Announcements: Warm-up Exercise:

Transcription

1 Fri Apr Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise

2 Here's a relaively simple 2-ank problem o illusrae he ideas Exercise 1) Find differenial equaions for solue amouns, above, using inpu-oupu modeling. Assume solue concenraion is uniform in each ank. If 0 = b 1, 0 = b 2, wrie down he iniial value problem ha you expec would have a unique soluion. answer (in marix-vecor form) 4 2 = = b 1 b 2

3 Geomeric inerpreaion of firs order sysems of differenial equaions. The example on page 1 is a special case of he general iniial value problem for a firs order sysem of differenial equaions x = F, x x 0 = x 0 We will see how any single differenial equaion (of any order), or any sysem of differenial equaions (of any order) is equivalen o a larger firs order sysem of differenial equaions. And we will discuss how he naural iniial value problems correspond. Why we expec IVP's for firs order sysems of DE's o have unique soluions x From eiher a mulivariable calculus course, or from physics, recall he geomeric/physical inerpreaion of x as he angen/velociy vecor o he parameric curve of poins wih posiion vecor x, as varies. This picure should remind you of he discussion, bu ask quesions if his is new o you Analyically, he reason ha he vecor of derivaives x compued componen by componen is acually a limi of scaled secan vecors (and herefore a angen/velociy vecor) is x x 1 1 x lim 0 1 = lim provided each componen funcion is differeniable. Therefore, he reason you expec a unique soluion o he IVP for a firs order sysem is ha you know where you sar (x 0 = x 0 ), and you know your "velociy" vecor (depending on ime and curren locaion) you expec a unique soluion! (Plus, you could use somehing like a vecor version of Euler's mehod or he Runge-Kua mehod o approximae i! You jus conver he scalar quaniies in he code ino vecor quaniies. And his is wha numerical solvers do.) =,

4 Exercise 2) Reurn o he page 1 ank example = 4 2 = = 9 0 = 0 2a) Inerpre he parameric soluion curve, T o his IVP, as indicaed in he pplane screen sho below. ("pplane" is he siser program o "dfield", ha we were using in Chapers 1-2.) Noice how i follows he "velociy" vecor field (which is ime-independen in his example), and how he "paricle T moion" locaion, is acually he vecor of solue amouns in each ank, a ime. If your sysem involved en coupled anks raher han wo, hen his "paricle" is moving around in 10. 2b) Wha are he apparen limiing solue amouns in each ank? 2c) How could your smar-alec younger sibling have old you he answer o 2b wihou considering any differenial equaions or "velociy vecor fields" a all?

5 Firs order sysems of differenial equaions of he form x = A x are called linear homogeneous sysems of DE's. (Think of rewriing he sysem as x A x = 0 in analogy wih how we wroe linear scalar differenial equaions.) Then he inhomogeneous sysem of firs order DE's would be wrien as x A x = f or x = A x f Noice ha he operaor on vecor-valued funcions x defined by L x x A x is linear, i.e. L x y = L x L y L c x = c L x. SO! The space of soluions o he homogeneous firs order sysem of differenial equaions x A x = 0 is a subspace. AND he general soluion o he inhomogeneous sysem x A x = f will be of he form x = x P x H where x P is any single paricular soluion and x H is he general homogeneous soluion. Exercise 3) In he case ha A is a consan marix (i.e. enries don' depend on ), consider he homogeneous problem x = A x. Look for soluions of he form x = e v, where v is a consan vecor. Show ha x = e v solves he homogeneous DE sysem if and only if v is an eigenvecor of A, wih eigenvalue, i.e. A v = v. Hin In order for such an x o solve he DE i mus be rue ha and Se hese wo expressions equal. x = e v A x = A e v = e A v

6 Exercise 4) Use he idea of Exercise 3 o solve he iniial value problem of Exercise 2!! Compare your soluion x o he parameric curve drawn by pplane, ha we looked a a couple of pages back. Exercise 5) Lessons learned from ank example Wha condiion on he marix A n uniquely solve every iniial value problem x = A x x 0 = x 0 n n will allow you o using he mehod in Exercise 3-4? Hin Chaper 6. (If ha condiion fails here are oher ways o find he unique soluions.)