Getting a system of equations and the using ODE45
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Hi, I have a coefficient matrix -
Xi=
1.0e+03 *
0.3002 0.3832 0.0577
0 0 0
0 0 0
0 0 0
-0.1425 -0.1576 -0.0162
0 0 0
0 0 0
-0.0352 -0.0601 -0.0140
0 0 0
0 0 0
-0.0941 -0.1304 -0.0245
0.1394 0.1897 0.0296
0.7197 0.8981 0.1662
0.0058 0.0024 0.0061
-1.0007 -1.2808 -0.2519
0 0 0
-0.0211 -0.0250 -0.0060
0.3760 0.4933 0.0964
-0.0242 -0.0253 0.0097
0 0 0
%
To get the system of ODEs I first defined the variables, and then multiplied it by transpose(Xi) to get the system-
syms x y z xx xy xz yy yz zz xxx xxy xxz xyy xyz xzz yyy yyz yzz zzz
var = [1;x;y;z;xx;xy;xz;yy;yz;zz;xxx;xxy;xxz;xyy;xyz;xzz;yyy;yyz;yzz;zzz]
A = transpose(Xi)
Xdot = A*var
where Xdot is the column containing the derivatives, ie. Xdot = [x' ; y' ; z']. The code above gives me the output:
Xdot =
(4905784495475543*xxy)/35184372088832 - (6622198076817333*xxx)/70368744177664 - (1253707074885461*xx)/8796093022208 + (6330728071069091*xxz)/8796093022208 + (6526091537672905*xyy)/1125899906842624 - (550152702477171*xyz)/549755813888 - (4953639297776579*yy)/140737488355328 - (743004146196139*yyy)/35184372088832 + (6615295418414269*yyz)/17592186044416 - (3404894971584935*yzz)/140737488355328 + 2640157789935161/8796093022208
(6675905246377611*xxy)/35184372088832 - (4586552884284477*xxx)/35184372088832 - (2773298922331439*xx)/17592186044416 + (987449040685507*xxz)/1099511627776 + (2666532593051851*xyy)/1125899906842624 - (5632811008410909*xyz)/4398046511104 - (8464188990339351*yy)/140737488355328 - (7048704279715849*yyy)/281474976710656 + (4339015521236507*yyz)/8796093022208 - (7129115399973291*yzz)/281474976710656 + 3370686532933275/8796093022208
(2082177599331991*xxy)/70368744177664 - (863749208859729*xxx)/35184372088832 - (4564685191010399*xx)/281474976710656 + (5847277141854391*xxz)/35184372088832 + (6877937169110265*xyy)/1125899906842624 - (8862390418139439*xyz)/35184372088832 - (491773253308733*yy)/35184372088832 - (3353395911004425*yyy)/562949953421312 + (6782944326528325*yyz)/70368744177664 + (5481194070111587*yzz)/562949953421312 + 2029864570192639/35184372088832
My question is
a) Is there a better way to define the system of ODEs?
b) How do I solve this system using ODE45? I do not want to manually input the equations, as my coefficient matrix will vary if I have a different dataset, so I cannot manually input equations everytime.
c) If I have 5 variables, say, x1,x2,x3,x4,x5 can I still use ODE45 to solve a system of ODEs containing 5 equations?
2 Comments
Torsten
on 31 Jul 2022
As far as I can see, you have 3 equations for 19 unknowns. What are the missing 16 equations ?
Accepted Answer
Torsten
on 31 Jul 2022
Edited: Torsten
on 31 Jul 2022
fun = @(x) Xi.'*[1;x(1);x(2);x(3);x(1)^2;x(1)*x(2);x(1)*x(3);x(2)^2;x(2)*x(3);x(3)^2;x(1)^3;x(1)^2*x(2);x(1)^2*x(3);x(1)*x(2)^2;x(1)*x(2)*x(3);x(1)*x(3)^2;x(2)^3;x(2)^2*x(3);x(2)*x(3)^2;x(3)^3];
tspan = [0 1]; % time interval of integration
x0 = [1; 1; 1]; % Initial conditions
[T,X] = ode45(fun,tspan,x0) % solver call
5 Comments
More Answers (1)
Walter Roberson
on 31 Jul 2022
I recommend that you read the first example for odeFunction() to see the flow to turn an array of equations into something that can be evaluated numerically by ode45()
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