-
Notifications
You must be signed in to change notification settings - Fork 1
/
FDEGL.m
178 lines (153 loc) · 5.73 KB
/
FDEGL.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
%----------------------------------------------%----------------------------------------------%
% Grunwald-Letnikov Scheme %
%----------------------------------------------%----------------------------------------------% %%%%% %%%%%
%----------------------------------------------%
% Setting the Parameters %
%----------------------------------------------%
alpha =1 ; % Fractional Order
T = 18; % Total Time
h = 0.05; % Step
N = T/h; % Number of Steps
k = 5; % Number of Corrections8
k01 = 0.3; % Parameter of E:uation
k2 = 0.6; % Parameter of E:uation
y0 = [5,0]; % Initial Condition8
V = 1;
dim = 2; % Number of Equations
numd = 3; % Number of Doses
params = [k01,k2,V];
%----------------------------------------------%
% Initializing Matrices %
%----------------------------------------------%
y = zeros(N,dim); % Solution Matrix
t = zeros(1,N); % Time Matrix
bi = zeros(1,N); % Fractional Coefficient matrix
doses = [0,0,0]; % Doses Matrix
tdoses = [0,0,0]; % Dosing Times
temp = zeros(k,dim);
sum = zeros(1,dim);
psi = zeros(1,dim);
gf = zeros(k,dim);
yf = zeros(N+1,dim);
tf = zeros(N+1,1);
%----------------------------------------------%
% Fractional Coefficients %
%----------------------------------------------%
omega0=1;
for i=1:N
if i==1
omega(i)=omega0*(1-(1-alpha)/(i));
else
omega(i)=omega(i-1)*(1-(1-alpha)/(i));
end
end
for j=1:N
t(1,j)=h*j;
end
%----------------------------------------------%
% Numerical Algorithm for Solver %
%----------------------------------------------%
for n=1:N
sum=zeros(1,dim);
if n>1
for q=1:(numel(tdoses)-1)
if tdoses(q)<=t(n) && t(n)<=tdoses(q+1)
temp(1,:)=y(n-1,:)+doses(q);
elseif tdoses(numd)<=t(n)
temp(1,:)=y(n-1,:)+doses(end);
end
end
y(n,:)= temp(1,:);
for j=1:n-1
ffv = ff(t(j),y(j,:),params);
sum(:)=sum(:)+omega(n-j).*ffv(:);
end
else
temp(1,:) = y0(:);
y(n,:)=y0(:);
end
for q=1:(numel(tdoses)-1)
if tdoses(q)<=t(n) && t(n)<=tdoses(q+1)
psi(:)=y0(:)+h^alpha.*sum(:)+doses(q);
elseif tdoses(numd)<=t(n)
psi(:)=y0(:)+h^alpha.*sum(:)+doses(end);
end
end
for i=2:k
for q=1:(numel(tdoses)-1)
if tdoses(q)<=t(n) && t(n)<=tdoses(q+1)
temp(i,:) = y(n,:)+doses(q);
elseif tdoses(numd)<=t(n)
temp(i,:) = y(n,:)+doses(end);
end
end
temp(i,:) = y(n,:);
ffv(:) = ff(t(n),temp(i,:),params);
gf(i,:) = temp(i,:)-h^(alpha).*ffv(1,:)-psi(1,:);
ffvd = ffdot(t(n),temp(i,:),params);
gdf = eye(dim)-h^(alpha).*ffvd;
y(n,:)=temp(i,:)-gf(i,:)/(gdf);
end
end
tf(1) = 0;
tf(2:N+1) = t(1:N);
yf(2:N+1,:) = y(1:N,:);
yf(1,:) = y0 (:);
yf(2:end,:) = y(:,:);
%----------------------------------------------%
% ODE15s for Comparison %
%----------------------------------------------%
sol=ode15s(@(t,y)rate(t,y,k01,k2),[0 T],y0)
ycl=deval(sol,tf);
%----------------------------------------------%
% Plots %
%----------------------------------------------%
figure(1)
plot(tf,yf,'o','linewidth',2)
hold on;
plot(tf,ycl/V,'linewidth',2)
hold on;
xlabel('t','Interpreter','latex')
ylabel('y','Interpreter','latex')
string_title={'\textbf{Two Compartment Model for $a=$}'+string(a)};
title(string_title,'Interpreter','latex');
legend('\textbf{GL Scheme y1}','\textbf{GL Scheme y2}','\textbf{ode15s y1}','\textbf{ode15s y2}','Interpreter','latex')
grid on;
hold on;
%----------------------------------------------%
% Defining the Model %
%----------------------------------------------%
function f=ff(t,y,params)
%----------------------------------------------%
% Model Parameters %
%----------------------------------------------%
k1 = params(1);
kd = params(2);
%----------------------------------------------%
% Model Equations %
%----------------------------------------------%
f(1) = -k1*y(1);
f(2) = k1*y(1)-kd*y(2);
end
function fdot=ffdot(t,y,params)
%----------------------------------------------%
% Model Parameters %
%----------------------------------------------%
k1 = params(1);
k2 = params(2);
%----------------------------------------------%
% Model Jackobian %
%----------------------------------------------%
fdot(1,1)=-k1;
fdot(1,2)=0;
fdot(2,1)=k1;
fdot(2,2)=-k2;
end
%----------------------------------------------%
% ODE15s Function %
%----------------------------------------------%
function dydt = rate(~,y,k1,k2)
dydt(1) = -k1*y(1);
dydt(2)=k1*y(1)-k2*y(2);
dydt=[dydt(1),dydt(2)]';
end