parenrightBigg e 1 75 t 7 3962 parenleftBigg 4472 8944 parenrightBigg e 125 t

Parenrightbigg e 1 75 t 7 3962 parenleftbigg 4472

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parenrightBigg e - 1 . 75 t + 7 . 3962 parenleftBigg 0 . 4472 0 . 8944 parenrightBigg e - 0 . 125 t + parenleftBigg 16 16 parenrightBigg . The MatLab’s numerical solver, ode23 , extends easily to systems of 1 st order differential equa- tions. Below we show how to both use the numerical solver and the exact solution above to graph the u ( t ). First the right hand side of System (1) is made into a MatLab function 1 function yp = greenhouse ( t , y ) 2 % Greenhouse DE ( rhs ) 3 yt1 = - (13/8) * y (1) +(3/4) * y (2) + 14; 4 yt2 = (1/4) * y (1) - (1/4) * y (2) ; 5 yp = [ yt1 , yt2 ] ’ ; 6 end The function plotting the numerical and exact solutions is given by 1 mytitle = ’ Greenhouse/Rockbed ’ ; % Title 2 xlab = ’ $t$ hrs ’ ; % X - l a b e l 3 ylab = ’ Temperature ( $ˆ \ circ$C ) ’ ; % Y - l a b e l 4 5 u0 = [ 5 , 2 5 ] ’ ; 6 [ t , u ] = ode23 ( @greenhouse , [ 0 , 1 0 ] , u0 ) ; % simulate heat with ode23 7 tt = linspa ce (0 ,10 ,200) ; 8 u1 = - 14.3077 * exp ( - 1.75 * tt ) +3.3077 * exp ( - 0.125 * tt ) +16; % so lutio n u1 9 u2 = 2.3846 * exp ( - 1.75 * tt ) +6.6154 * exp ( - 0.125 * tt ) +16; % so lutio n u2 10 11 plot ( t , u ( : , 1 ) , ’b - , ’ LineWidth ’ , 1 . 5 ) ; % Plot greenhouse a i r ( numeric ) 12 hold on % Plots Multiple graphs 13 plot ( t , u ( : , 2 ) , ’ r - , ’ LineWidth ’ , 1 . 5 ) ; % Plot greenhouse rocks ( numeric ) 14 plot ( tt , u1 , ’ c : ’ , ’ LineWidth ’ , 1 . 5 ) ; % Plot greenhouse air , u1 15 plot ( tt , u2 , ’m: ’ , ’ LineWidth ’ , 1 . 5 ) ; % Plot greenhouse rocks , u2 16 plot ( [ 0 1 0 ] , [ 1 6 1 6 ] , ’k : ’ , ’ LineWidth ’ , 1 . 5 ) ; % Plot equilibrium 17 grid % Adds Gr idlines 18 text (0 .6 ,1 1 , ’ $u 1$ ’ , ’ co lo r ’ , ’ blue ’ , ’ FontSize ’ , 1 4 , . . .
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19 ’FontName ’ , ’ Times New Roman ’ , ’ i n t e r p r e t e r ’ , ’ la tex ’ ) ; 20 text (3 ,22 , ’ $u 2$ ’ , ’ co lo r ’ , ’ red ’ , ’ FontSize ’ , 1 4 , . . . 21 ’FontName ’ , ’ Times New Roman ’ , ’ i n t e r p r e t e r ’ , ’ la tex ’ ) ; 22 legend ( ’ Air ( numeric ) ’ , ’ Rockbed ( numeric ) ’ , ’ Air ( exact ) ’ , . . . 23 ’ Rockbed ( exact ) ’ ,4) ; 24 25 a xis ( [ 0 10 0 3 0 ]) ; % Defines l i m i t s of graph 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 u 1 u 2 t hrs Temperature ( C) Greenhouse/Rockbed Air (numeric) Rockbed (numeric) Air (exact) Rockbed (exact) The graph shows how well the numerical routine ode23 in MatLab tracks the solution to System (1). As we saw in the lecture notes, the heat transfers rapidly into the air compartment, then slowly the solution tends toward the equilibrium solution. Phase Portrait - 2D This final section shows how to create two dimensional phase portraits and direction fields . You begin by downloading the MatLab files for pplane and dfield by John Polking from Rice University. The current version is pplane8 , which is invoked by having this m-file in your current directory and typing pplane8 in the command window of MatLab.
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