Iterative of Nonlinear Functions

2025 11월 2
2 답변
업데이트 시간: 2025 11월 3
조회 수: 13 (30일)
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0 개 추천

Ran in:
Someone please help, whenever i run my code I keep getting the same errors. My code and a screenshoot of the error page is attached.
% Given Values
k = 120;
alpha = 3.91e-6;
Ti=1073.15;
Tw=288.15;
Ta=288.15;
hw=100;
ha=10;
dt=10;
n_steps=60;
l=0.1;
tau=0.00391;
T1=zeros(n_steps,1);
T1(1,1)=Ti;
T2=zeros(n_steps,1);
T2(1,1)=Ti;
T3=zeros(n_steps,1);
T3(1,1)=Ti;
T4=zeros(n_steps,1);
T4(1,1)=Ti;
T5=zeros(n_steps,1);
T5(1,1)=Ti;
T6=zeros(n_steps,1);
T6(1,1)=Ti;
T7=zeros(n_steps,1);
T7(1,1)=Ti;
T8=zeros(n_steps,1);
T8(1,1)=Ti;
T9=zeros(n_steps,1);
T9(1,1)=Ti;
T10=zeros(n_steps,1);
T10(1,1)=Ti;
T11=zeros(n_steps,1);
T11(1,1)=Ti;
T12=zeros(n_steps,1);
T12(1,1)=Ti;
T13=zeros(n_steps,1);
T13(1,1)=Ti;
T14=zeros(n_steps,1);
T14(1,1)=Ti;
T15=zeros(n_steps,1);
T15(1,1)=Ti;
for i=1:(n_steps-1)
syms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
eqn1 = l*k*(ha+hw)*(Ta-T1(i))+(T6-T1(i))+(T2(i)-T1(i))==(t1-T1(i))/tau;
eqn2 = ha*l*(Ta-T2(i)+((T7(i)-T2(i))+(T3(i)-T2(i))+(T1(i)-T2(i))))==(t2-T2(i))/tau;
eqn3 = ha*l/k*(Ta-T3(i))+((T4(i)-T3(i))+(T8(i)-T3(i)+(T2(i)-T3(i))))==(t3-T1(i))/tau;
eqn4 = ha*l*(Ta-T4(i)+((T9(i)-T4(i))+(T3(i)-T4(i))+(T5(i)-T4(i))))==(t4-T4(i))/tau;
eqn5 = l*k*(ha+hw)*(Ta-T5(i))+(T10-T5(i))+(T4(i)-T5(i))==(t5-T5(i))/tau;
eqn6 = hw*l/k*(Ta-T6(i))+((T1(i)-T6(i))+(T7(i)-T6(i))+(T11(i)-T6(i)))==(t6-T6(i))/tau;
eqn7 = (T6(i)-T7(i))+(T8(i)-T7(i))+(T2(i)-T7(i))+(T12(i)-T7(i))==(t7-T7(i))/tau;
eqn8 = (T7(i)-T8(i))+(T9(i)-T8(i))+(T3(i)-T8(i))+(T13(i)-T8(i))==(t8-T8(i))/tau;
eqn9 = (T8(i)-T9(i))+(T10(i)-T9(i))+(T4(i)-T9(i))+(T14(i)-T9(i))==(t9-T9(i))/tau;
eqn10 = hw*l/k*(Ta-T10(i))+((T5(i)-T10(i))+(T9(i)-T10(i))+(T15(i)-T10(i)))==(t10-T10(i))/tau;
eqn11 = hw*l/k*(Ta-T11(i))+((T6(i)-T11(i))+(T12(i)-T11(i)))==(t11-T11(i))/tau;
eqn12 = hw*l/k*(Ta-T12(i))+((T13(i)-T12(i)+(T7(i)-T12(i))+(T11(i)-T12(i))))==(t12-T12(i))/tau;
eqn13 = hw*l/k*(Ta-T13(i))+((T12(i)-T13(i)+(T8(i)-T13(i))+(T14(i)-T13(i))))==(t13-T13(i))/tau;
eqn14 = hw*l/k*(Ta-T14(i))+((T13(i)-T14(i)+(T5(i)-T14(i))+(T9(i)-T14(i))))==(t14-T14(i))/tau;
eqn15 = hw*l/k*(Ta-T15(i))+((T14(i)-T15(i))+(T10(i)-T15(i)))==(t15-T15(i))/tau;
s = solve([eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10 eqn11 eqn12 eqn13 eqn14 eqn15],...
[t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15]);
nodal_temperatures = double(struct2array(s));
T1(i+1,1) = s(1);
T2(i+1,1) = s(2);
T3(i+1,1) = s(3);
T4(i+1,1) = s(4);
T5(i+1,1) = s(5);
T6(i+1,1) = s(6);
T7(i+1,1) = s(7);
T8(i+1,1) = s(8);
T9(i+1,1) = s(9);
T10(i+1,1) = s(10);
T11(i+1,1) = s(11);
T12(i+1,1) = s(12);
T13(i+1,1) = s(13);
T14(i+1,1) = s(14);
T15(i+1,1) = s(15);
end
Error using catMany>checkDimensions (line 53)
Dimensions of objects being concatenated must match.

Error in catMany (line 12)
[resz, ranges] = checkDimensions(sz, dim);
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Error in sym/horzcat (line 20)
y = catMany(2, args{:});
^^^^^^^^^^^^^^^^^^^^^^^^
% Display the results for the final temperatures
disp('Final Temperatures at Step 60:');
disp([T1(60,1), T2(60,1), T3(60,1), T4(60,1), T5(60,1), T6(60,1), ...
T7(60,1), T8(60,1), T9(60,1), T10(60,1), T11(60,1), T12(60,1),...
T13(60,1), T14(60,1), T15(60,1)]);

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답변 (2개)

Torsten
Torsten 2025년 11월 2일
편집: Torsten 2025년 11월 2일
Ran in:

0 개 추천

Ran in:
Why do you use "solve" to get t1,...,t15 ? You can directly solve for the unknowns because you use the explicit Euler method to advance in time. That's why your time step size tau might have to be chosen smaller than 0.00391 for that your temperatures don't diverge. Or you simply made a mistake in setting up your equations.
% Given Values
k = 120;
alpha = 3.91e-6;
Ti=1073.15;
Tw=288.15;
Ta=288.15;
hw=100;
ha=10;
dt=10;
n_steps=60;
l=0.1;
tau=0.00391;
T1=zeros(n_steps,1);
T1(1,1)=Ti;
T2=zeros(n_steps,1);
T2(1,1)=Ti;
T3=zeros(n_steps,1);
T3(1,1)=Ti;
T4=zeros(n_steps,1);
T4(1,1)=Ti;
T5=zeros(n_steps,1);
T5(1,1)=Ti;
T6=zeros(n_steps,1);
T6(1,1)=Ti;
T7=zeros(n_steps,1);
T7(1,1)=Ti;
T8=zeros(n_steps,1);
T8(1,1)=Ti;
T9=zeros(n_steps,1);
T9(1,1)=Ti;
T10=zeros(n_steps,1);
T10(1,1)=Ti;
T11=zeros(n_steps,1);
T11(1,1)=Ti;
T12=zeros(n_steps,1);
T12(1,1)=Ti;
T13=zeros(n_steps,1);
T13(1,1)=Ti;
T14=zeros(n_steps,1);
T14(1,1)=Ti;
T15=zeros(n_steps,1);
T15(1,1)=Ti;
for i=1:(n_steps-1)
syms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
eqn1 = l*k*(ha+hw)*(Ta-T1(i))+(T6(i)-T1(i))+(T2(i)-T1(i))==(t1-T1(i))/tau;
eqn2 = ha*l*(Ta-T2(i)+((T7(i)-T2(i))+(T3(i)-T2(i))+(T1(i)-T2(i))))==(t2-T2(i))/tau;
eqn3 = ha*l/k*(Ta-T3(i))+((T4(i)-T3(i))+(T8(i)-T3(i)+(T2(i)-T3(i))))==(t3-T1(i))/tau;
eqn4 = ha*l*(Ta-T4(i)+((T9(i)-T4(i))+(T3(i)-T4(i))+(T5(i)-T4(i))))==(t4-T4(i))/tau;
eqn5 = l*k*(ha+hw)*(Ta-T5(i))+(T10(i)-T5(i))+(T4(i)-T5(i))==(t5-T5(i))/tau;
eqn6 = hw*l/k*(Ta-T6(i))+((T1(i)-T6(i))+(T7(i)-T6(i))+(T11(i)-T6(i)))==(t6-T6(i))/tau;
eqn7 = (T6(i)-T7(i))+(T8(i)-T7(i))+(T2(i)-T7(i))+(T12(i)-T7(i))==(t7-T7(i))/tau;
eqn8 = (T7(i)-T8(i))+(T9(i)-T8(i))+(T3(i)-T8(i))+(T13(i)-T8(i))==(t8-T8(i))/tau;
eqn9 = (T8(i)-T9(i))+(T10(i)-T9(i))+(T4(i)-T9(i))+(T14(i)-T9(i))==(t9-T9(i))/tau;
eqn10 = hw*l/k*(Ta-T10(i))+((T5(i)-T10(i))+(T9(i)-T10(i))+(T15(i)-T10(i)))==(t10-T10(i))/tau;
eqn11 = hw*l/k*(Ta-T11(i))+((T6(i)-T11(i))+(T12(i)-T11(i)))==(t11-T11(i))/tau;
eqn12 = hw*l/k*(Ta-T12(i))+((T13(i)-T12(i)+(T7(i)-T12(i))+(T11(i)-T12(i))))==(t12-T12(i))/tau;
eqn13 = hw*l/k*(Ta-T13(i))+((T12(i)-T13(i)+(T8(i)-T13(i))+(T14(i)-T13(i))))==(t13-T13(i))/tau;
eqn14 = hw*l/k*(Ta-T14(i))+((T13(i)-T14(i)+(T5(i)-T14(i))+(T9(i)-T14(i))))==(t14-T14(i))/tau;
eqn15 = hw*l/k*(Ta-T15(i))+((T14(i)-T15(i))+(T10(i)-T15(i)))==(t15-T15(i))/tau;
s = solve([eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10 eqn11 eqn12 eqn13 eqn14 eqn15],...
[t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15]);
s = double(struct2array(s));
T1(i+1,1) = s(1);
T2(i+1,1) = s(2);
T3(i+1,1) = s(3);
T4(i+1,1) = s(4);
T5(i+1,1) = s(5);
T6(i+1,1) = s(6);
T7(i+1,1) = s(7);
T8(i+1,1) = s(8);
T9(i+1,1) = s(9);
T10(i+1,1) = s(10);
T11(i+1,1) = s(11);
T12(i+1,1) = s(12);
T13(i+1,1) = s(13);
T14(i+1,1) = s(14);
T15(i+1,1) = s(15);
end
% Display the results for the final temperatures
disp('Final Temperatures at Step 60:');
Final Temperatures at Step 60:
disp([T1(60,1), T2(60,1), T3(60,1), T4(60,1), T5(60,1), T6(60,1), ...
T7(60,1), T8(60,1), T9(60,1), T10(60,1), T11(60,1), T12(60,1), ...
T13(60,1), T14(60,1), T15(60,1)]);
1.0e+39 * -2.9940 0.0017 0.7202 0.0017 -2.9940 0.0023 -0.0000 -0.0005 -0.0000 0.0023 -0.0000 0.0000 -0.0000 0.0023 -0.0000

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Jamari
Jamari 2025년 11월 2일
Thanks for the response. I recently ran the code you provide, first making some adjustments to equ 1-15, and i ran into a new issue.
% Given Values
k = 120;
alpha = 3.91e-6;
Ti=1073.15;
Tw=288.15;
Ta=288.15;
hw=100;
ha=10;
dt=10;
n_steps=60;
l=0.1;
tau=0.0039;
T1=zeros(n_steps,1);
T1(1,1)=Ti;
T2=zeros(n_steps,1);
T2(1,1)=Ti;
T3=zeros(n_steps,1);
T3(1,1)=Ti;
T4=zeros(n_steps,1);
T4(1,1)=Ti;
T5=zeros(n_steps,1);
T5(1,1)=Ti;
T6=zeros(n_steps,1);
T6(1,1)=Ti;
T7=zeros(n_steps,1);
T7(1,1)=Ti;
T8=zeros(n_steps,1);
T8(1,1)=Ti;
T9=zeros(n_steps,1);
T9(1,1)=Ti;
T10=zeros(n_steps,1);
T10(1,1)=Ti;
T11=zeros(n_steps,1);
T11(1,1)=Ti;
T12=zeros(n_steps,1);
T12(1,1)=Ti;
T13=zeros(n_steps,1);
T13(1,1)=Ti;
T14=zeros(n_steps,1);
T14(1,1)=Ti;
T15=zeros(n_steps,1);
T15(1,1)=Ti;
for i=1:(n_steps-1)
syms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
eqn1 = 2*l/k*((ha+hw)*(Ta-T1(i)))+2*(T6(i)+T2(i)+2*T1(i))==(t1-T1(i))/tau;
eqn2 = 2*ha*l/k*(Ta-T2(i))+2*(T7(i)-T2(i))+(T3(i)-T2(i))+(T1(i)-T2(i))==(t2-T2(i))/tau;
eqn3 = 2*ha*l/k*(Ta-T3(i))+(T2(i)-T3(i))+(T4(i)-T3(i))+2*(T8(i)-T3(i))==(t3-T3(i))/tau;
eqn4 = 2*ha*l/k*(Ta-T4(i))+2*(T9(i)-T4(i))+(T3(i)-T5(i))+(T5(i)-T4(i))==(t4-T4(i))/tau;
eqn5 = 2*l/k*((ha+hw)*(Ta-T5(i)))+2*(T10(i)+T4(i)+2*T5(i))==(t5-T5(i))/tau;
eqn6 = 2*hw*l/k*(Ta-T6(i))+2*(T7(i)*T6(i))+(T11(i)-T6(i))==(t6-T6(i))/tau;
eqn7 = (T2(i)-T7(i))+(T8(i)-T7(i))+(T12(i)-T7(i))+(T6(i)-T7(i))==(t7-T7(i))/tau;
eqn8 = (T3(i)-T8(i))+(T9(i)-T8(i))+(T13(i)-T8(i))+(T7(i)-T8(i))==(t8-T8(i))/tau;
eqn9 = (T4(i)-T9(i))+(T10(i)-T9(i))+(T14(i)-T9(i))+(T8(i)-T9(i))==(t9-T9(i))/tau;
eqn10 = 2*hw*l/k*(Ta-T10(i))+2*(T9(i)*T10(i))+(T15(i)-T10(i))==(t10-T10(i))/tau;
eqn11 = 4*hw*l/k*(Ta-T11(i))+2*(T6(i)-T11(i))+2*(T12(i)-T11(i))==(t11-T11(i))/tau;
eqn12 = 2*hw*l/k*(Ta-T12(i))+2*(T7(i)-T12(i))+(T13(i)-T12(i))+(T11(i)-T12(i))==(t12-T12(i))/tau;
eqn13 = 2*hw*l/k*(Ta-T13(i))+2*(T8(i)-T13(i))+(T14(i)-T13(i))+(T12(i)-T13(i))==(t13-T13(i))/tau;
eqn14 = 2*hw*l/k*(Ta-T14(i))+2*(T9(i)-T14(i))+(T15(i)-T14(i))+(T13(i)-T14(i))==(t14-T14(i))/tau;
eqn15 = 4*hw*l/k*(Ta-T15(i))+2*(T10(i)-T15(i))+2*(T14(i)-T15(i))==(t15-T15(i))/tau;
s = solve([eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10 eqn11 eqn12 eqn13 eqn14 eqn15],...
[t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15]);
s = double(struct2array(s));
T1(i+1,1) = s(1);
T2(i+1,1) = s(2);
T3(i+1,1) = s(3);
T4(i+1,1) = s(4);
T5(i+1,1) = s(5);
T6(i+1,1) = s(6);
T7(i+1,1) = s(7);
T8(i+1,1) = s(8);
T9(i+1,1) = s(9);
T10(i+1,1) = s(10);
T11(i+1,1) = s(11);
T12(i+1,1) = s(12);
T13(i+1,1) = s(13);
T14(i+1,1) = s(14);
T15(i+1,1) = s(15);
end
% Display the results for the final temperatures
disp('Final Temperatures at Step 60:');
disp([T1(60,1), T2(60,1), T3(60,1), T4(60,1), T5(60,1), T6(60,1), ...
T7(60,1), T8(60,1), T9(60,1), T10(60,1), T11(60,1), T12(60,1), ...
T13(60,1), T14(60,1), T15(60,1)]);
Torsten
Torsten 2025년 11월 2일
편집: Torsten 2025년 11월 2일
Ran in:
Ran in:
You get undefined results from "solve" for i >= 14:
% Given Values
k = 120;
alpha = 3.91e-6;
Ti=1073.15;
Tw=288.15;
Ta=288.15;
hw=100;
ha=10;
dt=10;
n_steps=60;
l=0.1;
tau=0.0039;
T1=zeros(n_steps,1);
T1(1,1)=Ti;
T2=zeros(n_steps,1);
T2(1,1)=Ti;
T3=zeros(n_steps,1);
T3(1,1)=Ti;
T4=zeros(n_steps,1);
T4(1,1)=Ti;
T5=zeros(n_steps,1);
T5(1,1)=Ti;
T6=zeros(n_steps,1);
T6(1,1)=Ti;
T7=zeros(n_steps,1);
T7(1,1)=Ti;
T8=zeros(n_steps,1);
T8(1,1)=Ti;
T9=zeros(n_steps,1);
T9(1,1)=Ti;
T10=zeros(n_steps,1);
T10(1,1)=Ti;
T11=zeros(n_steps,1);
T11(1,1)=Ti;
T12=zeros(n_steps,1);
T12(1,1)=Ti;
T13=zeros(n_steps,1);
T13(1,1)=Ti;
T14=zeros(n_steps,1);
T14(1,1)=Ti;
T15=zeros(n_steps,1);
T15(1,1)=Ti;
%for i=1:(n_steps-1)
for i = 1:14
syms t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
eqn1 = 2*l/k*((ha+hw)*(Ta-T1(i)))+2*(T6(i)+T2(i)+2*T1(i))==(t1-T1(i))/tau;
eqn2 = 2*ha*l/k*(Ta-T2(i))+2*(T7(i)-T2(i))+(T3(i)-T2(i))+(T1(i)-T2(i))==(t2-T2(i))/tau;
eqn3 = 2*ha*l/k*(Ta-T3(i))+(T2(i)-T3(i))+(T4(i)-T3(i))+2*(T8(i)-T3(i))==(t3-T3(i))/tau;
eqn4 = 2*ha*l/k*(Ta-T4(i))+2*(T9(i)-T4(i))+(T3(i)-T5(i))+(T5(i)-T4(i))==(t4-T4(i))/tau;
eqn5 = 2*l/k*((ha+hw)*(Ta-T5(i)))+2*(T10(i)+T4(i)+2*T5(i))==(t5-T5(i))/tau;
eqn6 = 2*hw*l/k*(Ta-T6(i))+2*(T7(i)*T6(i))+(T11(i)-T6(i))==(t6-T6(i))/tau;
eqn7 = (T2(i)-T7(i))+(T8(i)-T7(i))+(T12(i)-T7(i))+(T6(i)-T7(i))==(t7-T7(i))/tau;
eqn8 = (T3(i)-T8(i))+(T9(i)-T8(i))+(T13(i)-T8(i))+(T7(i)-T8(i))==(t8-T8(i))/tau;
eqn9 = (T4(i)-T9(i))+(T10(i)-T9(i))+(T14(i)-T9(i))+(T8(i)-T9(i))==(t9-T9(i))/tau;
eqn10 = 2*hw*l/k*(Ta-T10(i))+2*(T9(i)*T10(i))+(T15(i)-T10(i))==(t10-T10(i))/tau;
eqn11 = 4*hw*l/k*(Ta-T11(i))+2*(T6(i)-T11(i))+2*(T12(i)-T11(i))==(t11-T11(i))/tau;
eqn12 = 2*hw*l/k*(Ta-T12(i))+2*(T7(i)-T12(i))+(T13(i)-T12(i))+(T11(i)-T12(i))==(t12-T12(i))/tau;
eqn13 = 2*hw*l/k*(Ta-T13(i))+2*(T8(i)-T13(i))+(T14(i)-T13(i))+(T12(i)-T13(i))==(t13-T13(i))/tau;
eqn14 = 2*hw*l/k*(Ta-T14(i))+2*(T9(i)-T14(i))+(T15(i)-T14(i))+(T13(i)-T14(i))==(t14-T14(i))/tau;
eqn15 = 4*hw*l/k*(Ta-T15(i))+2*(T10(i)-T15(i))+2*(T14(i)-T15(i))==(t15-T15(i))/tau;
s = solve([eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8 eqn9 eqn10 eqn11 eqn12 eqn13 eqn14 eqn15],...
[t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15]);
s = double(struct2array(s));
if i==14
s
end
T1(i+1,1) = s(1);
T2(i+1,1) = s(2);
T3(i+1,1) = s(3);
T4(i+1,1) = s(4);
T5(i+1,1) = s(5);
T6(i+1,1) = s(6);
T7(i+1,1) = s(7);
T8(i+1,1) = s(8);
T9(i+1,1) = s(9);
T10(i+1,1) = s(10);
T11(i+1,1) = s(11);
T12(i+1,1) = s(12);
T13(i+1,1) = s(13);
T14(i+1,1) = s(14);
T15(i+1,1) = s(15);
end
s = 1×15
1.0e+197 * 3.6785 0.0000 0.0000 0.0000 3.6784 NaN 1.8393 0.0000 1.8392 NaN 3.6785 0.0000 0.0000 0.0000 3.6784
<mw-icon class=""></mw-icon>
<mw-icon class=""></mw-icon>
% Display the results for the final temperatures
%disp('Final Temperatures at Step 60:');
%disp([T1(60,1), T2(60,1), T3(60,1), T4(60,1), T5(60,1), T6(60,1), ...
% T7(60,1), T8(60,1), T9(60,1), T10(60,1), T11(60,1), T12(60,1), ...
% T13(60,1), T14(60,1), T15(60,1)]);
Torsten
Torsten 2025년 11월 2일
편집: Torsten 2025년 11월 3일
Systems like yours usually take the form
dT_i/dt = h_ext * (T_ext - T_i) + sum_{j~=i} h_ij * (T_j - T_i)
with a symmetric matrix h_ij.
I cannot find this structure in your equations.

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Torsten
Torsten 2025년 11월 2일
이동: Torsten 2025년 11월 2일
Ran in:

0 개 추천

Ran in:
My suggestion: Implicit Euler method.
And maybe there is a mistake in equation (3) (see below).
% Given Values
k = 120;
alpha = 3.91e-6;
Ti=1073.15;
Tw=288.15;
Ta=288.15;
hw=100;
ha=10;
dt=10;
n_steps=60;
l=0.1;
tau=0.00391;
Tnew = sym('Tnew',[15 1]);
Told = sym('Told',[15,1]);
eqn(1) = l*k*(ha+hw)*(Ta-Tnew(1))+(Tnew(6)-Tnew(1))+(Tnew(2)-Tnew(1))==(Tnew(1)-Told(1))/tau;
eqn(2) = ha*l*(Ta-Tnew(2)+((Tnew(7)-Tnew(2))+(Tnew(3)-Tnew(2))+(Tnew(1)-Tnew(2))))==(Tnew(2)-Told(2))/tau;
%eqn(3) = ha*l/k*(Ta-Tnew(3))+((Tnew(4)-Tnew(3))+(Tnew(8)-Tnew(3)+(Tnew(2)-Tnew(3))))==(Tnew(3)-Told(3))/tau;
eqn(3) = ha*l/k*(Ta-Tnew(3))+((Tnew(4)-Tnew(3))+(Tnew(8)-Tnew(3)+(Tnew(2)-Tnew(3))))==(Tnew(3)-Told(1))/tau;
eqn(4) = ha*l*(Ta-Tnew(4)+((Tnew(9)-Tnew(4))+(Tnew(3)-Tnew(4))+(Tnew(5)-Tnew(4))))==(Tnew(4)-Told(4))/tau;
eqn(5) = l*k*(ha+hw)*(Ta-Tnew(5))+(Tnew(10)-Tnew(5))+(Tnew(4)-Tnew(5))==(Tnew(5)-Told(5))/tau;
eqn(6) = hw*l/k*(Ta-Tnew(6))+((Tnew(1)-Tnew(6))+(Tnew(7)-Tnew(6))+(Tnew(11)-Tnew(6)))==(Tnew(6)-Told(6))/tau;
eqn(7) = (Tnew(6)-Tnew(7))+(Tnew(8)-Tnew(7))+(Tnew(2)-Tnew(7))+(Tnew(12)-Tnew(7))==(Tnew(7)-Told(7))/tau;
eqn(8) = (Tnew(7)-Tnew(8))+(Tnew(9)-Tnew(8))+(Tnew(3)-Tnew(8))+(Tnew(13)-Tnew(8))==(Tnew(8)-Told(8))/tau;
eqn(9) = (Tnew(8)-Tnew(9))+(Tnew(10)-Tnew(9))+(Tnew(4)-Tnew(9))+(Tnew(14)-Tnew(9))==(Tnew(9)-Told(9))/tau;
eqn(10) = hw*l/k*(Ta-Tnew(10))+((Tnew(5)-Tnew(10))+(Tnew(9)-Tnew(10))+(Tnew(15)-Tnew(10)))==(Tnew(10)-Told(10))/tau;
eqn(11) = hw*l/k*(Ta-Tnew(11))+((Tnew(6)-Tnew(11))+(Tnew(12)-Tnew(11)))==(Tnew(11)-Told(11))/tau;
eqn(12) = hw*l/k*(Ta-Tnew(12))+((Tnew(13)-Tnew(12)+(Tnew(7)-Tnew(12))+(Tnew(11)-Tnew(12))))==(Tnew(12)-Told(12))/tau;
eqn(13) = hw*l/k*(Ta-Tnew(13))+((Tnew(12)-Tnew(13)+(Tnew(8)-Tnew(13))+(Tnew(14)-Tnew(13))))==(Tnew(13)-Told(13))/tau;
eqn(14) = hw*l/k*(Ta-Tnew(14))+((Tnew(13)-Tnew(14)+(Tnew(5)-Tnew(14))+(Tnew(9)-Tnew(14))))==(Tnew(14)-Told(14))/tau;
eqn(15) = hw*l/k*(Ta-Tnew(15))+((Tnew(14)-Tnew(15))+(Tnew(10)-Tnew(15)))==(Tnew(15)-Told(15))/tau;
vars = Tnew;
[A,b] = equationsToMatrix(eqn,vars);
Anum = double(A);
Tsol = zeros(15,n_steps);
Tsol(:,1) = Ti*ones(15,1);
for i = 2:n_steps
bnum = double(subs(b,Told,Tsol(:,i-1)));
Tsol(:,i) = Anum\bnum;
end
plot(Tsol(:,end))
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Jamari
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2025년 11월 3일

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