Differential equations. Differential equations

Transcription

1 Differential equations A differential equation (DE) describes ow a quantity canges (as a function of time, position, ) d - A ball dropped from a building: t gt () dt d S qx - Uniformly loaded beam: wx () dx wx x L EI EI - Scrödinger equation: r + Vrr Er (3) m - Heat conduction: c P Tr t ktr t t (4) - DEs can be classified in various ways: Ordinary DEs (ODEs) Partial DEs (PDEs) Only one independent variable Many independent variables Eqs () and () Eqs (3) and (4) Initial value problems(ivp) Boundary value problems (BVP) Function value(s) known at certain values of te variable(s) Function value(s) known on te domain borders Want function value(s) at arbitrary variable value(s) Want function values in oter point(s) in te domain Eqs () and (4) Eqs (), (3) and (4) In D: ft 0 y 0, compute ft In D: fx y, fx y, compute fx (two-point BVP) Scientific computing III 03: 3 Differential equations Differential equations Te difference between initial value problems and boundary value problems can be illustrated as below (according to NR, fig 90) boundary conditions boundary conditions initial values boundary values - In general boundary value problems are arder to solve tan initial value problems: BVPs need iteration IVP you only need to integrate te equations in time to reac te desired point - In tis capter we deal wit ordinary differential equations bot IVPs and BVPs - Probably tere will be no time partial differential equations Scientific computing III 03: 3 Differential equations

2 Te simplest ODE (IVP) is dy ft dt and its solution (starting from time t 0 ) yt c+ fd (Integration of te differential equation) t 0 - Te solution yt is unique up to te constant c ie we get a family of solutions - Te initial condition is needed to fix te value of te constant and to get a unique solution: yt 0 y 0 t Te general form of an IVP is dy fty dt Scientific computing III 03: 3 Differential equations 3 Solution metods for a single ODE are easily generalized to a group if ODEs (Use of vector notation makes it easier) y t f t y A y t y t, fty f t y, y 0 y n t f n t y A A n dy y' dt y' fty, yt 0 y 0 ODEs aving also iger derivatives can be transformed to groups of ODEs so tat metods presented ere can be applied to tese iger order ODEs: y n ft y y' y n yt 0 A y' t 0 A y n t 0 A n y ' y y ' y 3 y n ' y t 0 A y t 0 A y n t 0 A n ft y y y n Scientific computing III 03: 3 Differential equations 4

3 Numerical solution calls for discretization of te problem - Instead of computing yt at any t we obtain values t i in discrete points (We talk about time ere, but t can - of course - be any variable to be solved) - Te interval between points (in IVPs te time step) may vary as a function of time (variable time step) Tere several error sources in te integration of ODEs: - Discretization error: a too large time step decrease te time step - Instability: small errors introduced in te beginning grow exponentially cange te algoritm - Stiff ODEs: penomena wit different time scales step size selected according to te fastest processes Scientific computing III 03: 3 Differential equations 5 Someting general can be said about te existence of te solution of te IVP y' ft y - A function fty is said to satisfy a Lipscitz condition in te variable y in te set D R if L 0 so tat fty ft y Ly y wenever t y t y D - L is called te Lipscitz constant of f - A set D R is said to be convex if wenever t y t y D te point t + t y + y also belongs to D for eac, 0 - Suppose fty is defined in a convex set D R If L 0, suc tat fty L, for all ty D ten y f satisfies te Lipscitz condition on D in variable y wit Lipscitz constant L - Suppose tat D ty a t b y and tat fty is continuous on D If f satisfies te Lipscitz condition on D in variable y ten te IVP y' fty, a t b, ya as a unique solution yt for a t b Scientific computing III 03: 3 Differential equations 6

4 - Example: y' + tsinty, 0 t y0 0 - Keeping t constant we apply te mean value teorem to function fy + tsinty : df dy fy fy , y, y y y y y t fy cost fy fy fy fy y y y y fy 4 y y - Tus L 4 and te IVP as a unique solution - Wat about te stability of te IVP? - Te IVP y' ft y, a t b, ya is said to be a well-posed problem (or a stable problem) if A unique solution exists For any 0, k 0 wit te property tat, wenever 0 and t, a unique solution zt to te problem z' ft z + t, a t b, za + 0 (PB) (perturbed problem) exists wit zt yt k, t a t b - One can sow tat te IVP y' ft y is a well-posed problem if ) fty is a continuous function of t and y for all t y D and ) it satisfies te Lipscitz condition fty ft y Ly y, t y t y D Scientific computing III 03: 3 Differential equations 7 - Example: D ty 0 t y y' y + t + y0 - Since y + t +, te function ft y y + t + satisfies te Lipscitz condition y - Morever, f is continuous te problem is well posed - Let s see wat we get wen we perturb te problem: dz ---- z + t + +, 0 t, z0 + dt 0 - Solutions to te original and perturbed problems are yt e t + t, zt + 0 e t + t + - One can now easily see tat if and 0 ten yt zt 0 e t 0 + e t t Scientific computing III 03: 3 Differential equations 8

5 - Te above discussion concerns te stability of te problem itself - A problem is stable (well-posed) if te solution depends continuously on te initial value and on te rigt-and side function f - Te IVP could, owever, be stable but ill-conditioned - Condition number gives te sensitivity of te solution to te initial data - A large condition number means an ill-conditioned problem - Assume we ave a perturbed IVP: y' t; ftyt ;, yt 0 ; y 0 +, t a b - Let s write zt ; yt ; yt, were yt is te solution of te unperturbed IVP Tis gives us ftyt z' t; ftyt ; ftyt zt ; and zt y 0 ; - Te above approximation is valid for small (we ave a well-posed problem) - From te equation we can readily solve z: zt ; exp - Tis means tat if t ft'yt' dt' y t 0 ftyt , t a b y ten zt ; probably remains bounded by as t increases In tis case te problem is well-conditioned - An example of an ill-conditioned problem is: y' y+ gt, y 0 y 0, 0 - Now we can calculate zt ; e t indicating tat te cange in yt becomes increasing large as t increases Scientific computing III 03: 3 Differential equations 9 - As anoter example of an ill-conditioned problem consider y' 00y 0e t, y0 IVP: y' ay + be t - Te solution is yt e t - Perturbed problem: y' 00y 0e t, y0 +, as te solution yt ; e t + e 00t Solution: y0 y 0 fty ay + be t f y yt a y a+ 0 + ay 0 + be at be t zt ; exp t ft'yt' dt' y 0 t 00y 0e t exp dt' y e 00t 0 Scientific computing III 03: 3 Differential equations 0

6 Stability of te solution of an IVP y' t ft y: - Solution is stable if 0 0, suc tat every solution of te problem ŷ tat fulfills also fulfills for all t0 y0 ŷ0 yt ŷt - Solution is asymptotically stable if it is stable and lim yt ŷt 0 t - A simple example: y' y Solution is yt y0 e t - If 0 all solutions grow and diverge from eac oter exponentially: unstable solution - If 0 all solution converge to zero: asymptotically stable solution - If is complex ( a + ib) ten stability depends on Re: yt y0 e a + ibt - If Re 0 ten we get an oscillatory solution tat is stable but no asymptotically stable In addition to te stability definitions given above, te numerical metod used to solve te IVP can also be stable or unstable Scientific computing III 03: 3 Differential equations Euler s metod is te simplest possible metod to solve an IVP It is not used muc in practical problems but its derivation illustrates te concepts needed in deriving more advanced metods Te IVP we are trying to solve is te same as before dy fty, a t b, ya dt Te problem is discretized by computing te y values in mes points t 0 t t N in interval a b: t i a + i, i 0N - Te distance between te consecutive points b a N is called te step size - Assume te solution yt as two continuous derivatives - We will expand it as a Taylor series yt i + yt i t i+ t i y' t i t, i + t i y'' i i t i t i+ or yt i+ yt i + y' t i + -- y '' i - Since yt is te solution to te IVP we can write te expansion as yt i+ yt i + ft i yt i + -- y '' i Scientific computing III 03: 3 Differential equations

7 - In Euler s metod we simply drop te nd order term and get te iteration formula y 0, y i + y i + ft i y i, i 0 N - So we get an approximation to te solution y i yt i - Te algoritm is extremely simple Set b a N, t a, y For i N do Set y y + ft y Set t a + i Scientific computing III 03: 3 Differential equations 3 - As a simple example take dyt sint y0 0 0 t 3 dt - Te exact solution is yt cost - Te Euler algoritm in awk: BEGIN { 05; y0; tm30; ntm/; print 0,y; for (i0;i<n;i++) { ti*; yy+*sin(t); t(i+)* print t,y; } } cost Euler solution - In te figure one can see te grapical interpretation of te alrgoritm - Wen te problem is well-posed ft i y i y' t i ft i yt i - Euler s metod is easily generalized to groups of equations: y k + y k + ft k y k Scientific computing III 03: 3 Differential equations 4

8 - Wat about te error beavior of te Euler s metod? - By subtracting te exact solution from te Euler iteration result we get y'' yt k + y k+ yt k y k ft k yt k ft k y k k , () k t k t k + - Te left and side of tis equation is te total error of te approximation y k + and it consists of Errors cumulated from te previous step at t k Error formed in computing te current step (local error) - Error is called local because if we appened to get te exact value in te previous step ( k ) ten te only error tat is left is y'' k l k+ - In general te local error is defined as l n ŷt n y n and ŷ is te solution of te local problem: y' ˆ t ftŷ t ŷt n y n cumulated local - Due to te local error y'' k O Euler s metod is of te order one - In general te order of a metod is p if te local error beaves as O p + Scientific computing III 03: 3 Differential equations 5 - Using te mean value teorem we can estimate te term in te brackets in (): ft k yt k ft k y k ft k k yt, y k y k k yt k y k - Substituting tis to eq () we get were yt k + y k+ yt k y k ft k k y'' yt y k y k k y'' yt k y k + f y t k k k f y t k k ft k k y - So we get te total error at step k + as e k + e k + f y t k k + l k + error at previous step amplification factor local error - As far as + f y t k k te total error in Euler s metod does not grow - If + f y t k k te error grows during te iteration witout limits Scientific computing III 03: 3 Differential equations 6

9 - Let s look at te test problem y' y, were is a complex constant - Te solution is yt y0 e t - Applying Euler s metod we get y k y k + y k + y k y0 - Let now Re 0 so tat te solutions are asymptotically stable and approac zero - Tis means tat te numerical approximations y k must not grow In Euler s metod tis means + + k - Te absolute stability region D of a numerical metod consists of tose complex numbers z tat fulfill te condition y k y k, k Generally for te Euler s metod: + f y - Absolute stability interval is te part of te real axis tat belongs to D - Te absolute stability region of Euler s metod is te circle D z C z + and te interval is 0 Scientific computing III 03: 3 Differential equations 7 - Example: solving y' 00y + 00, y0 y 0 using Euler s metod - Exact solution is yt y 0 e 00t + and it is clearly stable - Euler s metod gives a difference equation y k + y k + 00y k y k + 00 wit solution y k y Let te initial value be y 0 - Exact solution is now yt e 00t + - Euler s metod gives y k 00 k + - We see immediately tat if 00 te solution y k grows witout limit wen k increases Scientific computing III 03: 3 Differential equations 8

10 - Te error formula of Euler s metod contains bot in te error amplification term and in te local error term: y'' e k + e k + f y t k k k In practical computations sould be as large as possible in order to speed up te calculations so small tat te error amplification factor < so small tat te local error term is acceptable - Te derivatives are difficult to compute but te local term can be estimated based on te iterations: y y a '' k ' y k ' t k t k - By requiring tat te local error fulfills te condition y a '' we get te upper limit to te step size y a '' Scientific computing III 03: 3 Differential equations 9 - Even if it would be possible - from te point of view of te consumed CPU time - to decrease te step size indefinitely we wouldn t get better and better results: finally te round-off errors sow up - Te figure on te rigt sows te relative error of te solution of te IVP y' sint y t 3 using step sizes and using single precision floating point numers - You probably remember te same penomenon in computing te derivative of a function using simple difference quotient Scientific computing III 03: 3 Differential equations 0

11 and develop iger order metods We can go furter in te Taylor s series yt i+ yt i t i + t i y' t i t i + t i y'' t i + - Tese metods ave te advantage of iger-order local truncation error - However, one as to calculate te iger derivatives f n t y wic is time consuming in many practical problems An alternative to te Euler s metod is an implicit version of it (backward Euler s metod): - Integrate y' t fty over one time step t k + yt k + yt k + ftyt t k - By approximating te intergral by t k + t k ft k y k we get te Euler s metod - However, we can also use t k+ t k ft k + y k + wen we get te backward Euler: y k + y k + ft k + y k+ - Te word implicit means tat new point y k + is on bot sides of te iteration formula It as to be solved by numerical metods - One can sow tat tis metods as a error amplification factor f y wic means tat wen applied to a stable linear problem wit constant coefficients wen Re 0, we get for all 0 dt Scientific computing III 03: 3 Differential equations Te metods described above were presented to illustrate te important concepts in numerical integration of IVPs Tey are seldom used in practical applications Probably te most often used metod to solve IVPs is te Runge-Kutta (RK) metod (or one of te many RK metods) It offers te iger orders witout te need to compute derivatives of f Te only drawback is tat te number of function evaluations of f per time step may be large We need a couple of results concerning te Taylor s teorem of a function of two variables - Suppose fy t as all its partial derivatives up to n + continuous on D ty a t b c y d - Let t 0 y 0 D - For every t y D tt 0 y y 0 wit fty P n ty + R n t y P n ty ft 0 y 0 t t 0 ft 0 y y y t 0 ft 0 y y t t 0 y y ft0 t y ft0 y y 0 + t t 0 y y 0 ft0 y ty n n t t0 n! j n j y y 0 j n + + t n j y j ft 0 y 0 j 0 R n ty n n + t t0 n+! j n + y y 0 j n + t n + j y j j - In oter words P n is te Taylor polynomial and R n is te remainder term f Scientific computing III 03: 3 Differential equations

12 - Let s see ow by using te RK metod we could increase te order of te Euler s metod by one - Tis means tat we ave to approximate te term Higer order Taylor metod: T ty fty + -- f ' t y y 0 yt 0 wit error no greater tan O - We take te approximation to ave te form a y i + y i + T n t y ft + y+ i i - Since dfty f' ty fty + fty y' t and y't dt t y T ty fty -- fty fty y' t t y - Now we expand ft + y+ into a Taylor polynomial of degree one about ty - Te coefficients a,, and can be determined by setting fty fty + -- f ' ty a ft + y+ fty -- + fty + -- fty y' t a fty a (remember ) t y ft y a + + fty t y' f y and matcing te coefficients of f and its derivatives T n ft i y i + -- f ' t i y i n f n n+! a ft + y+ a fty + a fty + a t fty + a R y t + y+ R t + y f f + t y ty f t i y i Scientific computing III 03: 3 Differential equations 3 - Te results is a, --, -- fty and T t y f t+ -- y + -- fty R t + -- y + -- fty - If te nd partial derivatives of f are bounded ten te remainder term R is of order O - Te metod derived above is a nd order (and two-stage) RK metod and te algoritm itself is simple: y 0 y0 y i+ y i + ft i + -- y i + -- ft i y i - Tis as a name of its own: te midpoint metod Scientific computing III 03: 3 Differential equations 4

13 - However, te most used RK metod is te 4 t order version: y 0 y0 k ft i y i k k ft i + -- y i k k 3 ft i + -- y i k 4 ft i + y i + k 3 y i + y i + -- k 6 + k + k 3 + k 4 - Note tat in tis algoritm te function f is evaluated 4 times per time step - In general a s-stage RK metod can be written as s y i + y i + b j k j j k j ft i + c j y i + a jl k l l c j s l a jl Scientific computing III 03: 3 Differential equations 5 - One can describe te RK metod as follows: Take an Euler step of lengt c from t i y i Te derivative at te new point is k Return to te original point and take a new Euler step ( c 3 ) using te weigted average of k and k as te derivative We get a new derivative k 3 - In tis way a number of approximation are computed for te derivative and tey all are used in te end to compute te step taken from t i y i - Te number of stages in a RK metod is not directly te order of te metod: Order Stages s 7 3 s 7 - Te table above explains te popularity of te four-stage RK metod Scientific computing III 03: 3 Differential equations 6

14 Most practical routines for integration of an IVP require te user to give an upper limit to te error in te end result - Te program ten adjusts te time step accordingly - Tis means tat te algoritm someow as to be able to estimate te error - One way to do tis is to compute te solution at te same time wit step sizes and - Assuming tat te local error introduced by te larger step is twice tat of te smaller step one can get an estimate for te error by l n -- ỹ n y n, p were ỹ n and y n are te solution at time n - However, computing wit two time steps is an extra effort - In te RK metod te error estimate can be computed by solving te IVP wit two RK solvers simultaneously - Te two solvers can be cosen in suc a way tat te derivatives needed by te lower order solver are already computed by te iger order metod - Tis pair of solvers is called te embedded Runge-Kutta pair - Tey are sometimes also called Runge-Kutta-Felberg metods - If te RK metod is written in te form y n + y n + Ft n y n f ; ten te metod is stable if F fulfills te Lipscitz condition Ftyf ; Ftzf ; L y z, for all a t b, y z Scientific computing III 03: 3 Differential equations 7 Below is a comparison of te maximum error of various one-step metods - A one-step metod is a metod tat only uses y i to compute te next step y i + - Te IVP was y' 5ty ,, t t y 0 t 0 - Te exact solution is yt t - Te trapezoidal metod is y k + y k + -- ft k + y k + + ft k y k From J Haataja et al, Numeeriset menetelmät käytännössä, CSC, 999 Scientific computing III 03: 3 Differential equations 8

15 In multistep (MS) metods one uses also information from earlier time steps tan only te previous one - Ie we want to use also y k, y k, ft k y k, ft k y k, to approximate te next point y k + - In general a r step linear multistep metod can be written as f j ft j y j were te coefficients fulfill te conditions, r + 0, r Metods can be derived by approximating te integral in wit a polynomial and by integrating it j y k + j j f k + j j r + j r + t k + yt k + yt k + ftyt dt Pt a 0 + a t + a t + + a s t s t k Scientific computing III 03: 3 Differential equations 9 - Te unknown coefficients a i are determined by fixing te polynomial to certain values y i and f i, i k + kk - Te good point in te MS metods is tat te function f is only evaluted once per time step - On te oter and, in te beginning of te iteration tere are no previous data points available - Tese can be generated by perfoming a couple of steps using a one-step metod - Also, canging te time step is not so straigtforward in MS metods - MS metods are divide to explicit (open) metods, 0 and implicit (closed) metods, 0 Scientific computing III 03: 3 Differential equations 30

16 Of te explicit MS metods te Adams-Basford (AB) is maybe te most well known - It is of te form 0 y k + y k + j f k + j j r + - Te unknown new iteration point does not sow up in te RHS of te equation so tis is an open metod - Below are te five first AB metods y k+ y k + f k y k + --3f k f k y k+ y k f k 6f k + 5f k y k+ y k f 4 k 59f k + 37f k 9f k 3 y k+ y k f 70 k 774f k + 66f k 74f k 3 + 5f k 4 - One can sow tat an AB metod wit r steps is of te order O r - Te stability interval of te AB metods are rater narrow, so tat tey are not suitable for stiff problems Scientific computing III 03: 3 Differential equations 3 Te implicit Adams-Moulton (AM) metod is more suitable for stiff problems: y k + y k + j f k + j j r + - Te five first AM metods are y k+ y k + f k + y k+ y k + -- f k + + f k y k+ y k f k f k f k y k+ y k f 4 k + + 9f k 5f k + f k y k+ y k f 70 k f k 64f k + 06f k + 9f k 3 - Wen r te order of te AM metod is r+ - Te absolute stability intervals of te metods are Scientific computing III 03: 3 Differential equations 3

17 - Te AM metods give better results tan te AB metods - However, te drawback of te AM metod is tat due to its implicity one as to solve te new iteration point from a group of equations - A combination metod often used is so called predictor-corrector metod - In it te AB metod produces a initial prediction to te new y value wile te AM metod is used to correct it more precise - Te correction may be done many times but not until te final convergence - For example te pair of 3 rd degree: yp k + y k f k 6f k + 5f k yc k + y k ft k + yp k+ + 8f k f k Scientific computing III 03: 3 Differential equations 33 r So called backward differentiation formulae (BDF) metods are well suited for stiff ODEs - Tey are derived by requiring a polynomial going troug te previous iteration points: r P r t yt n j l jn t, were l jn t is te Lagrange interpolation polynomial j - Because P is (an approximate) solution Pr ' tn + y' t n + ft n + yt n + - Combining te above equations and solving yt n + we get r y n + j y n j + ft n+ y n +, j 0 were j and are obtained by expanding te Lagrange interpolation polynomial in te first equation above - Te metod is implicit and te order is r - Below are te five first BDF metods: y k+ y k + f k + (Tis is backward Euler) 4 y k+ -- y 3 k -- y 3 k + -- f 3 k + y k y k y k y k f k + y k y 5 k y 5 k y 5 k y 5 k f 5 k y k y 37 k y 37 k y 37 k y 37 k y 37 k f 37 k + Scientific computing III 03: 3 Differential equations 34

18 Stiff ODEs are tose were tere are two or more disparate time scales - An example: y' 00y + 00t + 0, y0 y 0 - Solution: yt y 0 e 00t + t + - By varying te initial value y 0 sligtly causes large deviations in te initial beavior of te solution - Euler metod is results very sensitive to te initial value: even a small error during te iteration causes te iteration to pick a solution wit a different y 0 ; particularly one wit y 0 - Morever, wit large enoug time steps we get outside te stability interval - Demo Exact solutions for y Stiffness is difficult to define exactly: ) Process described by te ODE contains disparate timescales ) A group of ODEs is stiff if te eigenvalues of te Jacobian J ( J ij f i y j ) as eigenvalues tat differ greatly in magnitude 3) Wen using an explicit metod a muc smaller time step must be used as wic would be dictated by te accuracy criteria Scientific computing III 03: 3 Differential equations 35 - Example (from NR, section 66) - Exact solution u' 998u + 998v v' 999u 999v u e t e 000t v e t + e 000t - Using Euler s metod we ave to use really small steps: - Altoug te smootness of te function would allow a longer time step outside te initial transient beavior (term e 000t ) - Wat about aving a small step in te beginning and increasing it at - say - t 0 u u t exact t Scientific computing III 03: 3 Differential equations 36

19 - Below is te results of a version of Euler wic canges te stepsize after a certain time: u t - Tis does not elp because te instability of te transient is still tere toug te solution itself is a smootly varying function - Or to put it in anoter way: In order te solution to be stable we must use a smaller time step tan te one based only on te accuracy criteria Scientific computing III 03: 3 Differential equations 37 - For comparison consider te following equation tat is not stiff but as a rater similar solution: u' u + v v' u v - And te solution wit initial conditions u0, v0 0 is u v -- + e t -- e t - Using Euler wit gives reasonable results: u 0 3 exact 0 0 t Scientific computing III 03: 3 Differential equations 38

20 - For stiff ODEs te implicit metods work better - Take a simple example: y' cy, c 0 - Euler s iteration gives us y n + y n + y' n cy n - Te metod is unstable if -- c - Now using implicit metod (backward Euler) we get y n + y n + y' n + or y y n n + c - Tis metod is stable for all values of - So wit implicit metods we can use larger time steps in considering te stability of te metod - However, te accuracy may require to use smaller stepsize Scientific computing III 03: 3 Differential equations 39 Practical tools: GSL: routines for many stepping algoritms and for timestep control (including various Runge-Kutta and implicit metods) Matlab: also many routines: say elp funfun to get a list Maxima: symbolic solver, example solving te last example equation: Scientific computing III 03: 3 Differential equations 40

21 Practical tools (continued) - SLATEC library includes (at least) te following algoritms: DDEABM: Adams-Basfort metod DDEBDF: backward differentiation; for stiff problems DDERKF: Runge-Kutta-Felberg metod - User sould, in addition, provide te function tat calculates te RHS of te equation(s) and for te backward differentiation metod Jacobian J of te problem needed: y ' y ' y N ' f t y y N f t y y N f N ty y N J f f y y f yn f f f y y yn f N f N y y f N yn Wat metod to use? For non-stiff ODEs explicit RK, or predictor-corrector and AM if computing te derivative is expensive For stiff ODEs BDF and implicit RK - Now a nice demo Scientific computing III 03: 3 Differential equations 4 Differential equations: boundary value problems Boundary value problems (BVPs) are arder to solve because te conditions ave to be fulfilled in more tan one point - Te general form of D BVP is y' fx y gya yb 0 ( y and g ave N components) - Linear boundary conditions (BC) can be written in te form B a ya + B b yb b - If te BCs are separable tey can be written as ya b a yb b b - Sometimes te DE and BC contain a parameter : y' fx y gya yb 0 and te objective is to determine tose values of for wic te system as non-trivial solutions (eigenvalues and eigenfunctions) - Eg Scrödinger equation -- + V r r Er - In D a particle constrained by infinite potential walls: -- '' x + Vx x Ex, wic can be written as a system of st order DEs a 0 b 0 x x Vx Ex ' x Sources: J Haataja et al, Numeeriset menetelmät käytännössä, CSC, 999; R L Burden, J D Faires, Numerical Analysis, PWS-KENT, 989 Scientific computing III 03: 3 Differential equations 4

22 Differential equations: boundary value problems - In te sooting metod (SM) we utilize metods developed for IVPs - We integrate te DE starting from a cosen initial condition: y' fx y, ya s - We ave to coose s so tat te solution yx; s fulfills te BC gya ; s yb; s 0 - Te solution of te original BVP is tus yx; s * were s * is te solution of te group of equations Fs gs yb; s 0 - Tis generally a non-linear group of equations Scientific computing III 03: 3 Differential equations 43 Differential equations: boundary value problems - A common form of BVPs is te nd order equation y'' fxyy', a x b, ya yb, - Tis is converted to a pair of st order DEs as ( y y, y' y ) y ' y y ' fxy y - Existence of te solution: - Assume tat te function f and its derivatives f y, f y' are continuous in D xyy' a x b y y' - Now, if for all for all x y y' D te following apply a) fxyy' 0 and y b) M suc tat fxyy' M y' ten te abovementioned BVP as a unique solution Scientific computing III 03: 3 Differential equations 44

23 Differential equations: boundary value problems - An example: y'' + e xy + siny' 0, x, y 0, y 0 or y'' fxyy', fxyy' e xy siny' Now fxyy' xe xy 0, fxyy' y y' cosy' So, te equation as a unique solution Solution of te IVP y'' + e xy + siny' 0 y 0 y' c wit values c c 06 c 0569 c 04 Scientific computing III 03: 3 Differential equations 45 Differential equations: boundary value problems - Te linear form of te nd order BVP can be written as y'' px y' + qx y+ rx, a x b, ya, yb () - For tis form te existence of a unique solution requires tat a) px, qx, rx are continuous in a b and b) qx 0 in a b - Consider two IVPs based on te BVP above: y'' px y' + qx y+ rx, x a b, ya, y' a 0, solution y x y'' px y' + qx y, x a b, ya 0, y' a, solution y x y y - It is easy to sow tat te unique solution to te original equation () can be expressed in terms of y x and y x: yx assuming y b 0 y y x b y, y b x a y y b x - A demo using GSL No need to worry about tis One can sow tat if y is te solution of y'' px y' + qx y wit y a y b 0 ten y x 0 x ab Scientific computing III 03: 3 Differential equations 46

24 Differential equations: boundary value problems - In te multiple sooting metod (MSM) te interval is divided into smaller parts and te SM is applied to every part so tat te parts join to eac oter continuously - Using te metod you end up solving a (possibly nonlinear) group of equations - Divide te interval a b to parts a x x x m b - Let y' xx ; k s k be te solution to te IVP y' fx y, yx k s k on te interval x x k x k + - In te MSM metod we pursue te vectors s k, k m tat satisfy te continuity and boundary conditions yx k + ; x k s k s k +, k m gs ybx ; m s m 0 - Te unknown initial values s s s s m T are obtained from te nonlinear system of equations Fs yx ; x s s yx 3 ; x s s 3 0 gs yx m ; x m s m Scientific computing III 03: 3 Differential equations 47 Differential equations: boundary value problems Te finite difference metod is not based on solving te corresponding IVP but te BVP is discretized to a finite mes and derivatives are approximated by finite differences - An example sows ow te metod works - Te BVP is y'' fxyy' ya yb - Divide interval a b to m parts: b a , x m i a + i, i 0 m - Approximate te derivatives in te BVP by finite differences: y y' x n n+ y n y y'' x n n + y n + y n - Now we get te difference equation y n + y n + y n y f x n y n+ y n n , n m y 0, y m JHaataja et al, Numeeriset menetelmät käytännössä, example 75 Scientific computing III 03: 3 Differential equations 48

25 - In matrix form Ay fy Differential equations: boundary value problems A f y x y y y 0 0 y f y x y 3 y , y y, fy 3 f y x 3 y 4 y y m f y x m y m y m m Note tat tis is a nonlinear group of equations Scientific computing III 03: 3 Differential equations 49 ifferential equations: boundary value problems - Tere are also metods based on expansion of te solution in terms of basis functions; eg collocation metod (CM), Galerkin s metod - Take for example te equation y'' fxyy', ya, yb - We expand te solution in terms of basis functions N : yx N i c i i x - Te basis set can be cosen in an appropriate way considering te boundary conditions - In te CM metod te above expansion is substituted in te equation and we require tat it solves te equation in N points x j witin te interval a b: N N N c i i '' x j f x j c i i x j c i i ' x j, j N i i i - From tis equation we can determine te coefficients c c c N Scientific computing III 03: 3 Differential equations 50

26 Differential equations: boundary value problems Practical tools: Matlab: see elp funfun for more information NAG: many routines for BVPs SLATEC: many routines for BVPs NETLIB: see te package ode (ttp://wwwnetliborg/ode/) Scientific computing III 03: 3 Differential equations 5