Ordinary differential equations

Transcription

1 Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties of ODE. Solving ODE. Vector spces. Dynmic systems.

2 Modelling

3 Modelling Describing the behvior of system. Understnd it. Predict its future behviour. Be ble to mnipulte the system How to do this? Wht to tke into ccount nd wht not. Hving knowledge of the system Knowing physicl lws Using mth s working lnguge to describe the system Simultion A simultion is set of equtions tht describes system They do not need to hve solid conceptul bckground. The grde of trust of simultion is not known when you try to pply it to different conditions

4 Modelling By using some knowledge of physicl processes involved in the system, modeling strtegy tries to describe the system. Could be less exct thn simultion. It is dptble to different situtions gt F = my & y( t) = x0 + v0t + Why is model useful? It llows better understnding of system. It mkes system more predictble It llows engineering the system It could drive the experiments It llows the control of system.

5 Engineering cycle Ordinry differentil equtions

6 Introduction Why should we use differentil equtions? Whenever deterministic reltionship involving some continuously chnging quntities (modeled by functions) nd their rtes of chnge (expressed s derivtives) is known or postulted. Previous concepts Wht is function? A function is reltion between two sets of elements How is it represented in mth? f : x f ( x)

7 f ( t) = 3 f ( x) = x +

8 f ( x) = ( x ) f ( x) = x e

9 f ( x) = log x x < f ( x) = 0 x

10 x f ( x) = Asen( ) T x f ( x) = Asen( + ϕ) T K=0;τ=;n= K=;τ=vr;n= K=;τ=;n= K=;τ=;n=vrible

11 K=0;τ=;n= K=;τ=vr;n= K=;τ=;n= K=;τ=;n=vrible Functions of severl vribles It is function which depends on severl vribles: Sclr: z=f(x,y) Vectoril:y i =F i (x j ) R n m f ( xi ) R m

12 Liner function A liner function is function which fulfills: f ( x by) f ( x) bf ( y) Wht is derivtive? Hving two vribles, relted by function, the derivtive gives the vrition of one of them when the remining is chnged f ( t t) f ( t) df ( t) lim t 0 t

13 Tylor series development A serie is summtion of terms. The Tylor serie development of function f(x) is the pproximtion of this function by power serie: f ( x) f ( ) f ' ( t) ( t! ) f ''( t) ( t! )... n f '( t) ( t n! ) n ( t n ) The error of function is of n+ order if we develop the power serie up to the n power. f ( t) t e de t e t

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17 0 f ( t) k t n 3

18 Remrk α e i = cosα + i sinα The Tylor development mkes to conclude tht the complex exponentil is equivlent to the sum of sinus nd cosinus. Wht is differentil eqution? Wht is n eqution? Wht is differentil eqution? t: vrible x: vrible dependent λ i : prmeters of the function Hving vrible x, n eqution express mthemticl reltion between tht vrible nd some other which re known. Eqution which reltes function with its derivtes. dx = f ( t x, { λ λ...}), If the function f depends on more thn one vrible then the differentil eqution is clled prtil differentil eqution(pde)

19 In order to solve differentil eqution, we should trnsform the problem in problem in which we cn integrte function. Integrtion is the opossite to derivtion.if we substrct infinitesiml terms in derivtive we perform sum of infinitesiml terms in n integrl. When we solve n indefined integrl, there is costnt of integrtion tht we should fix using the conditions of the defined problem: f ( x) dx g( x) K The solution of n ODE is function x(t) which is defined but constnt nd is unique. Solution of differentil eqution The solution of differentil eqution is n eqution which llows to know the vlue of the dependent vrible s function of the independent ones given the vlue of the dependent vrible for defined vlue of the independent one. Initil conditions of the problem: the independent vrible is the time. Contour conditions of the problem: the independent vrible is nother one. X(t=0) Y(t=0) X(t) Y(t)

20 Simple exmples df df ( t) f ( t) f ( t) f ( t) e t t K K d f df f ( t) t ( f ( t) ) f ( t) K sin( wt K) Types of differentil equtions dx = kx st order dx = kx Liner d x dx + = kx nd order dx = kx Non liner dx = kx Autonomous d x dx + = kx ODE dx = kx + sen (t ) Non Autonomous y( x, t) y x t c ) (, = ky t x PDE

21 How to solve differentil eqution? In some cses there is possibility of solving nlyticlly the differentil eqution. In most of the cses this is not possible nd severl techniques hve been developed to solve the problem in n pproximted wy. Some of these techniques re included in the brnch of mthemtics known s numericl methods Liner diferentil equtions A liner differentil eqution is: n Y (t) Solutions n d f ( t) Liner n n d f ( t)... 0 f ( t) g( t) n Superposition principle α Y (t) + β Solution!! Y (t) Y (t)

22 Liner diferentil equtions The problem is reduced to solve the chrcteristic eqution n z n n z n... 0z The generl solution of the system will hve the form: 0 f ( t) i e z t i Non liner Rel system Lineriztion Liner Non liner equtions Simultion Liner equtions

23 Lineriztion method A function f(x) could be pproximted by: f ( x) f ( ) f' ( t) ( t! ) f''( t) ( t! )... n f '( t) ( t n! ) n ( t n ) First order pproximtion or lineriztion: f '( t) f ( x) = f ( ) + ( t ) + θ ( t! Bsed on this there is procedure to obtin liner eqution which hs similr behvior to the equtions we wnt to solve, round some specil points. ) Prcticl issues of the LP The working point should be close to n equilibrium point in such wy tht the first order derivtives will be 0. The liner model will be more ccurte ner the equilibrium point. The equilibrium point should be selected s close s possible to the working point.

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25 Numericl solution of ODE We hve come to know tht there re solutions just for few ODE. Themeningof derivtiveisthesubstrctionof very close terms. If we do not hve n exct solution of differentil eqution, we cn try to obtin n pproximted solution of it, substrcting by hnd these terms time nd gin. Very stupid devices cn perform very stupid opertion but with n incredible speed Numericl solution of ODE Lter on n exmple of very simple lgorithm to solve numericlly n ODE is given y '( t) = f ( t; y( t)) y ( t0) = y0 We substitute the derivtive term by its pproximted vlue: y( t + h) y( t) y '( t) y( t + h) y( t) + hf ( t, y( t)) h Selecting proper vlue for h, we hve time step with the vlue t 0 =t 0,t =t 0 +h,,t n =t n- +h y = y + hf ( t ; y n+ n n n )

26 Numericl solution of ODE This method is clled explicit Euler method nd it is one of the simplest methods to obtin the solution of ODE. The obtined solution is only n pproximtion to the rel solution, nd the goodness of the solution depends on the kind of the problem nd on the selection of the different model prmeters like h. Although the power of computing hs incresed lot in the lst yers, there re too mny problems which require mny computing time to be solved. Alwys you should tke cre when you solve problem numericlly!! Usefulness of differentil equtions When describing systems, it is usully very useful to know not only the vlue of the vrible but lso the evolution of it. d x dx AAM b x 0 F d x m

27 An introduction to vectors nd mtrix Wht is vector?. Any mthemtic set of things whose sum gives us nother thing of the set.. It is defined multipliction property between these things nd numbers. In more rigurous wy v 3 v v Bse of vector spce Minimum number of vectors which generte ll the spce Dimension of the spce The number of vectors of the bse

28 Mtrix Roughly, it is set of numbers Determinnt of squre mtrix k j i ijk A 3 ε = Some mtrix properties d c b d c b = + d h c g b f e d c b h g f e fd eb hc g fd eb fc e d c b h g f e

29 Linerity c b d x y x cx by dy Eigenvlues nd eigenvectors c b d x y x y 0 0 Clcultion of eigenvlues r Av r r = λv v ( A λi ) = 0 [ A λi ] 0 det =

30 Systems of liner diferentil equtions As in stndrd lgebric equtions, there cn be problems in which insted of ODE we hve system of differentil equtions. If the system is liner we cn pply ll the developed lgebric methods for vectoril spces. A system of n ODE of n order is equivlent to system of n+ ODE of n- order Exmple df df f f 3 4 f f df df 3 4 f f df ' df ' f ' f ' df ' df ' 0 0 f ' f '

31 Dynmicl system A system tht evolves with time is dynmicl system. Stte vrible: mgnitudes whose evolution in time we wnt to know. Evolution vrible: the stte vribles evolution s function of them. Prmeters: constnt mgnitudes of the system under certin conditions Evolution lws: the reltion between the stte vribles nd the evolution vribles. Exmple [ Y ] d = α Y + n [ U ] + K β[ ] γ

32 Dynmicl systems We re going to describe utonomous systems, which fulfill the following form: dfi Fi ( fi; i ) Autonomous mens not explicit dependence in time Any ODE of order higher thn cn be chnged in n equivlent system of equtions of first order Chnge of vribles d y dy k + k + k0 y = C x ( t ) = x ( t ) = y dy dx = x ( t) dx C k0x ( t) kx = k ( t)

33 Phse spce: It is representtion of the solution of ODE in which f is represented ginst f. It represents ll the possible sttes of the system. Stbility At this point fundmentl concept is the fix point or equilibrium point dfi 0 F ( f ; α ) = 0 = i i i Stbility If ll the solutions of dynmicl system which strt out ner n equilibrium point x e remin ner x e, then xe is Lypunov stble. Even more, if ll the solutions which strt out ner xe converge to xe, then xe is symptoticlly stble.

34 Stbility Liner systems dfi = Af i Obtin eigenvlues of A Re(λ) i >0: The system will be not stble Re(λ) i <0: The system will be stble Stbility Linerizble systems dfi Fi ( fi; i ) dfi = F( f ; α) + JΔf Δf = ie i i f f i ie F f J =... Fn f n F... fn Fn... f n Jcobin mtrix As we re in stble point F=0, the eqution could be written s dfi The criteri re the sme ones s for = JΔfi liner systems

35 Stbility All these procedures re quite simple but here they re explined for very prticulr cses. An study of the stbility of system is usully not so simple. Severl mthemticlly more ccurte definitions hve been done by mthemticins: Stbility in the sense of Lipunov Stbility in the sense of Lgrnge If we re working with nonliner systems, the nlysis turns into more complicted one. Sensibility Sensibility nlysis is relted not only to the dependence of the results of the model in the initil conditions but lso in the vlue of the prmeters. A sensitive nlysis of problem could give us informtion bout: - The influence in the system of the different prmeters. - The cre we should tke when determining the initil conditions of problem. - If model of system is useful to predict it or not. - The influence of externl perturbtions.

36 Non linerity nd Chos Why could Non liner systems be so complicted? The liner property f ( x by) f ( x) bf ( y) This property sttes tht if we introduce smll vrition in the system the system, in principle, do not chnge too much But if we introduce other kind of dependences like product or power lws, this property is not vlid. Consequently, smll chnges could produce very different behviours of the system. Non linerity nd Chos Chos does not men rndomness in system, chos mens determined system but very difficult to be predicted. dx/ = s ( y - x ) dy/ = r x - y - xz dz/ = xy - b z

37 Bibliogrphy Elementry Differentil equtions. CH Edwrds, DE Penney. Prentice hll. Introduction to dynmicl systems: Theory, models nd pplictions. DG Luenberger. Wiley. Clculus, Vol. : One-Vrible Clculus with n Introduction to Liner Algebr. T Apostol. Wiley