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Quiz

MAT 443 Study Guide - Quiz Guide: Nondimensionalization


Department
Mathematics
Course Code
MAT 443
Professor
akman
Study Guide
Quiz

Page:
of 4
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 time1, and Bhas the dimension of Ntime1.
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 =B
Adt
Thus, the equation becomes du
=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.
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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) = 1x(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) = 1x
rx+s,r, s > 0
(h) x(n+ 1) = x(n)eax(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) = xeax,a > 0
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