Bucknell University Using ODE45 MATLAB Help

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1 Bucknell University Using ODE45 MATLAB Help MATLAB's standard slver fr rdinary differential equatins (ODEs) is the functin de45. This functin implements a Runge-Kutta methd with a variable time step fr efficient cmputatin. de45 id designed t handle the fllwing general prblem dy = f(, t y) y( t ) = y [] dt where t is the independent variable (time, psitin, vlume) and y is a vectr f dependent variables (temperature, psitin, cncentratins) t be fund. The mathematical prblem is specified when the vectr f functins n the right-hand side f Eq. [], f(, t y), is set and the initial cnditins, y = y at time t, are specified. The ntes here apply t versins f MATLAB abve 5.0 and cver the basics f using the functin de45. Fr mre infrmatin n this and ther ODE slvers in MATLAB, see the n-line help. Cntents: Syntax fr de45... Integrating a single, first-rder equatin... 3 Getting the slutin at particular values f the independent variable... 4 Integrating a set f cupled first-rder equatins... 4 Integrating a secnd-rder initial-value prblem (IVP)... 7 Integrating an Nth-rder initial-value prblem... 8 Changing mdel parameters... 9 Integrating a secnd-rder bundary-value prblem (BVP)... Setting ptins in de45... Ging beynd de ENGR0 Using ODE45

2 Syntax fr de45 de45 may be invked frm the cmmand line via [t,y] = de45('fname', tspan, y0, pts) where fname name f the functin Mfile used t evaluate the right-hand-side functin in Eq. [] at a given value f the independent variable and dependent variable(s) (string). The functin definitin line usually has the frm functin dydt = fname(t,y) The utput variable (dydt) must be a vectr with the same size as y. Nte that the independent variable (t here) must be included in the input argument list even if it des nt explicitly appear in the expressins used t generate dydt. tspan y0 pts t y -element vectr defining the range f integratin ([t tf]) thugh variatins are pssible. vectr f initial cnditins fr the dependent variable. There shuld be as many initial cnditins as there are dependent variables. a MATLAB structure variable that allws yu t cntrl the details f cmputatin (if yu want t). This argument is ptinal and, if nt prvided, de45 will use default values (see the examples belw). Value f the independent variable at which the slutin array (y) is calculated. Nte that by default this will nt be a unifrmly distributed set f values. Values f the slutin t the prblem (array). Each clumn f y is a different dependent variable. The size f the array is length(t)-by-length(y0) Specific examples f using de45 nw fllw. Mfiles fr these examples are in the bdy f this dcument and shuld als be available in the flder that cntains this dcument. ENGR0 Using ODE45

3 Integrating a single, first-rder equatin The height f fluid in a tank (h(t)) whse utlet flw is dependent n the pressure head (height f fluid) inside the tank and whse inlet flw is a functin f time may be mdeled via the equatin dh dt = α() t β h h( 0 ) = h [] Find the slutin, h(t), fr 0< t < 30 if the fllwing values fr the parameters are given. Input flw: α( t) = sin( t) β = h = Step : Identify f(, t y) and write a MATLAB functin Mfile t evaluate it. In this case, we have time as the independent variable and the tank height as the (single) dependent variable. Thus, we have f(, t y) f(, t h) = α() t β h [3] Frm the given infrmatin fr this prblem, the required Mfile, named tankfill.m, is functin dhdt = tankfill(t,h) % RHS functin fr tank-fill prblem A = 0 + 4*sin(t); H = *sqrt(h); dhdt = A - H; % alpha(t) % beta*sqrt(h) % ef - tankfill.m Step : Use de45 t slve the prblem The initial cnditin has the height at fr t = 0 and we want t integrate until t = 30. The fllwing set f cmmands shw explicitly hw the slutin is put tgether. >> tspan = [0 30]; (integratin range) >> h0 = ; (initial cnditin, h(0)) >> [t,h] = de45('tankfill',tspan,h0); (slve the prblem) Step 3: Lk at the slutin The slutin can be viewed via the plt cmmand as in >> plt(t,h) The "curve" is a little chppy thugh it is accurate t the default relative tlerance (0.00). Nte that the places where the slutin is given are nt unifrmly spread ut. See the next sectin fr imprving appearances. ENGR0 Using ODE45 3

4 Getting the slutin at particular values f the independent variable de45 uses a variable-step-length algrithm t find the slutin fr a given ODE. Thus, de45 varies the size f the step f the independent variable in rder t meet the accuracy yu specify at any particular pint alng the slutin. If de45 can take "big" steps and still meet this accuracy, it will d s and will therefre mve quickly thrugh regins where the slutin des nt "change" greatly. In regins where the slutin changes mre rapidly, de45 will take "smaller" steps. While this strategy is gd frm an efficiency r speed pint f view, it means that the slutin des nt appear at a fixed set f values fr the independent variable (as a fixed-step methd wuld) and smetimes the slutin curves lk a little ragged. The simplest way t imprve n the density f slutin pints is t mdify the input tspan frm a -element vectr t an N-element vectr via smething like >> tspan = linspace(t,tf,500) ; and use this new versin in the input list t de45. Smther curves can als be generated by interplatin (spline interplatin usually wrks nicely). Fr example, if yu wanted a smther result frm the slutin fr the tank-fill prblem, yu might d the fllwing >> ti = linspace(tspan(),tspan(),300); (300 pints - yu culd use mre) >> hi = spline(t,h,ti); >> plt(t,h,,ti,hi); The interplated curve smthes ut the rugh edges caused by simply cnnecting the data pints (which is what plt des) and s makes the graph mre appealing, in a visual sense. Integrating a set f cupled first-rder equatins Chemical-kinetics prblems ften lead t sets f cupled, first-rder ODEs. Fr example, cnsider the reactin netwrk A B C [4] Assuming a first-rder reactin-rate expressin fr each transfrmatin, material balances fr each species lead t the fllwing set f ODEs: da = ka + kb dt db = ka kb kb 3 dt dc = kb 3 dt [5] with the initial cnditins, A( 0) = A, B( 0) = B, C( 0) = C. Since the equatins are cupled, yu cannt slve each ne separately and s must slve them simultaneusly. The system in Eq. [5] can be put in the standard frm fr de45 (Eq. []) by defining the vectrs y, y and f as ENGR0 Using ODE45 4

5 A A + ky ky y= B y( 0) = y = B f( t, y) = ky ( k+ k3) y C C ky 3 [6] Slving the system represented by Eq. [6] is a simple extensin f what was dne fr slving a single equatin. We'll demnstrate the slutin fr the fllwing situatin k = 5 k = k = A = B = C = 0 3 Step : Write a functin Mfile t evaluate the right-hand-side expressin The primary difference here, cmpared t the single-equatin case, is that the input variable y will be a vectr. The first element f y represents the cncentratin f species A at a time t, and the secnd and third elements representing the cncentratins f species B and C, respectively, at the same time, t. This rdering f variables is defined by Eq. [6]. There is n "right" rder t the variables but whatever rder yu d chse, use it cnsistently. We'll call the Mfile react.m. It lks like this: functin dydt = react(t,y) % Slve the kinetics example dydt = zers(size(y)); % Parameters - reactin-rate cnstants k = 5; k = ; k3 = ; A = y(); B = y(); C = y(3); We'll be explicit abut it here thugh yu can d the calculatins directly with the y-values. % Evaluate the RHS expressin dydt() = -k*a + k*b; dydt() = k*a - (k+k3)*b; dydt(3) = k3*b; % ef - react.m Nte that the input arguments must be t and y (in that rder) even thugh t is nt explicitly used in the functin. Step : Use de45 t slve the prblem N time interval is given s we'll pick ne (0 t 4) and see what the slutin lks like. If a lnger r shrter interval is needed, we can simply re-execute the functin with a new value fr the ending time. Fllwing the utline fr the single-equatin prblem, the call t de45 is, >> [t,y] = de45('react',[0 4],[ 0 0]); Nte that the initial cnditin is prvided directly in the call t de45. Yu culd als have defined a variable y0 prir t the call t de45 and used that variable as an input. ENGR0 Using ODE45 5

6 Take a mment t lk at the utputs. The number f pints at which the slutin is knwn is >> length(t) Als cnsider the shape f the utput variable y: >> size(y) Is the result as stated abve (i.e., is it length(t)-by-length(y0))? Step 3: Lk at the slutin If yu want t see the time-curse f all species, use the cmmand >> plt(t,y) The blue line will be the first clumn f y (species A). The green and red lines will be the secnd and third clumns f y (species B and C, respectively). If yu wanted t lk at nly ne species (fr example, species B), yu wuld give the cmmand >> plt(t,y(:,)) since the secnd clumn f y hlds the infrmatin n species B. ENGR0 Using ODE45 6

7 Integrating a secnd-rder initial-value prblem (IVP) A mass-spring-dashpt system can be mdeled via the fllwing secnd-rder ODE y + cy + ω y= g() t y( 0) = y, v( 0) = y ( 0 ) = v [7] In this mdel, c represents a retarding frce prprtinal t the velcity f the mass, ω is the natural frequency f the system and g(t) is the frcing (r input) functin. The initial cnditins are the initial psitin (y ) and initial velcity (v ). de45 is set up t handle nly first-rder equatins and s a methd is needed t cnvert this secndrder equatin int ne (r mre) first-rder equatins which are equivalent. The cnversin is accmplished thrugh a technique called "reductin f rder". We'll illustrate the slutin fr the particular set f cnditins c= 5 ω = y( 0) = v( 0) = 0 g() t = sin( t) Step: Define the cmpnents f a vectr p = [ p p ] T as fllws: p = y p = y [8] Step : Frm the first derivatives f each f the cmpnents f p Using the given differential equatin, we can write a system f first-rder equatins as p = y = p p = y = g() t cy ω y = gt () cp ω p [9] In writing the expressin fr the secnd cmpnent, we've used the gverning ODE (Eq. [7]). Step 3: Cast the prblem in the frmat needed t use de45. dp d p (, p) p = = f(, p) = () = p f t dt dt p gt cp p f (, t p) = t ω [0] Step 4: Cllect the initial cnditins int a single vectr ( 0) ( 0) p( 0) = p = p ( 0) = y ( 0) = y p y v [] Step 5: Apply de45 t slve the system f equatins The Mfile fr the RHS functin fr this prblem will be called spring.m. Here it is: functin pdt = spring(t,p) % Spring example prblem ENGR0 Using ODE45 7

8 % Parameters - damping cefficient and natural frequency c = 5; w = ; g = sin(t); % frcing functin pdt = zers(size(p)); pdt() = p(); pdt() = g - c*p() - (w^)*p(); % ef - spring.m The call t de45 is, fr a slutin interval f 0 t 0, >> p0 = [ 0]; (initial psitin and velcity) >> [t,p] = de45('spring',[0 0],p0); Step 6: Lk at the results If yu wanted t lk at nly the displacement, yu'd want t lk at the first clumn f p (see the definitin f p in the first step, Eq. [8]). Hence, yu wuld give the cmmand >> plt(t,p(:,)) An interesting plt fr these srts f prblems is the phase-plane plt, a plt f the velcity f the mass versus its psitin. This plt is easily created frm yur slutin via >> plt(p(:,),p(:,)) Phase-plane plts are useful in analyzing general features f dynamic systems. Integrating an Nth-rder initial-value prblem T use de45 t integrate an Nth-rder ODE, yu simply cntinue the prcess utlined in the sectin n integrating a nd-rder ODE. The first element f the vectr p is set t the dependent variable and then subsequent elements are defined as the derivatives f the dependent variable up t ne less than the rder f the equatin. Finally, the initial cnditins are cllected int ne vectr t give the frmat presented in Eq. []. Fr example, the 4th-rder equatin wuld generate the first-rder system a d 4 y b dy 3 c dy d dy + ey = 0 [] 4 3 dx dx dx dx p p d p p3 = dt p3 p4 p ( bp + cp + dp + ep )/ a [3] which, alng with an apprpriate set f initial cnditins wuld cmplete the set-up fr de45. ENGR0 Using ODE45 8

9 Changing mdel parameters In all the examples given abve, the parameter values were given specific variables in the Mfile used t evaluate the RHS functin (the mdel f the system). This is fine fr ne-sht cases and in instances where yu dn't anticipate a desire t change the parameters. Hwever, this situatin is nt fine where yu want t be able t change the parameters (e.g., change the damping cefficient in rder t see the result in a phase-plane plt). One apprach t changing parameters is t simply edit the file every time yu want t make a change. While having the advantage f simplicity, this apprach suffers frm inflexibility, especially as the number f parameters and as the frequency f the changes increase. T get ther parameters int the functin, yu need t use an expanded versin f the syntax fr de45, ne that allws ther infrmatin t be prvided t the derivative functin when de45 uses that functin. This is mst easily seen by an example (and by reading the n-line help n de45). Step : Write the Mfile fr the RHS functin s that it allws mre input variables. The parameter list starts after the dependent variable and after a required input called flag. Fr this example, we will re-write spring.m s that c and w are given their values via the functin definitin line. The altered Mfile is functin pdt = spring(t,p,flag,c,w) % Spring example prblem % Parameters c is the damping cefficient and % w is the natural frequency pdt = zers(size(p)); g = sin(t); % frcing functin pdt() = p(); pdt() = g - c*p() - (w^)*p(); % ef spring.m Step : Write a driver script that implements yur lgic and allws yu t set values f the parameters fr the prblem. The script created here will d the fllwing. Implement a lp that a. asks fr values f the parameters b. slves the ODE c. plts the phase-plane view f the slutin. Exits if n inputs are given Certainly mre sphisticated scripts are pssible but this has the essence f the idea. The script is called dspring.m and it is: %DOSPRING Interactive pltting f the phase-plane while % infinite lp C_SPRING = input('damping cefficient [c]: '); ENGR0 Using ODE45 9

10 end if isempty(c_spring) % hw t get ut break end W_SPRING = input('natural frequency [w]: '); [t,p] = de45('spring',[0 0],[ 0],[],C_SPRING,W_SPRING); plt(p(:,),p(:,)) title('phase-plane plt f results') xlabel('psitin') ylabel('velcity') % es - dspring.m Nte the additins t the call t de45. First, a placehlder fr the ptins input is inserted (an empty array) s that default ptins are used. Then, the parameters fr the mdel are prvided in the rder that they appear in the definitin line f the RHS functin. Try the script ut and mdify it (e.g., yu culd add the frequency and/r amplitude f the frcing functin as smething t be changed). ENGR0 Using ODE45 0

11 Integrating a secnd-rder bundary-value prblem (BVP) de45 was written t slve initial-value prblems (IVPs). Hence it cannt be (directly) used t slve, fr example, the fllwing prblem derived frm a mdel f heat-transfer in a rd: d y dy y = 0 y(0) = = 0 [4] dx dx x= since the value f the derivative at x = 0 is nt specified (it is knwn at x =, thugh). Equatin [4] is a bundary-value prblem (BVP) and is cmmn in mdels based n transprt phenmena (heat transfer, mass transfer and fluid mechanics). All is nt lst because ne way t slve a BVP is t pretend it is an IVP. T make up fr the lack f knwledge f the derivative at the initial pint, yu can guess a value, d the integratin and then check yurself by seeing hw clse yu are t meeting the cnditins at the ther end f the interval. When yu have guessed the right starting values, yu have the slutin t the prblem. This apprach is smetimes called the "shting methd" by analgy t the ballistics prblem f landing an artillery shell n a target by specifying nly it's set-up (pwder charge and angle f the barrel). Step : Set up the prblem s that de45 can slve it Using the apprach f turning a secnd rder equatin int a pair f cupled first-rder equatins, we have d p p dx p = p 0 = p( ) [5] v where v has been used t represent the (unknwn) value f the derivative at x = 0. The Mfile used t evaluate the RHS is as fllws functin dpdx = htrd(x,p) % Ht-rd prblem illustrating the shting methd dpdx = zers(size(p)); dpdx() = p(); dpdx() = p(); % ef - htrd.m Step : Guess a value f the initial slpe and integrate t x = The prblem will be iterative s it's nt likely that the first guess will be right. Frm the physics f the prblem, the end f the rd (at x = ) will be clder than the place we are starting frm (x = 0) and s we'll guess a negative value fr the initial slpe. >> v = -; >> [x,p] = de45('htrd',[0 ],[ v]); The value f the derivative at x = is the last value in the secnd clumn f p (why?). Thus, we can check the accuracy f the first guess via ENGR0 Using ODE45

12 >> p(length(x),) which I fund t be That s t lw (it shuld be zer). Step 3: Iterate until the bundary cnditin at x = is met Yu can use brute frce here if yu have nly ne prblem r yu culd finesse it by hking the whle thing up t fzer and have fzer d the guessing. Here are my brute-frce results: Value f v Slpe at x = The trend is bvius and s the initial slpe is arund (the exact value is -tanh() = ). Using fzer wuld be a gd alternative if this prblem were t be slved many times ver. Step 4: Lk at the results Even thugh we are guessing the initial slpe t slve the prblem, it is the slutin, y(x), that we are really interested in. This slutin is in the first clumn f p and may be viewed via >> plt(x,p(:,)) Setting ptins in de45 The input pts is a MATLAB structure variable that cn be used t cntrl the perfrmance f the varius ODE-slvers in MATLAB. The mst cmmn ptin that yu ll likely want t alter is the accuracy t which slutins are cmputed. T make this prcess easy, a pair f functins are available deset fr creating and changing ptins and deget fr displaying infrmatin n ptins. T see what the current settings are, try the cmmand >> deset Default values fr any setting are dented by the braces, {}. MATLAB uses tw accuracy measures fr slving ODEs the relative tlerance (RelTl in pts) and the abslute tlerance (AbsTl in pts). Each step in the integratin is taken s that it satisfies the cnditin Errr at step j max( RelTl y, AbsTl ) k where the subscript k ranges ver all the cmpnents f the slutin vectr at time step j. T alter the default settings, use cmmands such as >> ldopts = deset; >> newopts = deset(ldopts, RelTl,e-6) Infrmatin n the settings fr the ther ptins is available in the n-line help. jk k ENGR0 Using ODE45

13 Ging beynd de45 The slver de45 is nt the be-all and end-all f ODE-slvers. While de45 shuld be yur first chice fr integratin, there are prblems that the functin perfrms prly n r even fails n. In such cases, there are fallback slvers that are available. All these slvers use the same syntax as de45 (see page ) but have ptins fr handling mre difficult r sphisticated prblems. Here are sme suggestins fr handling nn-standard ODE prblems: If accuracy yu desire is nt btainable via de45, try the functin de3. This slver uses a variable rder methd that may be able t imprve ver what de45 des. If de45 is taking t lng t cmpute a slutin, yur prblem may be stiff (i.e., it invlves a system with a wide range f time cnstants). Try the functin de5s. If yur system f equatins has the frm d M y = f(, t y) dt where M is a (typically nn-singular) matrix, try the functin de5s. Yu ll find mre infrmatin n these functin in the n-line help and dcumentatin. Fr example, try the n-line functin reference (available thrugh the cmmand helpdesk) n any f the slvers nted abve. ENGR0 Using ODE45 3