Phys102 Lecture06!08!10Fall DiffEquations

8/14/2019 Phys102 Lecture06!08!10Fall DiffEquations

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Computational Lab in Physics:Solving Differential Equations.

Steven Kornreich www.beachlook.com

Frequently in Physics, we are faced with solving a differential equation. Findingsolutions analytically can be difficult. We can use numerical methods to give us agood idea of the solution in many cases.


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Logistic Map, Cobweb diagram


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Logistic Map Discussion

As the control parameter ris increased in the interval 3< r < 3.5699456

there is period doubling pitchfork bifurcations increase the number of m-cycle periods by a factor of 2at each bifurcation.

Fixed points obey theequation x= f (m) (x)

f (m) (x) = f(f(f(f(x)))), m-

times


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Examples of differential eqs.Newtons Law:

Maxwells Eqs:Gausss Law

No monopolesFaradays LawAmpere-Maxwell

Pendulum:

Schrodinger Eq:

d r F m

dt

E

B

B E

t E

B J t

d g dt l

i V

t m

r

r r

r r

r r

r r r

hh


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Ordinary Differential Eqs.: 1-variable

Order : highest derivativeLinear: each term has only 1 st power of function and derivatives

no 2 nd or higher powersno cross-terms

Homogeneous: no function of the

independent variable appears by itself.Physics: 2 nd order eqs: need two initialconditions.


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Simple and Driven Harmonic oscillator

SHO

Driven HarmonicOscillator

d x t m kx t

dt

d x t m kx t F t

dt


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Example: xy+y=0

Analyticsolution:

dy x ydxdy dx y x

y x AC

y x

Numericalsolution:

Start from apoint x 0 , goingup to x f , and use

step size h .

dy ydx x

y x y x h y x h

x


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Comparison of Numerical/Analyticvoid eulerExample1() {

TF1* analytic = new TF1("analytic","1/x",0.1,5);analytic->SetNpx(1000);TCanvas* euCnv1 = new TCanvas("euCnv1","ODE",500,500);analytic->Draw();gPad->SetLogy();

double xmin=0.1;double xmax=5.0;double step=0.01;

int Nsteps=static_cast((xmax-xmin)/step);const int maxPoints = 1000;if (Nsteps>maxPoints) {

cout


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Harmonic oscillator solution

Transform 2 nd order eq.into two coupled linearequations:

Solution, with initialconditions

v(0)=0x(0)=A

Set m=k=1 forsimplicity.

d x t m kx t dt

dxm mv

dt

dvm kxdt

x t A t

v t A t k m


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Numerical Solution

Equations become:dx

v x v t dt dv

x v x

x t t x t t v t

v t t v t t t t dt

x


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ROOT macro for SHO with SimpleEuler

void eulerExample2() {TF1* analytic = new

TF1("analytic","sin(x+TMath::Pi()/2.0)",0.1,20);analytic->SetNpx(1000);TCanvas* euCnv2 = new TCanvas("euCnv2","SHO-

Euler",500,500);analytic->Draw();

double tmin=0.0;double tmax=20.0;double step=0.01;int Nsteps=static_cast((tmax-tmin)/step);const int maxPoints = 2001;if (Nsteps>maxPoints) {

cout


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Result of Simple Euler Method for SHO

Analytic solutionx(t)=sin(t+ /2) inBLACK

Numerical solutions:x(t) in RED

v(t) in BLUEProblem:Amplitude of numericalsolution grows!


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Spring is gaining energy?!As particle propagates from say, x=0 to x=1:

Force is evaluated at x(t i)Average displacement of spring during propagation:x(t i+ t/2).Velocity value used is also larger than averagevelocity over the displacement.

Result:restoring force is weaker than it should be.Acts on the particle a shorter amount of time.Displacement overshoots the true maximum.

When returning to zero, force is overestimatedCycle repeats.

xi xi+1

x(t i+ t/2)

x


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Modified Euler Method

Simple Euler:derivative at the beginning of interval

Modified Eulerderivative at the midpoint of interval

For first problem:

y x h y x hy xh

y x h y x hy x

dy ydx x

h h y x y x y x

y x h y x h y x h x h


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Simple vs Modified Euler: 1/x

Much better agreement with Modified Euler Method.


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Modified Euler Method for SHO

Need to calculate x and v atmidpoint .

t t v v t v t x t

t t x x t x t v t

x t t x t t vv t t v t t x


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Simple vs Modified Euler: SHO

Works better!


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For smaller step size, can still haveproblems

With step size = 0.3, obviousproblems as t grows.


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HomeworkFrom Chapter 15Exercise (5): Duffing Oscillator

Damped, nonlinear (pendulum at NLO), driven oscillator.Your program should make a plot of the resulting equation xvs t as well as the v vs x.

Use Modified Euler Method, (ignore the last sentence on page201)Instead of instructions in page 202, you can use a TGraph forthis, as in the examples shown in lecture.

Extra Problem (Not in Book)Solve the full pendulum equation of motion (noapproximations), using the modified Euler method.

Use initial angle = /2, initial angular speed = 0. Take gravity= 9.81 m/s 2 and l=1 m.Make a plot of angle vs time.In that same plot, compare the plot to the SHO of frequencyg/l and amplitude /2 which starts at the same angle andsame angular speed.

Phys102 Lecture06!08!10Fall DiffEquations

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