Ecuaciones diferenciales en derivadas parciales

Transcription

1 Ecuacones dferencales en dervadas parcales Pedro Corcuera Dpto. Matemátca Aplcada y Cencas de la Computacón Unversdad de Cantabra corcuerp@uncan.es

2 Obetvos Revsón de las ecuacones dferencales en dervadas parcales y técncas de solucón Ec. dervadas parcales

3 Dferencacón Ec. dervadas parcales

4 Dferencacón de funcones contnuas he mathematcal defnton of a dervatve begns wth a dfference approxmaton: y ( ) f ( x ) x f x x x and as x s allowed to approach zero, the dfference becomes a dervatve: f ( x x) f ( x ) dy dx lm x x Ec. dervadas parcales 4

5 Fórmulas de dferencacón de gran exacttud aylor seres expanson can be used to generate hgh-accuracy formulas for dervatves by usng lnear algebra to combne the expanson around several ponts. hree categores for the formula nclude forward fnte-dfference, backward fnte-dfference, and centered fnte-dfference. Ec. dervadas parcales 5

6 Aproxmacón por dferenca en adelanto f For a fnte ( x) ' Δx' f(x) lm Δx f ( x) f ( x Δx) f ( x) f Δx ( x x) f ( x) x x xδx Graphcal Representaton of forward dfference approxmaton of frst dervatve. Ec. dervadas parcales 6

7 Aproxmacón por dferenca en adelanto - eemplo he velocty of a rocket s gven by ν t ( t) ln 9.8t, t Where 'ν' s gven n m/s and 't' s gven n seconds. a)use forward dfference approxmaton of the frst dervatve of ν( t) to calculate the acceleraton at t 6s. Use a step sze of Δ t s. b)fnd the exact value of the acceleraton of the rocket. c) Calculate the absolute relatve true error for part (b). Ec. dervadas parcales 7

8 Aproxmacón por dferenca en adelanto - eemplo Soluton a) a ( t ) ( 6) ( t ) ( t ) ν ν t ν ν ( 8) ( 6) t 6 Δ t t t 6 t a 4 4 ν ( 8) ln 9.8( 8) 45. m/s 4 4 ( 8) 4 4 ν ( 6) ln 9.8( 6) m/s 4 ( 6) Hence ν ( ) ( 8) ν ( 6) a 6.474m/s b) he exact value of a( 6) can be calculated by dfferentatng 4 ( t 4 ν ) ln 9. t 4 t 8 4 as d a ( t) [ ν( t) ] Ec. dervadas parcales 8 dt 8

9 Aproxmacón por dferenca en adelanto - eemplo Knowng that d [ ( t) ] d ln and dt t dt t t t d 4 a( t) dt 4 t t t t t a ( 6) ( ) ( ) ( ) ( ) 9. 8 ( ) 9.674m/s c) he absolute relatve true error s rue Value - Approxmate Value t x.6967 % rue Value Ec. dervadas parcales 9

10 Efecto del tamaño de paso en el método de dferenca dvdda en adelanto f ( x) 9e 4x Value of h ' (.) f (.) E f a usng forward dfference method. ε a % Sgnfcant dgts E t ε t % E E Ec. dervadas parcales

11 Efecto del tamaño de paso en el método de dferenca dvdda en adelanto 9 f'(.) Number of tmes step sze halved, n Ea Number of tmes step sze halved, n Ea % Number of tmes step sze halved, n Least number of sgnfcant dgts correct Number of tmes step sze halved, n Ec. dervadas parcales

12 Efecto del tamaño de paso en el método de dferenca dvdda en adelanto Number of tmes step sze halved, n E t % 6 E t Number of tmes step sze halved, n Ec. dervadas parcales

13 Aproxmacón por dferenca en atraso Sabemos que For a fnte Δ ', f ( x) ' x f ( x) lm Δx f ( x Δx) f ( x) f Δx ( x x) f ( x) x If ' Δx' s chosen as a negatve number, f ( x) f ( x x) f ( x) x f ( x) f ( x Δx) Δx Ec. dervadas parcales

14 hs s a backward dfference approxmaton as you are takng a pont backward from x. o fnd the value of f ( x) at x x, we may choose another pont ' Δx' behnd as x. hs gves f ( x ) where Aproxmacón por dferenca en atraso f x ( x ) f ( x ) f x f ( x ) ( x ) x x Δx x x f(x) x-δx Graphcal Representaton of backward dfference approxmaton of frst dervatve. x x Ec. dervadas parcales 4

15 Obtencón de la adad a partr de las seres de aylor aylor s theorem says that f you know the value of a functon f at a pont x and all ts dervatves at that pont, provded the dervatves are contnuous between and, then f x x ( x ) f ( x ) f ( x )( x x ) Substtutng for convenence f f f f! Δx ( x ) f ( x ) f ( x ) f Δx! f x! Δ ( x ) f ( x ) f ( x ) ( ) x ( x ) f ( x ) ( ) ( x ) x ( x ) f ( x ) O( x) x ( x ) ( ) x x x x Ec. dervadas parcales 5

16 Obtencón de la adad a partr de las seres de aylor O( x) he term shows that the error n the approxmaton s of the order of x. It s easy to derve from aylor seres the formula for backward dvded dfference approxmaton of the frst dervatve. As shown above, both forward and backward dvded dfference approxmaton of the frst dervatve are accurate on the order of O( x). Can we get better approxmatons? Yes, another method s called the Central dfference approxmaton of the frst dervatve. Ec. dervadas parcales 6

17 Obtencón de la adc a partr de las seres de aylor From aylor seres f f ( x ) f ( x ) f ( x ) Δx ( x ) ( ) ( ) f x ( ) Δx Δ ( x ) f ( x ) f ( x ) Δx Subtractng equaton () from equaton () f f! f x! x ( ) ( ) ( ) f ( ) Δx Δ ( x ) f ( x ) f ( x )( x) f f ( x ) f!! ( x ) ( ) f x! ( x ) ( ) f x f ( x ) ( ) x ( x ) f x ( x ) f ( x ) x O! ( x) x x () () Ec. dervadas parcales 7

18 Obtencón de la adc a partr de las seres de aylor Hence showng that we have obtaned a more accurate formula as the error s of the order of O( x) f(x) x-δx x xδx x Graphcal Representaton of central dfference approxmaton of frst dervatve Ec. dervadas parcales 8

19 Aproxmacón por dferenca central - eemplo he velocty of a rocket s gven by ν t ( t) ln 9.8t, t Where 'ν' s gven n m/s and 't' s gven n seconds. a)use central dvded dfference approxmaton of the frst dervatve of ν( t) to calculate the acceleraton at t 6s. Use a step sze of Δ t s. b)calculate the absolute relatve true error for part (a). Ec. dervadas parcales 9

20 Soluton Aproxmacón por dferenca central - eemplo ( t ) ( t ) ν ν t 6 Δ t t t t t 6 ν ν t t t 6 a) a( t ) 4 ( ) ( 8) ( 4) 8 a ν ( 8) ln 9.8( 8) 4 4 ( 8) 45. m/s 4 4 ν ( 4) ln 9.8( 4) 4 4 ( 4) 4.4 m/s Hence ν ( ) ( 8) ν ( 4) a m/s 4 4 b) he absolute relatve true error s knowng that the exact value at s a( 6 ) m/s t.6957 % t 6s Ec. dervadas parcales

21 Efecto del tamaño de paso en el método de dferenca dvdda central f ( x) 9e 4x Value of h ' (.) f (.) E f a usng forward dfference method. ε a % Sgnfcant dgts E t ε t % E E E-5.5E E E E-6 7.6E E E E-6.9E E-7 Ec. dervadas parcales

22 Efecto del tamaño de paso en el método de dferenca dvdda central 95 f'(.) 85 Number of steps nvolved, n E(a) Number of tmes the step sze s halved, n - E(a),% Least number of sgnfcant dgts correct Number of steps nvolved, n Number of steps nvolved, n Ec. dervadas parcales

23 Efecto del tamaño de paso en el método de dferenca dvdda central.8 E(t) Number of steps nvolved, n E t % Number of tmes step sze halved,n Ec. dervadas parcales

24 Comparacón de métodos adad, adat, adc he results from the three dfference approxmatons are gven n the followng table. Summary of a (6) usng dfferent dvded dfference approxmatons ype of Dfference Approxmaton Forward Backward Central a( 6) ( m / s ) t % Ec. dervadas parcales 4

25 Obtencón del valor de la dervada con una toleranca específca In real lfe, one would not know the exact value of the dervatve so how would one know how accurately they have found the value of the dervatve. A smple way would be to start wth a step sze and keep on halvng the step sze untl the absolute relatve approxmate error s wthn a pre-specfed tolerance. ake the example of fndng at ν t 6 v ( t) t for ( t ) ln 9. 8 t usng the backward dvded dfference scheme. Ec. dervadas parcales 5

26 Obtencón del valor de la dervada con una toleranca específca he values obtaned usng the backward dfference approxmaton method and the correspondng absolute relatve approxmate errors are gven n the followng table. t v ( t) % a one can see that the absolute relatve approxmate error decreases as the step sze s reduced. At t.5 the absolute relatve approxmate error s.655%, meanng that at least sgnfcant dgts are correct n the answer. Ec. dervadas parcales 6

27 Fórmulas de dferenca fnta en adelanto Ec. dervadas parcales 7

28 Fórmulas de dferenca fnta en atraso Ec. dervadas parcales 8

29 Fórmulas de dferenca fnta centrada Ec. dervadas parcales 9

30 Sstemas de ecuacones lneales Ec. dervadas parcales

31 Matrces A matrx conssts of a rectangular array of elements represented by a sngle symbol (example: [A]). An ndvdual entry of a matrx s an element (example: a ) Ec. dervadas parcales

32 Matrces especales Matrces where mn are called square matrces. here are a number of specal forms of square matrces: Symmetrc 5 [ A] Dagonal a [ A] a a Identty [ A] Upper rangular a a a [ A] a a a Lower rangular a [ A] a a a a a Banded a a a [ A a a ] a a a 4 a 4 a 44 Ec. dervadas parcales

33 Multplcacón matrcal he elements n the matrx [C] that results from multplyng matrces [A] and [B] are calculated usng: c n k a k b k Ec. dervadas parcales

34 Representacón de Algebra Lneal Matrces provde a concse notaton for representng and solvng smultaneous lnear equatons: a x a x a x b a x a x a x b a x a x a x b a a a a a a a a a x x x b b b [A]{x} {b} Ec. dervadas parcales 4

35 Solucón con Matlab MALAB provdes two drect ways to solve systems of lnear algebrac equatons [A]{x}{b}: Left-dvson x A\b Matrx nverson x nv(a)*b he matrx nverse s less effcent than left-dvson and also only works for square, non-sngular systems. Ec. dervadas parcales 5

36 Matrz nversa Recall that f a matrx [A] s square, there s another matrx [A] -, called the nverse of [A], for whch [A][A] - [A] - [A][I] he nverse can be computed n a column by column fashon by generatng solutons wth unt vectors as the rght-hand-sde constants: { } A [ ] x { } A [ ] x [ A] [ x x x ] { } A [ ] x Ec. dervadas parcales 6

37 Matrz nversa y sstemas estímulo - respuesta Recall that LU factorzaton can be used to effcently evaluate a system for multple rght-hand-sde vectors - thus, t s deal for evaluatng the multple unt vectors needed to compute the nverse. Many systems can be modeled as a lnear combnaton of equatons, and thus wrtten as a matrx equaton: [ Interactons] { response} { stmul} he system response can thus be found usng the matrx nverse. Ec. dervadas parcales 7

38 Normas vectorales y matrcales A norm s a real-valued functon that provdes a measure of the sze or length of mult-component mathematcal enttes such as vectors and matrces. Vector norms and matrx norms may be computed dfferently. Ec. dervadas parcales 8

39 Normas vectorales For a vector {X} of sze n, the p-norm s: X p n x p Important examples of vector p-norms nclude: / p p p p : sum of the absolute values : Eucldan norm (length) : maxmum magntude X X X n X x e max n x n x Ec. dervadas parcales 9

40 Normas matrcales Common matrx norms for a matrx [A] nclude: column - sum norm A Frobenus norm A f a row - sum norm A spectral norm ( norm) n max a n n n n max a n ( ) / A µ max Note - µ max s the largest egenvalue of [A] [A]. Ec. dervadas parcales 4

41 Número de condcón de una matrz he matrx condton number Cond[A] s obtaned by calculatng Cond[A] A A - In can be shown that: X Cond [ A ] A A X he relatve error of the norm of the computed soluton can be as large as the relatve error of the norm of the coeffcents of [A] multpled by the condton number. If the coeffcents of [A] are known to t dgt precson, the soluton [X] may be vald to only t-log (Cond[A]) dgts. Ec. dervadas parcales 4

42 Comandos Matlab MALAB has bult-n functons to compute both norms and condton numbers: norm(x,p) Compute the p norm of vector X, where p can be any number, nf, or fro (for the Eucldean norm) norm(a,p) Compute a norm of matrx A, where p can be,, nf, or fro (for the Frobenus norm) cond(x,p) or cond(a,p) Calculate the condton number of vector X or matrx A usng the norm specfed by p. Ec. dervadas parcales 4

43 Método teratvo: Gauss - Sedel he Gauss-Sedel method s the most commonly used teratve method for solvng lnear algebrac equatons [A]{x}{b}. he method solves each equaton n a system for a partcular varable, and then uses that value n later equatons to solve later varables. For a x system wth nonzero elements along the dagonal, for example, the th teraton values are found from the - th teraton usng: x b a x a x a x b a x a x a x b a x a x a Ec. dervadas parcales 4

44 he Jacob teraton s smlar to the Gauss-Sedel method, except the -th nformaton s used to update all varables n the th teraton: a) Gauss-Sedel b) Jacob Iteracón de Jacob Ec. dervadas parcales 44

45 Convergenca he convergence of an teratve method can be calculated by determnng the relatve percent change of each element n {x}. For example, for the th element n the th teraton, ε x a, x % he method s ended when all elements have converged to a set tolerance. x Ec. dervadas parcales 45

46 Domnanca dagonal he Gauss-Sedel method may dverge, but f the system s dagonally domnant, t wll defntely converge. Dagonal domnance means: a > n a Ec. dervadas parcales 46

47 Relaacón o enhance convergence, an teratve program can ntroduce relaxaton where the value at a partcular teraton s made up of a combnaton of the old value and the newly calculated value: x new new ( ) old λx λ x where λ s a weghtng factor that s assgned a value between and. <λ<: underrelaxaton λ: no relaxaton <λ : overrelaxaton Ec. dervadas parcales 47

48 Ecuacones dferencales en dervadas parcales Ec. dervadas parcales 48

49 Ecuacones en dervadas parcales Ordnary Dfferental Equatons have only one ndependent varable dy x 5y e, y() 5 dx Partal Dfferental Equatons have more than one ndependent varable u x u y x y subect to certan condtons: where u s the dependent varable, and x and y are the ndependent varables. Ec. dervadas parcales 49

50 Clasfcacón de EDPs de orden u u u A B C D x x y y where A, B, andc are functons of x and y, and. D s a functon of u u xyu,, and,. x y can be: Ellptc f B 4AC < Parabolc f B 4AC Hyperbolc f B 4AC > Ec. dervadas parcales 5

51 Eemplos de EDPs de orden Ellptc Parabolc Hyperbolc A, B, C x t x y k y c A, B x A k, B, C, C t y c Laplace equaton Heat equaton Wave equaton Ec. dervadas parcales 5

52 Eemplo físco de una PDE elíptca y t W l r L b x Schematc dagram of a plate wth specfed temperature boundary condtons he Laplace equaton governs the temperature: x y Ec. dervadas parcales 5

53 Dscretzando la PDE elíptca y (, n) t l (, ) x x y L m y r W n (, ) y (, ) x (, ) (, ) (,) b (m,) x (, ) x ( x, y) ( x x, y) ( x, y) ( x ( x) x, y) y ( x, y) ( x, y y) ( x, y) ( x, ( y) y y) Ec. dervadas parcales 5

54 Dscretzando la PDE elíptca y (, n) t l (, ) x y r (, ) y (, ) x (, ) (, ) (,) b (m,) x (, ) x ( x, y) ( x x, y) ( x, y) ( x x, y) ( x), x,, ( x), y ( x, y) ( x, y y) ( x, y) ( x, y y) ( y), y,, ( y), Ec. dervadas parcales 54

55 Substtutng these approxmatons nto the Laplace equaton yelds: f, the Laplace equaton can be rewrtten as (Eq. ) there are several numercal methods that can be used to solve the problem: Drect Method Gauss-Sedel Method Leberman Method Ec. dervadas parcales 55 Dscretzando la PDE elíptca y x ( ) ( ),,,,,, y x x y 4,,,,,

56 Eemplo : Método drecto Consder a plate.4m. m that s subected to the boundary condtons shown below. Fnd the temperature at the nteror nodes usng a square grd wth a length of.6m by usng the drect method. y C W 75. m C C 5.4m L C x Ec. dervadas parcales 56

57 Eemplo : Método drecto We dscretze the plate by takng, L W m 4 n 5 x y x y. 6m he nodal temperatures at the boundary nodes are gven by: y C,5,5, 5, 5 4, 5 75 C,4,,,,,4, 4,4 4, 4,,, 4,,,, 4,,,, 4,,,, 4, 5 C C x, 4,,,5 75,,,,4,,,,4 5,,,,,, Ec. dervadas parcales 57

58 Eemplo : Método drecto the equaton for the temperature at the node (,) y,5,5, 5, 5 4, 5,4,4, 4,4 4, 4,,,, 4,,,,, 4, and,,,, 4,,,,, 4,,,,, 4,,,,4, 4,,, 4,,4, x Ec. dervadas parcales 58

59 Eemplo : Método drecto We can develop smlar equatons for every nteror node leavng us wth an equal number of equatons and unknowns. For ths problem the number of equatons generated s Ec. dervadas parcales 59

60 he corner nodal temperature of,5, 4,5, 4,,, are not needed o get the temperature at the nteror nodes we have to wrte Equaton for all the combnatons of and,,..., m ;,..., n and and and and 4 and and and and 4 and and and and 4 Eemplo : Método drecto 4,,, 5, 4,,,, 4,,4, 75, 4,4, 4 75, 4,,,,, 4,,,,, 4,,4,,4, 4,4, 4, 4,, 5,, 4,,,, 4,, 4,4, 4, Ec. dervadas parcales 6

61 Eemplo : Método drecto We can use Excel and matrx operatons to solve the lnear equatons system,,,,4,,,,4,,,,4 RHE -4-5, , , , , , , , , , , , Ec. dervadas parcales 6

62 Método Gauss-Sedel Recall the dscretzed equaton,,,, 4, hs can be rewrtten as,,,, For the Gauss-Sedel Method, ths equaton s solved teratvely for all nteror nodes untl a pre-specfed tolerance s met. 4, Ec. dervadas parcales 6

63 Eemplo : Método Gauss-Sedel Consder a plate.4m. m that s subected to the boundary condtons shown below. Fnd the temperature at the nteror nodes usng a square grd wth a length of.6m usng the Gauss-Sedel method. Assume the ntal temperature at all nteror nodes to be C. y C W 75. m C C 5 C x.4m L 6

64 Eemplo : Método Gauss-Sedel Dscretzng the plate by takng, L W m 4 n 5 x y x y. 6m he nodal temperatures at the boundary nodes are gven by: y C,5,5, 5, 5 4, 5 75 C,4,,,,,4, 4,4 4, 4,,, 4,,,, 4,,,, 4,,,, 4, 5 C C x, 4,,,5 75,,,,4,,,,4 5,,,,,, Ec. dervadas parcales 64

65 Eemplo : Método Gauss-Sedel Now we can begn to solve for the temperature at each nteror node usng,,,,,,,,,4;,,,4,5 4 Assume all nternal nodes to have an ntal temperature of zero. Iteraton : and.5º C and, and 6.565º C and 4, and 5.96 º C and, and 4.98º C and,4 and.5º C and,, and.788º C and º C,.44º C, º C, 8.574º C, 6.969º C,, º C Ec. dervadas parcales 65

66 Eemplo : Método Gauss-Sedel Iteraton : we take the temperatures from teraton and calculate the present prevous approxmated error.,, ε a, present,, º C 7.7%,,, º C.49%, 4,, º C ε a 54.49%,,, 4.8º C ε a 4.9%,,4, 6.864º C 44.8%,,,.8594 º C 6.%, 4, ε a, ε a,,,4 ε a, ε a, º C, 56.5º C, º C, 56.45º C, º C, º C,4 ε a ε a ε a,,4, ε a, ε a, ε a,4 8.58% 4.46% 4.44%.7% 57.44% 6.% Ec. dervadas parcales 66

67 Eemplo : Método Gauss-Sedel Node,,,,4,,,,4,,,,4 emperature Dstrbuton n the Plate ( C) Number of Iteratons Ec. dervadas parcales 67

68 Eemplo : Método Gauss-Sedel n Excel he numercal soluton of Laplace equaton at a pont s the average of four neghbors Example for cell S8: (S7S9R88)/4, 4,,,, Enter the boundary condtons n the approprate cells. Copy and paste to cover the cells where values of the potental are to be calculated. hs calculaton contans a "crcular reference. Ec. dervadas parcales 68

69 Eemplo : Método Gauss-Sedel n Excel o allow crcular references and enable teratons: Fle Optons Formulas On the "Calculatons optons" form select "Enable teratve calculaton" We can ncrease the Maxmum Iteratons ( s the deafult) and reduce the Maxmum Change (. s the default). Iteratons wll stop when the maxmum teraton s reached or the change s less than the maxmum change. F9 to recalculate. Ec. dervadas parcales 69

70 Eemplo : Método Gauss-Sedel n Excel Color cell based on value o acheve the cell color based on value: Inco Estlos Formato condconal Escalas de color Más reglas We can chose a color scale wth blue for mmmum, whte or gray for mdpont and red for maxmum. Ec. dervadas parcales 7

71 Eemplo : Método Gauss-Sedel n Excel Plottng the results Normally we use the chart type Surface or Contour. Ec. dervadas parcales 7

72 Eemplo : Uso de Solver Fnte Dfference Soluton x\y (-4*D5D4D6C5E5)^ Resduals-squared x\y sum.7745e- SUMA(D7:J) Ec. dervadas parcales 7

73 Método de Leberman Recall the equaton used n the Gauss-Sedel Method,,,,, Because the Gauss-Sedel Method s guaranteed to converge, we can accelerate the process by usng overrelaxaton. In ths case, he λ s known as the overrelaxaton parameter" and s n the range < λ <. 4 relaxed new old, λ, ( λ),, Ec. dervadas parcales 7

74 Condcones de contorno alternatvas In the past examples, the boundary condtons on the plate had a specfed temperature on each edge. What f the condtons are dfferent? For example, what f one of the edges of the plate s nsulated. In ths case, the boundary condton would be the dervatve of the temperature. Because f the rght edge of the plate s nsulated, then the temperatures on the rght edge nodes also become unknowns. y C. m 75 C Insulated 5.4 m C x Ec. dervadas parcales 74

75 Condcones de contorno alternatvas he fnte dfference equaton n ths case for the rght edge for the nodes ( m, ) for,,.. n m, m, m, m, 4 m, However the node ( m, ) s not nsde the plate. he dervatve boundary condton needs to be used to account for these addtonal unknown nodal temperatures on the rght edge. hs s done by approxmatng the dervatve at the edge node ( m, ) as x m m,, ( x) m, y 75 C. m C Insulated 5.4 m C x Ec. dervadas parcales 75

76 Rearrangng ths approxmaton gves us, We can then substtute ths nto the orgnal equaton gves us, Recall that s the edge s nsulated then, Substtutng ths agan yelds, Ec. dervadas parcales 76 Condcones de contorno alternatvas m m m x x,,, ) ( 4 ) (,,,,, m m m m m x x, m x 4,,,, m m m m

77 Ecuacones en dervadas parcales parabólcas he general form for a second order lnear PDE wth two ndependent varables and one dependent varable s u A x u B x y C u y u y he crtera for an equaton of ths type to be consdered parabolc: B 4AC Examne the heat-conducton equaton gven by α x t where A α, B, C, D, E, F, G thus we can classfy ths equaton as parabolc. where α u D x k ρc E Ec. dervadas parcales 77 Fu G k thermal conductvty of rod materal, ρ densty of rod materal, C specfc heat of the rod materal.

78 Eemplo de una EDP parabólca Consder the flow of heat wthn a metal rod of length L, one end of whch s held at a known hgh temperature, the other end at a lower temperature. Heat wll flow from the hot end to the cooler end. We'll assume that the rod s perfectly nsulated, so that heat loss through the sdes can be neglected. We want to calculate the temperature along the length of the rod as a functon of tme. Ec. dervadas parcales 78

79 Dscretzacón de una EDP Parabólca For a rod of length L dvded nto n nodes he tme s smlarly broken nto tme steps of x t L n Hence corresponds to the temperature at node,that s, x ( )( x) and tme t ( )( t) x x x Schematc dagram showng nteror nodes Ec. dervadas parcales 79

80 Solucón EDP Parabólca: Método explícto L n If we defne x we can then wrte the fnte central dvded dfference approxmaton of the left hand sde at a general nteror node ( ) as where ( ) s the node number along x, the tme. he tme dervatve on the rght hand sde s approxmated by the forward dvded dfference method as, t t x ( x) x x, Schematc dagram showng nteror nodes Ec. dervadas parcales 8

81 Substtutng these approxmatons nto the governng equaton yelds α ( x) t Solvng for the temp at the tme node gves t ( ) α ( x) choosng, t λ α we can wrte the equaton as, Solucón EDP Parabólca: Método explícto ( x) λ ( ) we can be solved explctly: for each nternal locaton node of the rod for tme node n terms of the temperature at tme node. If we know the temperature at node, and the boundary temperatures, we can fnd the temperature at the next tme step. We contnue the process untl we reach the tme at whch we are nterested n fndng the temperature. Ec. dervadas parcales 8

82 Eemplo EDP Parabólca: Método explícto Consder a steel rod that s subected to a temperature of C on the left end and 5 C on the rght end. If the rod s of length.5m,use the explct method to fnd the temperature dstrbuton n the rod from t and t 9 seconds. Use x. m, t s. W kg m K m Gven: k, ρ, he ntal temperature of the rod s C. J C 49 kg K 4 5 C 5 C.m Ec. dervadas parcales 8

83 Eemplo EDP Parabólca: Método explícto Number of tme steps Recall, k α ρ C t t fnal tntal 9 54 t 5.49 m / s hen, λ α.49 ( x) (.) Boundary Condtons C All nternal nodes are at C for sec: C C C Interor nodes C 4 C 5 C C for all,,, t C, for all,,,4 We can now calculate the temperature at each node explctly usng the equaton formulated earler, ( ) λ Ec. dervadas parcales 8

84 Eemplo EDP Parabólca: Método explícto Nodal temperatures vs. me t sec C C C Interor nodes C 4 C 5 C 5 t sec C Boundary Condton 5.9 C C C. C 5 C Boundary Condton 4 5 t 6sec C 59.7 C 4.75 C.889 C 4.44 C 5 C 5 t 9sec C C 9. C 7.66 C 4.87 C 5 C 5 Ec. dervadas parcales 84

85 Eemplo EDP Parabólca: Método explícto tme t (sec) me-dependent emperature Dstrbuton n a Brass Rod (emperature values n bold are constant) length, cm heat capacty of brass, cal/g/deg,9 (hcap) thermal conductvty of brass, cal/sec/cm/deg,6 (k) densty of brass, g/cm 8,4 (rho) Coeffcent e n general PDE, k/(hcap*rho),(e) x (Dx) t (Dt) fe*dt/(dx^), (f) Dstance x (cm) ,9,,,,,,,, 44,,8,,,,,,, 5,6 8,,6,,,,,, 4 56,5 4,4 7,,,,,,, 5 6, 9,,9,8,4,,,, 6 6,,4 4, 4,7,,,,, 7 65,5 6,9 7,4 6,6,9,4,,, 8 67,5 9,9, 8,6,,8,,, 9 69, 4,5,9,6 4,,,,, 7,5 44,8 5,,5 5,,9,5,, 7,8 46,9 7,5 4,4 6,6,6,9,, 7,9 48,7 9,6 6, 7,8,,,4, 7,8 5,4,4 7,8 9, 4,,6,6, 4 74,7 5,9, 9,4, 4,9,,8, 5 75,5 5, 4,8,,5 5,8,6,, 6 76, 54,5 6,,4,7 6,6,,,5 7 76,9 55,7 7,7,8,9 7,5,7,6,6 8 77,5 56,8 9, 5, 5, 8,4 4,,,7 9 78, 57,8 4, 6,4 6, 9, 4,9,,9 78,6 58,7 4,5 7,6 7,, 5,5,7, 79, 59,6 4,6 8,8 8,,9 6,,, 79,6 6,4 4,6 9,9 9,,7 6,7,5,4 8, 6, 44,6,9,,6 7,,8,6 4 8,4 6,9 45,6,9,,4 7,9 4,,8 5 8,8 6,6 46,5,9, 4, 8,5 4,6, emperature ºC emperature ºC emperature at x5cm me sec. emperature dstrbuton along the length of the rod Loc aton on rod cm. t sec. t 5 sec. t sec. Ec. dervadas parcales 85

86 Usng the explct method, we were able to fnd the temperature at each node, one equaton at a tme. However, the temperature at a specfc node was only dependent on the temperature of the neghborng nodes from the prevous tme step. hs s contrary to what we expect from the physcal problem. he mplct method allows us to solve ths and other problems by developng a system of smultaneous lnear equatons for the temperature at all nteror nodes at a partcular tme. he second dervatve s approxmated by the CDD and the frst dervatve by the BDD scheme at tme level at node ( ) as Ec. dervadas parcales 86 Solucón EDP Parabólca: Método mplícto t x α ( ), x x t t,

87 Substtutng these approxmatons nto the heat conducton equaton yelds Rearrangng yelds gven that he rearranged equaton can be wrtten for every node durng each tme step. hese equatons can then be solved as a smultaneous system of lnear equatons to fnd the nodal temperatures at a partcular tme. Ec. dervadas parcales 87 Solucón EDP Parabólca: Método mplícto t x α ( ) t x α ) ( λ λ λ ( ) x t α λ

88 Eemplo EDP Parabólca: Método mplícto Consder a steel rod that s subected to a temperature of C on the left end and 5 C on the rght end. If the rod s of length.5m,use the mplct method to fnd the temperature dstrbuton n the rod from t and t 9 seconds. Use x. m, t s. W kg m K m Gven: k, ρ, he ntal temperature of the rod s C. J C 49 kg K 4 5 C 5 C.m Ec. dervadas parcales 88

89 Eemplo EDP Parabólca: Método mplícto Number of tme steps Recall, k α ρ C t t fnal tntal 9 54 t 5.49 m / s hen, λ α.49 ( x) (.) Boundary Condtons C All nternal nodes are at C for sec: C C C Interor nodes C 4 C 5 C C for all,,, t C, for all,,,4 We can now form the system of equatons for the frst tme step by wrtng the approxmated heat conducton equaton for each node λ ( λ λ) Ec. dervadas parcales 89

90 Nodal temperatures when For the frst tme step we can wrte four such equatons wth four unknowns, expressng them n matrx form yelds he above coeffcent matrx s tr-dagonal, so specal algorthms (e.g.homas algorthm) can be used to solve. he soluton s gven by Ec. dervadas parcales 9 Eemplo EDP Parabólca: Método mplícto sec t

91 Nodal temperatures when: Ec. dervadas parcales 9 Eemplo EDP Parabólca: Método mplícto sec t sec t sec 9 t

92 Usng the mplct method our approxmaton of was of accuracy, whle our approxmaton of was of accuracy. One can acheve smlar orders of accuracy by approxmatng the second dervatve, on the left hand sde of the heat equaton, at the mdpont of the tme step. Dong so yelds he frst dervatve, on the rght hand sde of the heat equaton, s approxmated usng the forward dvded dfference method at tme level, Ec. dervadas parcales 9 Solucón EDP Parabólca: Método Crank-Ncolson x ( x) O t ) ( t O ( ) ( ), x x x α t t,

93 Substtutng these approxmatons nto the governng equaton for heat conductance yelds gvng where Havng rewrtten the equaton n ths form allows us to dscretze the physcal problem. We then solve a system of smultaneous lnear equatons to fnd the temperature at every node at any pont n tme. Ec. dervadas parcales 9 Solucón EDP Parabólca: Método Crank-Ncolson ( ) ( ) t x x α ) ( ) ( λ λ λ λ λ λ ( ) x t α λ

94 Eemplo EDP Parabólca: Método Crank-Ncolson Consder a steel rod that s subected to a temperature of C on the left end and 5 C on the rght end. If the rod s of length.5m,use the Crank-Ncolson method to fnd the temperature dstrbuton n the rod from t and t 9 seconds. Use x. m, t s. W kg m K m Gven: k, ρ, he ntal temperature of the rod s C. J C 49 kg K 4 5 C 5 C.m Ec. dervadas parcales 94

95 Eemplo EDP Parabólca: Método Crank-Ncolson Number of tme steps Recall, k α ρ C t t fnal tntal 9 54 t 5.49 m / s hen, λ α.49 ( x) (.) Boundary Condtons C All nternal nodes are at C for sec: C C C Interor nodes C 4 C 5 C 5 λ 5 5 C for all,,, t C, for all,,,4 We can now form the system of equatons for the frst tme step by wrtng the approxmated heat conducton equaton for each node λ λ λ λ λ ( ) ( ) Ec. dervadas parcales 95

96 Nodal temperatures when For the frst tme step we can wrte four such equatons wth four unknowns, expressng them n matrx form yelds he above coeffcent matrx s tr-dagonal, so specal algorthms (e.g.homas algorthm) can be used to solve. he soluton s gven by Ec. dervadas parcales 96 Eemplo EDP Parabólca: Método Crank-Ncolson sec t

97 Nodal temperatures when: Ec. dervadas parcales 97 Eemplo EDP Parabólca: Método Crank-Ncolson sec t sec 6 t sec 9 t

98 Comparacón de métodos: temperaturas en 9 seg. he table below allows you to compare the results from all three methods dscussed n uxtaposton wth the analytcal soluton. Node Explct Implct Crank- Ncolson Analytcal Ec. dervadas parcales 98

99 Ecuacones en dervadas parcales hperbólcas he general form for a second order lnear PDE wth two ndependent varables and one dependent varable s u A x u B x y C u y he crtera for an equaton of ths type to be consdered hyperbolc: B 4AC > he wave equaton (oscllatory systems) gven by y t k y x u y where A, B, C thus we can classfy ths equaton as hyperbolc. u D x g where k w E Ec. dervadas parcales 99 Fu G tenson, g gravtatonal constant, w weght/unt W/L,Wweght, Llength

100 Eemplo de una EDP hperbólca A strng of certan length and weght s under a fxed tenson. Intally the md-pont of the strng s dsplaced some dstance from ts equlbrum poston and released. We want to calculate the dsplacement as a functon of tme at fxed ntervals along the length of the strng. Ec. dervadas parcales

101 Once agan, we can solve the problem by replacng dervatves by fnte dfferences. whch, when rearranged, yelds If we set, the above equaton s smplfed to When employng the smplfed equaton, the value of s determned by the expresson.o begn the calculatons (value at t ), t s requred values of the functon at t and also a value at t -. We can get a value for the functon at t - by makng use of the fact that the functon s perodc. We can use for the frst row. Ec. dervadas parcales Solucón EDP hperbólca: Método explícto ( ) ( ) x w g t ( ) ( ) ( ) ( ) x t w g x t w g ) ( ( ) ( ) x w t g t w g x t /

102 Eemplo EDP hperbólca A strng 5 cm long and weghng.5 g s under a tenson of kg. Intally the md-pont of the strng s dsplaced.5 cm from ts equlbrum poston and released. We want to calculate the dsplacement as a functon of tme at 5 cm ntervals along the length of the strng, usng equaton x g / w From equaton t the must be 8.8 x -5 seconds. t Ec. dervadas parcales

103 Eemplo EDP hperbólca he Wave Equaton:Vbraton of a Strng length, cm 5 (L) tenson, g () weght,g,5 (Wt) weght per unt length, g/cm, (w) gravtatonal constant, cm/sec 98 (g) Dx 5 (Dx) Dt 8,79E-5 (Dt) tme t (sec) Dstance x (cm) ,,,,4,5,4,,, 8,8E-5,,,,4,4,4,,,,8E-4,,,,,,,,,,6E-4,,,,,,,,,,5E-4,,,,,,,,, 4,4E-4,,,,,,,,, 5,E-4 -, -, -, -, -, -, -, -, -, 6,E-4 -, -, -, -, -, -, -, -, -, 7,E-4 -, -, -, -, -, -, -, -, -, 7,9E-4 -, -, -, -,4 -,4 -,4 -, -, -, 8,8E-4 -, -, -, -,4 -,5 -,4 -, -, -, 9,7E-4 -, -, -, -,4 -,4 -,4 -, -, -,,E- -, -, -, -, -, -, -, -, -,,E- -, -, -, -, -, -, -, -, -,,E- -, -, -, -, -, -, -, -, -,,E-,,,,,,,,,,4E-,,,,,,,,,,5E-,,,,,,,,,,6E-,,,,,,,,,,7E-,,,,4,4,4,,,,8E-,,,,4,5,4,,,,8E-,,,,4,4,4,,,,9E-,,,,,,,,,,E-,,,,,,,,,,E-,,,,,,,,,,E-,,,,,,,,,,E- -, -, -, -, -, -, -, -, -,,4E- -, -, -, -, -, -, -, -, -,,5E- -, -, -, -, -, -, -, -, -,,5E- -, -, -, -,4 -,4 -,4 -, -, -,,6E- -, -, -, -,4 -,5 -,4 -, -, -,,7E- -, -, -, -,4 -,4 -,4 -, -, -,,8E- -, -, -, -, -, -, -, -, -, Dsplacement (cm.) Dsplacement evoluton at 5 cm me (sec.) Ec. dervadas parcales