How to solve system of first order ODE's by forming matrix

Question

Bathala Teja 2021-8-30

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https://ww2.mathworks.cn/matlabcentral/answers/1443124-how-to-solve-system-of-first-order-ode-s-by-forming-matrix

评论： Bathala Teja 2021-8-31

采纳的回答： Wan Ji

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Q:

dy1/dt = 2*y1+5*y2+y3

dy2/dt = 7*y1+9*y2

dy3/dt = 4*y1+y2+3*y3

y1(0) = 0, y2(0)= 0, y3(0) = 0;

i want to plot y1(t), y2(t) and y3(t) waveforms over time 0 to 10sec.

i tried the below code

syms y1(t) y2(t) y3(t)
Y = [y1(t);
    y2(t);
    y3(t)];
A = [2 5 1;
    7 9 0;
    4 1 3];
diff(Y, t) = A.*Y;
tspan = [0 10];
Y(0)=zeros(1.,3);
[t, Y] = dsolve(diff(Y, t), tspan, Y(0));
Y1 = Y(:, 1);
plot(t, Y1);
Y2 = Y(:, 1);
plot(t, Y2);
Y3 = Y(:, 1);
plot(t, Y3);

But iam getting an error like this

>> Untitled5

Error using sym/subsindex (line 864)

Invalid indexing or function definition. Indexing must follow MATLAB indexing. Function arguments must be

symbolic variables, and function body must be sym expression.

Error in sym/privsubsasgn (line 1151)

L_tilde2 = builtin('subsasgn',L_tilde,struct('type','()','subs',{varargin}),R_tilde);

Error in sym/subsasgn (line 972)

C = privsubsasgn(L,R,inds{:});

Error in Untitled5 (line 9)

diff(Y, t) = A.*Y;

Anyone please tell where i went wrong. I want to solve with matrices(since my main code have 20 ODE's to solve) only, please suggest me in this way only.

Thanks in advance.

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Answer 1

Wan Ji 2021-8-30

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https://ww2.mathworks.cn/matlabcentral/answers/1443124-how-to-solve-system-of-first-order-ode-s-by-forming-matrix#answer_776859

编辑：Wan Ji 2021-8-30

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Do not use syms, because without syms, the code runs faster

A = [2 5 1;
    7 9 0;
    4 1 3];
odefun = @(t,Y)A*Y;
tspan = [0 10];
Y0=zeros(3,1);
[t, Y] = ode45(odefun, tspan, Y0);
Y1 = Y(:, 1);
hold on
plot(t, Y1);
Y2 = Y(:, 1);
plot(t, Y2);
Y3 = Y(:, 1);
plot(t, Y3);
legend('Y1','Y2','Y3')

As the initial conditions are all zero, the final results become zero for Y1 Y2 and Y3

IF you want to use syms

syms y1(t) y2(t) y3(t)
A = [2,5,1;7,9,0;4,1,3];
eq = diff([y1;y2;y3])==A*[y1;y2;y3];
conds = [y1(0)==0; y2(0)==0; y3(0)==0];
[y1,y2,y3] = dsolve(eq,conds)
tspan = 0:0.1:10;
y_1 = eval(subs(y1,t,tspan));
y_2 = eval(subs(y2,t,tspan));
y_3 = eval(subs(y3,t,tspan));
figure(1)
clf
hold on
plot(tspan, y_1);
plot(tspan, y_2);
plot(tspan, y_3);
legend('Y1','Y2','Y3')

Then the result becomes

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Wan Ji 2021-8-30

编辑：Wan Ji 2021-8-30

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Hi,

As to pose a question. @Bathala Teja, you should show more details of your variables rather than just simplify it to something a constant matrix. The time of those who answer these questions is limited, do not make fun of them.

I need to say I really enjoy helping others, but I just like to anwer those questions that I see, the hidden is in your heart not in mine. So show me your true colors, my friend. As for the ODE, I have suggested you not to mix symbolic solutions of ode with numerical solutions. Here I just write down the symbolic solution to your question(in fact, answer 2 to this question).

As you use

Bi = sin(teta).*([1 2 3;
    4 5 6;
    7 8 9])

Do not kid me, this matrix is singular!!!!!! and has no inverse.

And I see you subsitute teta (theta exactly) with Y(3), why not directly use y3?

A = subs(Ai, teta, Y(3))
B = subs(Bi, teta, Y(3))

You drive crazy.

I just write what I believe

syms y1(t) y2(t) y3(t)
A = [2,5,1;7,9,0;4,1,3]*cos(y3);
B = sin(y3).*([1 0 0;0 5 0;0 0 9]); % since it's singular, I rewrite it here
V = [1;1;1];
Y = [y1;y2;y3];
eq = diff(Y)==inv(B)*(V-A*Y)
conds = [y1(0)==0; y2(0)==0; y3(0)==0];
[y1,y2,y3] = dsolve(eq,conds)
tspan = 0:0.1:10;
y_1 = eval(subs(y1,t,tspan));
y_2 = eval(subs(y2,t,tspan));
y_3 = eval(subs(y3,t,tspan));
figure(1)
clf
hold on
plot(tspan, y_1);
plot(tspan, y_2);
plot(tspan, y_3);
legend('Y1','Y2','Y3')

Actually, this does not have a solution, so I dont know what you are thinking about now, maybe homework, maybe something formidable.

Wan Ji 2021-8-30

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It doesnot matter how many odes there are, one thing that you should notice is that I always make the solution in two ways. One is symbolic, the other is numeric. Do not mix them together. Once again I remind you of that.

Though you have 27 odes, it is also very easy to get a numeric solution. I just give something a framework to help you understand how to solve odes

function dYdt = odefun(t, Y)
% input Y is a 27x1 matrix
% input t refers to time
% dYdt refers to the derivative of Y function with respect to time
% dYdt is a 27x1 matrix
B = []; % B is a 27x27 matrix, maybe include some Y items
% for example B(1,1) = Y(3); B(1,2) = Y(27)*Y(16);
V = []; % V is a 27x1 vector, maybe includes some Y items
A = []; % the same size as B, maybe include some Y items
dYdt = B\(V-A*Y); % that's enough
end

Now following is the main function command

tspan = [0 10];
Y0=zeros(27,1); % something a 27x1 matrix
[t, Y] = ode45(@(t,Y)odefun(t,Y), tspan, Y0);
% plot the results
Y1 = Y(:, 1);
hold on
plot(t, Y1);
Y2 = Y(:, 1);
plot(t, Y2);
Y3 = Y(:, 1);
plot(t, Y3);
legend('Y1','Y2','Y3')

Bathala Teja 2021-8-31

Happy to see your suggestion, but i already tried this way.

In the ODE function code i have to call (Since i cant include this function code in my original code where i get A and B as result) A and B matrices from my original code which are variable form (EX: A = [cos(theta) sin(2*theta+pi/3) ........]>>like this 27*27).

Problem is, i already used sym for theta in my main code. It is giving error whenever i call these variable matrices into function of ODE's.

I want to remind you onething, in my main code i already used sym for theta and formed A and B matrices interms of theta. Now i want to solve ODE's by including A and B in the eq. But the main problem comes here is putting a variable matrix(sym matrix) into ODE eq. But i can eliminate this theta variable by substituting last ODE solution(means Y27). I think its not possible to solve a ODE like this. Now iam trying to substitute t in place of theta, lets see how this works.

I think i have to proceed with syms, i cant do in numerical way. Iam totally tired of trying various ways.

Once again thanks for your reply.

Wan Ji 2021-8-31

replace theta with y(...), I hope you get the result soon

Bathala Teja 2021-8-31

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if you dont mind can you tell me how to form a variable array without using symbolic math toolbox.

Nr = 20
nr = rand(1, Nr)
p =2;
Nr = 20
for j = 1:(2*p)
    for n = (((j-1)*Nr/(2*p))+2):(j*(Nr/(2*p)))
        Aord = 1/(2*pi);
        nr(1, n) = nr(1, n)+cos((phi-theta-((n-j-1)*2)-((j-1)*2)))+Aord;
    end
end

Here nr is a array of Nr elements consists of theta and phi in it. How to form without using sym.

Thanks in advance.

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Answer 2

Bathala Teja 2021-8-30

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Firstly iam very happy for your suggestion, thank you very much.

First solution is fine. But i have some complexity in my main code.

My main code contains contains A and B matrices(both 3*3) and is interms of some variable teta.

I want to substitute y3 in place of teta and proceed solving ODE's.

My ODE is inv(B).*(V-(A*Y)), where V = column matrix of order 3.

i tried like this

syms teta
Ai = cos(teta).*([2 5 1;
    7 9 0;
    4 1 3])
Bi = sin(teta).*([1 2 3;
    4 5 6;
    7 8 9])
V = ones(3, 1)
odefun = @(t,Y) inv(B).*(V-(A*Y));
A = subs(Ai, teta, Y(3))
B = subs(Bi, teta, Y(3))
tspan = [0 10];
Y0=zeros(3,1);
[t, Y] = ode45(odefun, tspan, Y0);
Y1 = Y(:, 1);
hold on
plot(t, Y1);
Y2 = Y(:, 1);
plot(t, Y2);
Y3 = Y(:, 1);
plot(t, Y3);
legend('Y1','Y2','Y3')