MAT 442 Homework

Due on Jan. 22

Q1:

(i) Show the details of the non-dimensionalization described by Equations (1.6)-(1.8) in Murrays

book: The original equation is dN

dt =rBN(1 −N

KB)−BN 2

A2+N2, where A,KBhave the same

dimension as N,rBhas the dimension of time−1, and Bhas the dimension of Ntime−1.

Since A,Nhave the same dimension, and BN 2

A2+B2=B(N

A)2

1+ N

A)2, then we can set u=N

A, where u

is a non-demensional variable.

Then the orginial equation becomes

Adu

dt =rBAu(1 −Au

KB

)−Bu

1 + u2

A

B

du

dt =rBA

Bu(1 −A

KB

u)−u

1 + u2

Since KB,Ahave the same dimension, then set q=KB

A, then qis a non-dimensional constant.

Similarly, set r=ArB

B, then ris also s non-dimentional constant.

Then, the equation becomes du

B

Adt =ru(1 −u

q)−u

1 + u2

To non-dimensionalize what is left, it is reasonable to set a constant τsuch that B=τA

t⇒

τ=B

At, then dτ =B

Adt

Thus, the equation becomes du

dτ =ru(1 −u

q)−u

1 + u2

(ii) Choose ONE of the new dimensionless variables or parameters and explain its meaning in your

own words.

N

A: the proportion that how close the number of spurce budworms is close to the threshold number

of the predators are switched on.

1

Q2: For each diﬀerence equation below, answer the following questions, and explain your reasoning

in each case:

(i) What is the order of this DE?

(ii) Is it linear (including aﬃne) or nonlinear? If the equation is linear, then determine whether it is

(iii) homogeneous or nonhomogeneous.

(iv) Decide whether the equation is autonomous or nonautonomous.

(v) If the conditions are right, identify the recursive map f(x), as in Elaydi Section 1.2.

(a) x(n+ 3) + 4n3x(n+ 1) + sin(n)x(n)=3n+ 1

(i) the order of this DE is 3;

(ii) this is a linear equation;

(iii) this is a nonhomogeneous equation;

(iv) this DE is nonautonomous;

(v) there is no recursive map for this equation

(b) x(n+ 1) = x(n)(r+ 1 −x(n)),r > 0

(i) the order of this DE is 1;

(ii) this is a nonlinear equation;

(iii) this is a nonlinear equation;

(iv) this DE is autonomous;

(v) f(x) = x(r+ 1 −x),r > 0

(c) x(n) = x(n+ 1) + n5x(n+ 2)

(i) the order of this DE is 2;

(ii) this is a linear equation;

(iii) this is a homogeneous equation;

(iv) this DE is nonautonomous;

(v) there is no recursive map for this equation;

(d) x(n+ 2) = nx(n+ 1)/x(n)

(i) the order of this DE is 2;

(ii) this is a nonlinear equation;

(iii) this is a nonlinear equation;

(iv) this DE is nonautonomous;

(v) there is no recursive map for this equation;

(e) x(n+ 1) −x(n)=2

(i) the order of this DE is 1;

2

(ii) this is a linear equation;

(iii) this is a homogeneous equation

(iv) this DE is autonomous;

(v) f(x) = x+ 2

(f) x(n+ 1) −x(n) = sin(x(n)) + (x(n))2

(i) the order of this DE is 1;

(ii) this is a nonlinear equation;

(iii) this is a nonlinear equation;

(iv) this DE is autonomous;

(v) f(x) = x+sin(x) + x2

(g) x(n+ 1) = 1−x(n)

rx(n)+s,r, s > 0

(i) the order of this DE is 1;

(ii) it is a nonlinear equation because r, s > 0; it could be linear equation if and only if r+s= 0

(iii) this is a nonlinear equation;

(iv) this DE is autonomous;

(v) f(x) = 1−x

rx+s,r, s > 0

(h) x(n+ 1) = x(n)e−ax(n),a > 0

(i) the order of this DE is 1;

(ii) this is a linear equation;

(iii) this is a nonlinear equation

(iv) this DE is nonautonomous;

(v) f(x) = xe−ax,a > 0

3

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###### Document Summary

Q1: (i) show the details of the non-dimensionalization described by equations (1. 6)-(1. 8) in murrays. A2+n 2 , where a, kb have the same book: the original equation is dn dimension as n , rb has the dimension of time 1, and b has the dimension of n time 1. Since a, n have the same dimension, and bn 2 is a non-demensional variable. ( n. A )2 , then we can set u = n. 1+ n dt = rbn (1 n. Since kb, a have the same dimension, then set q = kb. B , then r is also s non-dimentional constant. A , then q is a non-dimensional constant. To non-dimensionalize what is left, it is reasonable to set a constant such that b = a. 1 + u2 (ii) choose one of the new dimensionless variables or parameters and explain its meaning in your own words.

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