p x2 + y 2 y Suppose that the physical units have been chosen so that MC = 1. Then the equations of motion governing the small body are d 2 dt 2 x(t) = d 2 dt 2 y(t) = x(t) (x(t) 2 + y(t) 2 ) 3 2 y(t) (x(t) 2 + y(t) 2 ) 3 2
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