(%i1) |
kill(all)$ |
[email protected],19/12/2016
general equation (second-order linear equations)
discrete : second-order linear equation( ⇒ alternate approach)
-------------------------------------------------------------
p1*a(n+2)+p2*a(n+1)+p3*a(n)=0
amount pollution ⇒ simplification to understand principles.
Lake Ontari:a(n)
Lake Eri :b(n)
Lake Ontario connected with Lake Eri
system of equations :difference equations
a(n+1)=0.87*a(n)+0.38*b(n) '1'
b(n+1)=0.62*b(n) '2'
(%i2) |
eqn1:a(n+1)=a1*a(n)+b1*b(n)$ eqn2:b(n+1)=a2*a(n)+b2*b(n)$ |
b(n):each year: Lake Eri 38%(0.38) 'replaced' by rain
(%i4) |
a2:0; b2:1-0.38; |
(%i6) |
eqn1:a(n+1)=a1*a(n)+b1*b(n); eqn2:b(n+1)=a2*a(n)+b2*b(n); |
rem:replaced : 1-..
(%i8) |
a1:1-0.13; b1:0.38; |
(%i10) |
eqn1:a(n+1)=a1*a(n)+b1*b(n); eqn2:b(n+1)=a2*a(n)+b2*b(n); |
(%i11) | eqn_1:part(solve(eqn1,b(n)),1); |
(%i12) | eqn_2:ev(eqn_1,n=n+1); |
(%i13) | eqn_3:ev(eqn2,eqn_1,eqn_2); |
p1*a(n+2)+p2*a(n+1)+p3*a(n)=0
(%i14) | eqn_4:lhs(eqn_3)-rhs(eqn_3)=0; |
(%i15) | (0.62/0.38)/100; |
a(n+2) = x^2
a(n+1) = x
a(n) = 1
p1*x^2+p2*x+p3 = 0, solve ?
(%i16) | eqn_5:ev(eqn_4,a(n)=1,a(n+1)=x,a(n+2)=x^2); |
(%i17) | eqs:solve(eqn_5,x); |
(%i18) | 31.0/50.0; |
(%i20) |
pc1:float(part(eqs,1,2)); pc2:float(part(eqs,2,2)); |
(%i21) | eq_model:a(k)=c1*(pc1)^k+c2*(pc2)^k; |
given : a(0),a(1) to find c1,c2
(%i23) |
eq_model1:ev(eq_model,k=0); eq_model2:ev(eq_model,k=1); |
solve the system : a(0),a(1)
(%i24) | cf:solve([eq_model1,eq_model2],[c1,c2]); |
(%i26) |
cf1:part(cf,1,1); cf2:part(cf,1,2); |
(%i27) | model_lake:ev(eq_model,cf1,cf2); |