As far as I understand your document, L is a function of t, not a function of eta and t. Why do you solve for it as a vector of size N and not as a single value ?
Solving an equation consisting of a PDE and ODE
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Hello,
I am trying to solve a model equation for a cake filtration system ising methods of line. The equation contains both a PDE and an ODE. I am a beginner in the use of MATLAB. I have attached the description of the equation and a code that have tried putting together. The code seems to have several errors. Could anyone advise on how I could resolve the errors in the code or how better I could structure it
clc;clear;close all;
%Given parameters
espo=0.269; gamma=0.49; beta=0.09; Pa=1200; k0=3.4965*10^(-15); u=0.00089; s=0.2;
qlm=2*10^(-5);
A=(k0*Pa)/(u*beta);
B=(1-gamma)/beta;
t=1:1:1000; %time mesh
N=100; %number of grid points
z=linspace(0,1,N); %space mesh
dz=(1-0)/N;
%initial conditions
for i=1:N
esp0= espo;% for PDE esp(z,t=1 sec)
L0=(s/(esp0-s)*qlm*1);%for ODE L(t=1 sec)
IC=[esp0 L0];
end
%solve the function with ode15s
[t y]=ode15s(@fpde,t,IC,[],espo,A,B,s,z,dz,N,qlm);
%recalculation
%define values
esp=y(1:N,:);
L=y(N+1:2*N,:);
%boundary conditions
despdt(:,1)=2*(qlm/L)*((esp(1)-eps(0))./dz);
esp(:,N)=espo;
%function for PDE-ODE
function dydt=fpde(t,y,espo,A,B,s,z,dz,N,qlm)
dydt=zeros(length(y),1);
despdt=zeros(N,1);
dLdt=zeros(N,1);
%define values
esp=y(1:N);
L=y(N:2*N);
%boundary conditions
despdt(1)=2*(qlm/L)*((esp(1)-eps(0))./dz);
esp(N)=espo;
%main program
%interior
for i=2:N-1
L(i)=(s/((sum(esp(i),'all')+esp(N))./(N-1))-s)*qlm.*t; %calculation of cake length
dLdt(i)=L(i)./t; %growth of cake with time
despdt(i)=((A/L)*((esp(i)/espo)^B)*(esp(i+1)-2*esp(i)+esp(i-1))./dz^2)-((1/L(i))*((z*dLdt(i))+qlm)*(esp(i)-esp(i-1))./dz);
end
dydt=real([despdt;dLdt]);
end
?
16 个评论
Suzanne
2023-9-5
Yes, you are right. But the calculation of L depends on epsilon, and epsilon depends on L. So I was wondering how to solve them simultaneously, and also how to store L.
Torsten
2023-9-5
I think you made a coordinate transformation somewhere as eta = x/L and deduced the equation you call "the main equation" in your document. Could you include the original equation written in x and t ?
Suzanne
2023-9-5
Yes, I did. To fix the moving boundary. Let me include it, and resend the document.
Torsten
2023-9-5
Is the multiplication with t in the definition of L some empirical correlation ? It looks quite arbitrary to me.
Suzanne
2023-9-5
No. It is an arbitrary value. 't' represents the time. It is the time taken for the filtration process,i.e., cake to grow. Therefore, the cake thickness (L) increases with time (t)
Torsten
2023-9-5
编辑：Torsten
2023-9-5
The equation
L - s*q_lm*t/(1/L*integral_{x=0}^{x=L} eps_s dx - s) = 0
is an implicit equation for L. It has to be differentiated implicitly with respect to t to derive dL/dt. Did you do that somewhere ? You'll have to apply Leibniz integral rule for the integral:
Another way is to define this equation as an algebraic equation and let ode15i, e.g., compute the time-derivative for you. But usually, ode15i is not that easy to handle and often fails.
Suzanne
2023-9-5
Are you referring to using it in the code?
In the code, I calculated the epsilon in the expression of L by averaging all the epsilon at different grid points. This is because the PDE was with respect to eta, instead of x
Torsten
2023-9-5
The thing is that your equation dL/dt = L/t is incorrect if you define L by
L = s*q_lm*t/(1/L*integral_{x=0}^{x=L} eps_s dx - s)
You neglect the time derivative of the expression 1/L*integral_{x=0}^{x=L} eps_s dx.
Suzanne
2023-9-6
Actually, the equation is supposed to be for dLdt without t as shown below. I wrote it wrongly. It wasn't for L.
dLdt= s*q_lm/(1/L*integral_{x=0}^{x=L} eps_s dx - s)
Suzanne
2023-9-6
Originally, the expressions were for dLdt. So I justed directly added t to the expression and assigned it to L, which from your explanation is wrong.I have now adjusted the equations back to what they were as will seen in the attached document.
采纳的回答
Torsten
2023-9-6
编辑：Torsten
2023-9-6
The discretization can be taken from the attached document (formulae 3.1 (c) and 3.2 (c)).
%Given parameters
espo=0.269; gamma=0.49; beta=0.09; Pa=1200; k0=3.4965*10^(-15); u=0.00089; s=0.2;
qlm=2*10^(-5);
A=(k0*Pa)/(u*beta);
B=(1-gamma)/beta;
t=0:100:15000; %time mesh
N=5000; %number of grid points
z=linspace(0,1,N).'; %space mesh
dz=(1-0)/N;
%initial conditions
IC = [ones(N,1)*espo;s/(espo-s)*qlm];
[t,y] = ode15s(@(t,y)fpde(t,y,espo,A,B,s,z,dz,N,qlm),t,IC);
%define values
esp=y(:,1:N);
L=y(:,N+1);
figure(1)
plot(t,L)
figure(2)
hold on
for i = [ 1 3 5 10 20 30 40 50 60 70 80 90 150]
plot(linspace(0,L(i),N),esp(i,:))
end
hold off
xlim([0 0.001])
function dydt=fpde(t,y,espo,A,B,s,z,dz,N,qlm)
dydt=zeros(length(y),1);
despdt=zeros(N,1);
%define values
esp=y(1:N);
L=y(N+1);
dLdt = s/(trapz(z,esp)-s)*qlm;
despdt(1) = A/L^2*(1/((z(2)+z(1))/2-z(1))*(((esp(1)/espo)^B+(esp(2)/espo)^B)/2*...
(esp(2)-esp(1))/(z(2)-z(1))-esp(1)*L/A*qlm))+...
1/L*(z(1)*dLdt+qlm)*(esp(2)-esp(1))/(z(2)-z(1));
despdt(2:N-1) = A/L^2*(1./((z(3:N)+z(2:N-1))/2-(z(2:N-1)+z(1:N-2))/2).*...
((esp(3:N)/espo).^B+(esp(2:N-1)/espo).^B)/2.*(esp(3:N)-esp(2:N-1))./(z(3:N)-z(2:N-1))-...
((esp(2:N-1)/espo).^B+(esp(1:N-2)/espo).^B)/2.*(esp(2:N-1)-esp(1:N-2))./(z(2:N-1)-z(1:N-2)))+...
1/L*(z(2:N-1)*dLdt+qlm).*(esp(3:N)-esp(2:N-1))./(z(3:N)-z(2:N-1));
despdt(N) = 0;
dydt=[despdt;dLdt];
end
21 个评论
Suzanne
2023-9-6
编辑：Suzanne
2023-9-6
despdt(1) = A/L^2*(1/((z(2)+z(1))/2-z(1))*(-((esp(1)/espo)^B+(esp(2)/espo)^B)/2*...
(esp(2)-esp(1))/(z(2)-z(1))-esp(1)*L/A*qlm))+...
1/L*(z(1)*dLdt+qlm)*(esp(2)-esp(1))/(z(2)-z(1));
If I am not wrong, after the second *.for the expression despdt as shown above. Is that correct?
Torsten
2023-9-6
编辑：Torsten
2023-9-6
The flux at eta = 0 is -esp(1)*L/A*qlm instead of esp(1)*L/A*qlm, thus
despdt(1) = A/L^2*(1/((z(2)+z(1))/2-z(1))*(((esp(1)/espo)^B+(esp(2)/espo)^B)/2*...
(esp(2)-esp(1))/(z(2)-z(1))-(-esp(1)*L/A*qlm)))+...
1/L*(z(1)*dLdt+qlm)*(esp(2)-esp(1))/(z(2)-z(1));
instead of
despdt(1) = A/L^2*(1/((z(2)+z(1))/2-z(1))*(((esp(1)/espo)^B+(esp(2)/espo)^B)/2*...
(esp(2)-esp(1))/(z(2)-z(1))-esp(1)*L/A*qlm))+...
1/L*(z(1)*dLdt+qlm)*(esp(2)-esp(1))/(z(2)-z(1));
The slope changes to negative, but the result does not look satisfactory. I'm quite sure the code implements your equations correctly. Maybe you made some mistake in the choice of parameters or in the equations.
Suzanne
2023-9-6
I will play around with the parameters and also recheck the equations to ensure satisfactory results. Immediately I get satisafactory results, I will give the update.
Suzanne
2023-9-7
Torsten, I got the issue. It was the plotting. The x axis is supposed to have the z values.
Then to refine the plot into a curve-like, I have to recheck my dLdt equation and get the appropriate equation.
Torsten
2023-9-7
Torsten, I got the issue. It was the plotting. The x axis is supposed to have the z values.
Well, I plotted in the original coordinate system (from 0 to L(i)) and restricted the plot to 0 <= z <= 1e-3 because nothing happened for greater values of z. Do you think plotting in [0;1] for L is more appropriate ?
Suzanne
2023-9-7
Yes. But the section of the code below seems to be of eps over L for specified time, if I got it correctly
for i = [ 1 3 5 10 20 30 40 50 60 70 80 90 150]
plot(linspace(0,L(i),N),esp(i,:))
end
Torsten
2023-9-7
The question is: Do you want to plot esp for a specified eta over time or for a specified x over time ? I think plotting eps for a fixed x in the original coordinate system is more natural, isn't it ?
Suzanne
2023-9-7
The plot that I posted might be having z (x/L) constant. I think the major reason the original equation was changed fro dx to dz (deta) was to immobilize the moving boundary L which was changing.
Suzanne
2023-9-7
Yes. So instead of having this:
for i = [ 1 3 5 10 20 30 40 50 60 70 80 90 150]
plot(linspace(0,L(i),N),esp(i,:))
end
We have:
for i = [ 1 3 5 10 20 30 40 50 60 70 80 90 150]
plot(z,esp(i,:))
end
Torsten
2023-9-7
If you want to plot in the eta-coordinate system: yes.
But to be honest: I wouldn't be able to interpret the plots because they are strongly related with L. So you cannot compare the values for esp at constant eta, while you can do it at constant x.
更多回答（1 个）
Mrutyunjaya Hiremath
2023-9-5
clc; clear; close all;
% Given parameters
espo = 0.269;
gamma = 0.49;
beta = 0.09;
Pa = 1200;
k0 = 3.4965e-15;
u = 0.00089;
s = 0.2;
qlm = 2e-5;
A = (k0 * Pa) / (u * beta);
B = (1 - gamma) / beta;
% Time and space mesh
t = 1:1:1000; % Time mesh
N = 100; % Number of grid points
z = linspace(0, 1, N); % Space mesh
dz = z(2) - z(1); % Grid spacing
% Debug: Display N and length of z
fprintf('N = %d, length of z = %d\n', N, length(z));
N = 100, length of z = 100
% Initial conditions
esp0 = espo * ones(N, 1); % Initial condition for PDE
L0 = (s / (espo - s) * qlm) * ones(N, 1); % Initial condition for ODE (corrected)
IC = [esp0; L0]; % This should now be 200 x 1 if N=100
% Debug: Display length of IC
fprintf('Length of IC = %d\n', length(IC));
Length of IC = 200
% Solve the function with ode15s
options = odeset('RelTol',1e-6,'AbsTol',1e-8);
[t, y] = ode15s(@(t, y) fpde(t, y, espo, A, B, s, z, dz, N, qlm), t, IC, options);
Length of esp = 100, Length of L = 100
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Length of esp = 100, Length of L = 100
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Length of esp = 100, Length of L = 100
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Length of esp = 100, Length of L = 100
Current index i = 2
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Length of esp = 100, Length of L = 100
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Length of esp = 100, Length of L = 100
Current index i = 2
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Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current index i = 72
Current index i = 73
Current index i = 74
Current index i = 75
Current index i = 76
Current index i = 77
Current index i = 78
Current index i = 79
Current index i = 80
Current index i = 81
Current index i = 82
Current index i = 83
Current index i = 84
Current index i = 85
Current index i = 86
Current index i = 87
Current index i = 88
Current index i = 89
Current index i = 90
Current index i = 91
Current index i = 92
Current index i = 93
Current index i = 94
Current index i = 95
Current index i = 96
Current index i = 97
Current index i = 98
Current index i = 99
Length of esp = 100, Length of L = 100
Current index i = 2
Current index i = 3
Current index i = 4
Current index i = 5
Current index i = 6
Current index i = 7
Current index i = 8
Current index i = 9
Current index i = 10
Current index i = 11
Current index i = 12
Current index i = 13
Current index i = 14
Current index i = 15
Current index i = 16
Current index i = 17
Current index i = 18
Current index i = 19
Current index i = 20
Current index i = 21
Current index i = 22
Current index i = 23
Current index i = 24
Current index i = 25
Current index i = 26
Current index i = 27
Current index i = 28
Current index i = 29
Current index i = 30
Current index i = 31
Current index i = 32
Current index i = 33
Current index i = 34
Current index i = 35
Current index i = 36
Current index i = 37
Current index i = 38
Current index i = 39
Current index i = 40
Current index i = 41
Current index i = 42
Current index i = 43
Current index i = 44
Current index i = 45
Current index i = 46
Current index i = 47
Current index i = 48
Current index i = 49
Current index i = 50
Current index i = 51
Current index i = 52
Current index i = 53
Current index i = 54
Current index i = 55
Current index i = 56
Current index i = 57
Current index i = 58
Current index i = 59
Current index i = 60
Current index i = 61
Current index i = 62
Current index i = 63
Current index i = 64
Current index i = 65
Current index i = 66
Current index i = 67
Current index i = 68
Current index i = 69
Current index i = 70
Current index i = 71
Current...
% Function for coupled PDE-ODE system
function dydt = fpde(t, y, espo, A, B, s, z, dz, N, qlm)
dydt = zeros(2 * N, 1);
despdt = zeros(N, 1);
dLdt = zeros(N, 1);
% Extract variables
esp = y(1:N);
L = y(N+1:2*N);
% Debug: Display lengths
fprintf('Length of esp = %d, Length of L = %d\n', length(esp), length(L));
% Boundary conditions
despdt(1) = 2 * (qlm / L(1)) * ((esp(1) - espo) / dz);
esp(end) = espo;
% Interior points
for i = 2:N-1
fprintf('Current index i = %d\n', i); % Debug: Display current index
L(i) = (s / ((sum(esp(i)) + esp(N-1)) / (N-1)) - s) * qlm * t; % Debug: Check if t should be scalar
dLdt(i) = L(i) / t; % Debug: Check if t should be scalar
despdt(i) = ((A / L(i)) * ((esp(i) / espo)^B) * (esp(i+1) - 2 * esp(i) + esp(i-1)) / dz^2) - ((1 / L(i)) * ((z(i) * dLdt(i)) + qlm) * (esp(i) - esp(i-1)) / dz);
end
dydt = [despdt; dLdt];
end
1 个评论
Suzanne
2023-9-5
Hello Hiremath,
Thank you. But 'L' here refers to the cake thickness which grows with time, and hence a function of only t. It is only 'eps' that is a function of both 'z' and 't'. I am trying to get 'eps' at each grid point at different time. And also the change of L with time.
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